Topology Seminar

A large class of flows of interest have trajectories minimizing “actions.” Novikov theory extends Morse theory to relate the dynamics of such a generic smooth flow (vector field with a Lyapunov closed one form with Morse zeros) on a closed smooth manifold to the underlying topology of the manifold.

I have proposed an alternative to such theory which weakens the hypothesis (closed manifold, closed Morse one form) to compact ANR, topological closed one form, and is computer friendly based on ‘`topological persistence.’' In this talk I will explain the topology behind this approach and the new invariants involved in.

From topologists perspective $k$-regular maps were introduced by Karol Borsuk in the 1950s: A continuous map from $X$ to $\mathbb{R}^n$ is $k$-regular iff any set of pairwise distinct points in $X$ goes to linearly independent set of vectors. Actually $k$-regular maps were introduced by Chebyshev in XIXth century and studied in approximation and interpolation theory. There is an affine version of this concept, clearly $(k=2)$ related to embeddings. So in analogy with theory of embeddings one may ask: given $X$ and $k$, what is a minimal dimension of $\mathbb{R}^n$ receiving a $k$-regular map from $X$.

This question is interesting even for the case when $X=\mathbb{R}^d$. I will discuss topology used in the lower bounds for $n(d,k)$ and construction of (fairly explicit) $k$-regular maps giving upper bounds on $n(d,k)$.

Some graduate students have asked me about surgery theory and what it can do; this talk is an extraordinarily brief, high-level introduction.

Persistence is an important new topic in Computational Topology. In this talk I will explain what “Persistence” is, what is this good for, how can it be calculated and what are the new invariants involved in the measuring of persistence. This is a summary of work of Edelsbrunner, Letcher, Zomorodian, Carlsson. To the extent the time permits, or in a follow up lecture, a more refined version of persistence, whose calculation has the same degree of complexity but carry considerably more information will be described (joint work with Tamal Dey). The exposition is elementary and needs only basic concepts of simplicial complexes and homology.

(This is joint work with David Gay) I will discuss the existence and uniqueness theorems for wrinkled fibrations of arbitrary orientable, smooth $n$-manifolds ($n=4$ is the most interesting case) over orientable surfaces. Existence sometimes holds, and there is a natural set of moves relating different wrinkled fibrations for a given $n$-manifold. A wrinkled fibration is one in which the rank of the differential is 2 or is a curve of points of rank 1 which look locally like an arc cross an indefinite $k$-handle (and the curve is the arc cross the critical point of the k-handle). Furthermore, fibers are always connected.

Let $X$ be a space which satisfied $4k$-dimensional Poincaré Duality, and let $\sigma(X)$ be the signature of $X$. If $X$ is a manifold, then $\sigma(X)$ can be “disassembled”, i.e. $\sigma(X)$ is determined by a local invariant, the Hirzebruch $L$-polynomial. In this talk I’ll give an enriched version of $\sigma(X)$ which is defined in all dimensions, and for dim >4, the enriched version can be disassembled if and only if $X$ admits manifold structure. There is also a family version of this for fibrations

This talk concerns actions of (discrete) groups on finite-dimensional contractible simplicial complexes. I call a group G a ‘Smith group’ if every action of G on a finite-dimensional contractible simplicial complex has a fixed point. (The P A Smith theorem tells us that every finite p-group is a Smith group; there are no other finite Smith groups.)

Kropholler's hierarchy assigns an ordinal to a group, describing how simply it can be made to act on a finite-dimensional contractible simplicial complex. Finite groups are at stage 0 of the hierarchy and stage 1 contains all groups that act on a finite-dimensional contractible simplicial complex without a fixed point. Until our work, no group was known to lie in the hierarchy beyond stage 3.

We construct an infinite Smith group, and construct groups that show that for countable groups, Kropholler's hierarchy is as long as it possibly could be.

In the talks, I will describe some fixed-point theorems and explain some aspects of our group constructions. The first talk will focus on Smith groups and the second on Kropholler's hierarchy.

Joint with G Arzhantseva, M Bridson, T Januszkiewicz, P Kropholler, A Minasyan and J Swiatkowski.

The Galois action of $\mathbb{Z}/2$ on the field of complex numbers plays prominent role in algebraic topology. Its significance in various contexts was discovered by Atiyah (in $K$-theory), Karoubi (Hermitian $K$-theory) and Landweber (real cobordism MR). It also played an important role in the development of equivariant stable homotopy theory by Araki, Adams, May and others. In 1998, Po Hu and I did extensive work on MR, including a complete calculation of its coefficients, and development of what we called Real homotopy theory. Our work was discovered 10 years later by Hill, Hopkins and Ravenel, and played a central role in their recent solution of the Kervaire invariant 1 problem. Meanwhile, there is a baffling parallel between Real and motivic homotopy theory which was used by Morel and Voevodsky, Levine and others in their investigation of algebraic cobordism. Recently, Hu,Ormsby and myself combined Real and algebraic techniques in a solution of the homotopy completion problem for Hermitian K-theory for fields of characteristic 0. I hope to touch on the different aspects of this amazing story in my talk.

The Jones polynomial of links in 3-space is a specialization of the Tutte polynomial of corresponding plane graphs. There are several generalizations of the Tutte polynomial to graphs embedded into a surface. Some of them are related to the theory of virtual links. Although virtual link theory predicts some relations between these generalizations. I will report about the results obtained in this direction during my summer program "Knots and Graphs".

In particular I will compare three polynomials of graphs on surfaces and a relative version of the Tutte polynomial of planar graphs. The first polynomial, defined by M.Las Vergnas, uses a strong map of the bond matroid of the dual graph to the circuit matroid of the original graph. The second polynomial is the Bollobas-Riordan polynomial of a ribbon graph, a straightforward generalization of the Tutte polynomial. The third polynomial, due V.Krushkal, is defined using the symplectic structure in the first homology group of the surface.

The Borel cohomology groups $H_B^\star(G, \mathbb{Z})$ of a Lie group $G$ are based on cocycles, which are Borel maps. These Borel cohomology groups are known to be naturally isomorphic to the singular cohomology groups $H^\star(BG,\mathbb{Z})$ of the classifying space $BG$ of $G$, the domain of primary characteristic classes. We discuss the relationship between boundedness properties of cocycles in $H_B^\star(G, \mathbb{Z})$ and subgroup distortion in $G$ (joint work with Indira Chatterji, Yves de Cornulier and Christophe Pittet).

Virtual knot theory studies knots in thickened surfaces and has a combinatorial representation that is similar to the diagrams for classical knot theory. This talk is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial and categorifications of the arrow polynomial. The arrow polynomial (of Dye and Kauffman) is a natural generalization of the Jones polynomial, obtained by using the oriented structure of diagrams in the state sum. We will discuss a categorification of the arrow polynomial due to Dye, Kauffman and Manturov and will give an example (from many found by Aaron Kaestner) of a pair of virtual knots that are not distinguished by Khovanov homology (mod 2), or by the arrow polynomial, but are distinguished by a categorification of the arrow polynomial.

The linking number was first defined by Gauss in 1833, who wrote it as an integral which is supposed to compute the number of times one circle wraps around another in space. I will begin by discussing the classical linking number using a much simpler definition by taking a planar projection of the link and counting the number of times one component lies over the other. From this we will see exactly what it is the linking number counts, and this leads to two things. The first is the realization that the linking number is really a "relative" invariant. The second is a generalization, due to the speaker, of a linking "number" for arbitrary manifolds in an arbitrary manifold. I will also discuss Milnor’s higher-order linking numbers, which detect, for example, that classical links such as the Borromean rings are linked (despite being pairwise unlinked). This was generalized by Koschorke to higher-order linking of arbitrary spheres in Euclidean space, and the speaker generalized this to arbitrary manifolds. These higher-order invariants are also relative invariants in the same way the linking number is, and they admit a number of interesting geometric interpretations. Along the way, we will observe that the classical linking number is related to the stable homotopy groups of spheres, whereas the higher-order generalizations are related to a certain filtration of the unstable homotopy groups of spheres.

This talk will describe computational results related to the structure of power operations on complex oriented cohomology theories (localized at a prime $p$), making use of the amazing connection between complex orientations and the theory of formal group laws. After introducing the relevant concepts, we will describe results from joint work with Justin Noel showing that, for primes $p$ less than or equal to 13, orientations factoring non-trivially through the Brown-Peterson spectrum cannot carry power operations, and thus cannot provide $MU_{(p)}$-algebra structure. This implies, for example, that if E is a Landweber exact $MU_{(p)}$-ring whose associated formal group law is $p$-typical of positive height, then the canonical map $\mathrm{MU}_{(p)} \to E$ is not a map of $H_\infty$ ring spectra. It immediately follows that the standard $p$-typical orientations on $\mathrm{BP}$, $E(n)$, and $E_n$ do not rigidify to maps of $E_\infty$ ring spectra. We conjecture that similar results hold for all primes.

It is a classic fact that any graded group (abelian above dimension 1) can be realized as the homotopy groups of a space. However, the question becomes difficult if one includes the data of primary homotopy operations, known as a Pi-algebra. When a Pi-algebra is realizable, we would also like to classify all homotopy types that realize it.

Using an obstruction theory of Blanc-Dwyer-Goerss, we will describe the moduli space of realizations of certain 2-stage Pi-algebras. This is better than a classification: The moduli space provides information about realizations as well as their higher automorphisms.

Let ${\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ denote the 2-category of based topological spaces, base point preserving continuous maps, and based track classes of based homotopies. Let $W$ be a fixed space or spectrum and consider the 2-functor on ${\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ obtained by taking the smash product with $W$. The categorical full image of this functor is a 2-category denoted $W{\mathcal{T}\hspace{-0.3ex}op}_{\ast}$ and called the $W$-topology category. For $W$ a space the study of $W$-topology was initiated by Hardie, Marcum and Oda [1]. Of course $W$-topology and stable homotopy theory, while related, are distinct.

In the associated $W$-homotopy category the $W$-homotopy groups $\pi_{r}^W (X)$ have long been recognized as rather significant (but in other notation of course). For example, Barratt (1955) studied $\pi_{n}^W (S^{n})$ for $W=S^1 \cup_p e^2$ Toda (1963) considered the suspension order of a complex $Y_k$ having the same homology as the $(n-1)$-fold suspension $\Sigma^{n-1} P^{2k}$ of the real projective $2k$-space $P^{2k}$, namely the order of the identity class of $\pi_{1}^W (S^{1})$ when $W=Y_k$.

In [1] some non-trivial elements in $W$-homotopy groups were detected by making use of $W$-Hopf invariants. This talk focuses on a general proceedure for introducing Hopf invariants into $W$-topology. As an application, when $W$ is a mod $p$ Moore space, namely $W=S^1 \cup_p e^2$, we show that it is possible to detect elements in $\pi_{r+1}^W (\Omega S^{m+1})$ which have connection with known stable periodic families of the homotopy groups of spheres. In particular we prove nontriviality in $\pi_{r+1}^W (\Omega S^{m+1})$ of elements related to families discovered by Gray (1984) (for $p$ an odd prime) and by Oda (1976) (for $p=2$). This represents joint work with K. Hardie and N. Oda.

[1] K. Hardie, H. Marcum and N. Oda, The Whitehead products and powers in $W$-topology, Proc. Amer. Math. Soc. 131 (2003), 941–951.

Given a bounding class $B$, we construct a bounded refinement $BK(-)$ of Quillen’s $K$-theory functor from rings to spaces. $BK(-)$ is a functor from weighted rings to spaces, and is equipped with a comparison map $BK \to K$ induced by "forgetting control". In contrast to the situation with $B$-bounded cohomology, there is a functorial splitting $BK(-) \simeq K(-) \times BK^{rel}(-)$ where $BK^{rel}(-)$ is the homotopy fiber of the comparison map. For the bounding class $P$ of polynomial functions, we exhibit an element of infinite order in $PK^{rel}_0(Z[G])$ for $G$ the fundamental group of a certain 3-dimensional solvmanifold.This is joint work with Crichton Ogle.

Inspired by the idea of “persistence” (persistent homology) we introduce a new class of topological invariants for (tame) circle valued maps $f: X \to S^1$. They are Bar Codes and Jordan Cells. If $X$ is compact and $f$ is topologically tame (in particular a Morse circle valued function), they are algorithmically computable; moreover all topological invariants of interest in Novikov-Morse theory can be recovered from them; (for example the Novikov-Betti numbers of $(M, \xi)$, $\xi \in H^1(M;\mathbb{Z})$ representing the homotopy class of $f$, can be recovered from the bar codes while the usual Betti numbers of $M$ from bar codes and Jordan cells. A more subtle invariant like Reidemeister torsion is related to the Jordan cells. The definition of these invariants is based on representation theory of quivers ($=$oriented graphs). The above theory extends Novikov-Morse theory from Morse circle valued maps to tame maps $f:X \to S^1$ and even further to 1-cocycle which is the topological version of closed one form for smooth manifolds. This last extension is more elaborate and will be discussed later.

Inspired by the idea of “persistence” (persistent homology) we introduce a new class of topological invariants for (tame) circle valued maps $f: X \to S^1$. They are Bar Codes and Jordan Cells. If $X$ is compact and $f$ is topologically tame (in particular a Morse circle valued function), they are algorithmically computable; moreover all topological invariants of interest in Novikov-Morse theory can be recovered from them; (for example the Novikov-Betti numbers of $(M, \xi)$, $\xi \in H^1(M;\mathbb{Z})$ representing the homotopy class of $f$, can be recovered from the bar codes while the usual Betti numbers of $M$ from bar codes and Jordan cells. A more subtle invariant like Reidemeister torsion is related to the Jordan cells. The definition of these invariants is based on representation theory of quivers ($=$oriented graphs). The above theory extends Novikov-Morse theory from Morse circle valued maps to tame maps $f:X \to S^1$ and even further to 1-cocycle which is the topological version of closed one form for smooth manifolds. This last extension is more elaborate and will be discussed later.

Using techniques developed for studying polynomially bounded cohomology, we show that the assembly map for $K_*^t(\ell^1(G))$ is rationally injective for all finitely presented discrete groups $G$. This verifies the $\ell^1$-analogue of the Strong Novikov Conjecture for such groups. The same methods show that the Strong Novikov Conjecture for all finitely presented groups can be reduced to proving a certain (conjectural) rigidity of the cyclic homology group $HC_1^t(H^{CM}_m(F))$ where $F$ is a finitely-generated free group and $H^{CM}_m(F)$ is the “maximal” Connes-Moscovici algebra associated to $F$.

A fundamental question (still unanswered) is whether certain “periodic” functors on the category of discrete groups, such as the topological $K$-theory of $C^*(\pi)$ or the Witt theory of $\mathbb Z[\pi]$, can be extended to a functors on the category of basepointed topological spaces which depend (rationally) on more than just the fundamental group. Following earlier work of Burghelea-Fiedorowicz, Fiedorowicz-Vogt, and Vogell, I will propose a model for a functor $X\mapsto AH(X)$, which may be thought of as an Hermitian analogue of Waldhausen’s functor $X\mapsto A(X)$, and occurs as the $\mathbb Z/2$ fixed-point set of an involution defined on a certain model of $A(X)$. I will also explain how a suitable $\mathbb Z/2$-equivariant version of Waldhausen's splitting $Q(X_+)\to A(X)\to Q(X_+)$ verifies the Novikov conjecture for $\pi_1(X)$.

How many involutions on the $n$ torus have an isolated fixed point?

This is a report of joint work with Jim Davis and Qayum Khan.

We prove that there is only one involution on the $n$-torus, $T^n$, up to conjugacy, for which the fixed set contains an isolated point. But here, $n$ must be of the form $4k$ or $4k+1$ (or else, n must be $\leq 3$). In the other dimensions, we classify all such involutions, using surgery theory and the calculation of the groups $UNil_n(Z,Z,Z).$

We also introduce a Topological Rigidity Conjecture and we show that the above result is a consequence of it.

We discuss a question by Nori, which is to determine when a discrete Zariski dense subgroup in a semisimple Lie group containing a lattice has to be itself a lattice. This is joint work with Venkataramana.

There is a theory of random graphs due to Erdos and Renyi. Associated to any group and a graph there is a notion of its graph product. So, there also is a notion of a random graph products of groups. For example, by letting the group be Z/2, the graph product can be any right-angled Coxeter group. We compute the cohomological invariants of random graph products. This is joint work with Matt Kahle.

In any symmetric monoidal category $C$, the Picard group is the group of isomorphism classes of invertible objects. For the usual stable homotopy category, the only invertible objects are the sphere spectra $S^n$, with $n$ an integer. However, if $E_\star$ is a good (i.e., complex-orientable) homology theory, Mike Hopkins noticed that the $E$-local stable homotopy category could have a rich and curious Picard group—and that this group could give information about how homotopy theory of spectra reassembles from localizations. I’ll review this theory, revisit some of the curious examples, and report on recent calculations. This is joint work with Hans-Werner Henn, Mark Mahowald, and Charles Rezk.

The Hochschild chain and cochain complexes and the cyclic complex of an associative algebra serve as noncommutative analogs of classical geometric objects on a manifold, such as differential forms and multivector fields. These complexes are known to possess a very nontrivial and rich algebraic structure that is analogous to, and goes well beyond, the classical algebraic structures known in geometry. In this talk, I will give a review of the subject and outline an approach that is based on an observation that differential graded categories form a two-category up to homotopy.

Given a fibration p, we can ask when p is fiber homotopy equivalent to a topological fiber bundle with compact manifold fibers; assuming that the fibration p does admit a compact bundle structure, we can also ask to classify all such bundle structures on p. Similarly, given a map f between compact manifolds, we can ask when f is homotopic to a topological fiber bundle with compact manifold fibers, and assuming that the map f does fiber, we can ask to classify all of the different ways to fiber f. In this talk, we will begin by describing the space of all compact bundle structures on a fibration, which is nonempty if and only if p admits a compact bundle structure. We will then show that, as long as we are willing to stabilize by crossing with a disk, the obstructions to stably fibering a map f are related to the space of bundle structures on the fibration p associated to f.

This talk will describe (Waldhausen type) algebraic $K$-theoretic obstructions to lifting fibrations to fiber bundles having compact smooth/topological manifold fibers. The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions.

We consider free actions of large prime order cyclic groups on products of spheres. The equivariant homotopy type will be determined and the simple structure set discussed. Similar to lens spaces, the first $k$-invariant generally determines the homotopy type, however for homotopy equivalences between products of an even number of spheres the Whitehead torsion vanishes and the quotients are also simple homotopy equivalent. Unlike lens spaces which are determined by their Reidemeister torsion and $\rho$-invariant, the $\rho$-invariant vanishes for products of an even number of spheres and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and the set of normal invariants will also be discussed.

TBA

We will generalize Sullivan’s rational surgery realization theorem to the case when the fundamental group is finite; given a finite group action on a rational Poincaré duality algebra, does there exist a closed manifold realizing the algebra as its cohomology ring with the group acting freely on it?

The deRham cohomology of a Poisson manifold comes equipped with a canonical filtration. For symplectic manifolds this filtration is well understood and can be computed from the cup product action of the cohomology class represented by the symplectic form. In this talk we will discuss said filtration for the total space of sympletic fiber bundles. The latter constitute a class of Poisson manifolds closely related to the topology of the sympletic group of the typical fiber.

Bar codes and Jordan cells provide a new type of linear algebra invariants which can be used in topology. In joint work with Tamal Dey we have associated with any angle valued generic map $f : X \to S^1$, $X$ a compact nice space (ANR), $\kappa$ any field and any integer $r$, $0 \leq r \leq \dim X$, a collection of such bar codes and Jordan cells. They can be effectively computed in case $\kappa = \mathbb{C}$ or $\mathbb{Z}_2$, $X$ is a simplicial complex and f a simplicial map by algorithms implementable by familiar software (Matematica, Mapple or Matlab). In this lecture I will describe some joint work with S Haller.

1. We prove that the Jordan cells defined using $f$ are homotopy invariants of the pair $(X,\xi)$, $\xi ∈ H^1(X;\mathbb{Z})$ representing $f$.

2. We calculate the homology $H_∗(\tilde{X};\kappa)$ as a $\kappa[t^{−1},t]$ module, $\tilde{X}$ the infinite cyclic cover of $X$ induced by $\xi$, as well as and the Novikov homology and Milnor-Turaev torsion of $(X;\xi)$ in terms of bar codes and Jordan cells.

3. As a consequence we introduce Lefchetz zeta function of a pair $(X;\xi)$ which generalizes the familiar Lefschetz zeta function of a self map of a compact manifold and the Alexander polynomial of a knot, and relate this function to dynamics.

Bar codes and Jordan cells provide a new type of linear algebra invariants which can be used in topology. In joint work with Tamal Dey we have associated with any angle valued generic map $f : X \to S^1$, $X$ a compact nice space (ANR), $\kappa$ any field and any integer $r$, $0 \leq r \leq \dim X$, a collection of such bar codes and Jordan cells. They can be effectively computed in case $\kappa = \mathbb{C}$ or $\mathbb{Z}_2$, $X$ is a simplicial complex and f a simplicial map by algorithms implementable by familiar software (Matematica, Mapple or Matlab). In this lecture I will describe some joint work with S Haller.

1. We prove that the Jordan cells defined using $f$ are homotopy invariants of the pair $(X,\xi)$, $\xi ∈ H^1(X;\mathbb{Z})$ representing $f$.

2. We calculate the homology $H_∗(\tilde{X};\kappa)$ as a $\kappa[t^{−1},t]$ module, $\tilde{X}$ the infinite cyclic cover of $X$ induced by $\xi$, as well as and the Novikov homology and Milnor-Turaev torsion of $(X;\xi)$ in terms of bar codes and Jordan cells.

3. As a consequence we introduce Lefchetz zeta function of a pair $(X;\xi)$ which generalizes the familiar Lefschetz zeta function of a self map of a compact manifold and the Alexander polynomial of a knot, and relate this function to dynamics.

We will compare two different quantum 3-manifold invariants, both of which are given using a finite dimensional Hopf Algebra $H$. One is the Hennings invariant, given by an algorithm involving the link surgery presentation of a 3-manifold and the Drinfeld double $D(H)$; the other is the Kuperberg invariant, which is computed using a Heegaard diagram of the 3-manifold and the same $H$. We have shown that when $H$ has the property of being involutory, these two invariants are actually equivalent. The proof is totally algebraic and does not rely on general results involving categorical invariants. We will also briefly discuss some results in the case where $H$ is not involutory.

The LS category of a space X is a numerical invariant that measures the complexity of a space. While it is usually very hard to compute explicitly, there are estimates and approximating invariants that help us to understand category better. A big problem is to understand the effect of the fundamental group on category. Recently, extending work of Dranishnikov, Jeff Strom and the speaker have given an upper bound for category using Ralph Fox’s 1-category (and another approximating invariant). Using an interpretation of this 1-category given by Svarc, we have also been able to refine Bochner's bound on the first Betti number in the presence of non-negative Ricci curvature. Finally, the 1-category forms a bridge between the theorems of Yamaguchi and Kapovitch-Petrunin-Tuschmann on manifolds with almost non-negative sectional curvature.

By studying the example of smooth structures on $CP^2 \# 3(-CP^2)$, I will illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold.

I will describe some recent results on isometry groups of aspherical Riemannian manifolds and their universal covers. The general theme is that topological properties of an aspherical manifold often restrict the isometries of an arbitrary complete Riemannian metric on that manifold. These topological properties tend to be established by using a specific "nice" metric on the manifold.

I will illustrate this by explaining why on an irreducible locally symmetric manifold, no metric has more symmetry than the locally symmetric metric. I will also discuss why moduli space is a minimal orbifold and relate this phenomenon to symmetries of arbitrary metrics on moduli space.

I will discuss the geometry of certain abelian-by-cyclic groups and show how to establish the optimal top-dimensional isoperimetric inequality that holds in these groups. This is joint work with Noel Brady.

We construct a variant of DeRham cohomology and use it to prove that the Higson compactification of $R^n$ has uncountably generated $n^{\mbox{th}}$ integral cohmology. We also explain that there is, nevertheless, a way of using the Higson compactification to prove the Novikov conjecture for a large class of groups.

A compact Clifford-Klein form of the homogeneous space $J\backslash H$ is a compact manifold $J\backslash H/\Gamma$ constructed using a discrete subgroup $\Gamma$ of $H$. I will survey the existence problem for compact forms, with particular attention to the case when there is an action by a large group on $J\backslash H/\Gamma$. I will also make some remarks on a conjecture of Kobayashi on the scarcity of compact forms.

We will describe decompositions of finitely presented groups, using descriptions of smooth and of symplectic four-manifolds. Every finitely presented group admits a decomposition into a triple consisting of the fundamental groups of two compact complex Kähler surfaces with boundary and the fundamental group of a three manifold. We will exhibit various ways of obtaining similar decompositions of finitely presented groups into graphs, via descriptions of smooth 4-manifolds into Lefschetz fibrations. We distill this data into invariants, by considering the minimal number of edges these graphs may have. These ideas are related to the minimal euler characteristic of symplectic four-manifolds and the minimal genus of a Lefschetz fibration, seen as group invariants.

The study of fundamental groups of random two dimensional simplicial complexes calls attention to the small subcomplexes of such objects. Such subcomplexes have fewer triangles than some multiple of the number of their vertices. One gets that this condition with constant less than two on a connected complex (and all of its subcomplexes) implies that it is homotopy equivalent to a wedge of circles, spheres and projective planes. This analysis yields parameter regimes for vanishing, hyperbolicity and Kazhdanness of these groups. For clique complexes of random graphs there is a similar problem involving complexes with fewer edges than thrice the number of their vertices resulting in similar results on the fundamental groups of their clique complexes. This is based on joint work with Hoffman and Kahle.

In parallel with a classical definition due to Bieri and Eckmann, say an FP group G is an abelian duality group if H^p(G,Z[G^{ab}]) is zero except for a single integer p=n, in which case the cohomology group is torsion-free. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers.

While related, the two notions are inequivalent: for example, surface groups of genus at least 2 are (Poincaré) duality groups, yet they are not abelian duality groups. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen-Macaulay property for the presentation graph. Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a slightly more general, cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky.

I’ll outline the construction of smooth 4-manifolds which support locally CAT(0)-metrics (the metric version of non-positive curvature), but do not support any Riemannian metric of non-positive sectional curvature. This is joint work with Mike Davis and Tadeusz Januszkiewicz.

In the last couple of decades the study of representations of braid groups $B_n$ attracted attention from two rather different motivations. One deals with the linearity of the braid groups, that is, whether the $B_n$ can be faithfully represented.

This question was answered in the positive for the Lawrence-Krammer-Bigelow (LKB) representation independently by Krammer and Bigelow around 2001. The LKB representation is given by the natural action of $B_n$ in the second homology of a rank two free abelian cover of the two-point configuration space on the $n$-punctured disc. It is thus naturally a module over the ring of Laurent polynomials in two variables.

The other development is the construction of braid representations from quantum groups. One such class of $B_n$-representations of is constructed using quantum-$sl_2$, which is a one-parameter Hopf algebra deformation of the universal enveloping algebra of $sl_2$. The algebra admits a quasi-triangular R-matrix which can be used to represent $B_n$ on the $n$-fold tensor product of a Verma module with generic highest weight.

We prove that the latter representation, with some refinement of the ground ring, is isomorphic to the LKB representation where the two parameters corresponding to the generators of the Deck transformation group are identified with the deformation parameter of quantum-$sl_2$ and the generic highest weight of the Verma module respectively. We also show irreducibility of this representation over the fraction field of the ring of Laurent polynomials.

Time permitting we will discuss relations to other types of braid group representations that may shed light on this curious connection, as well as reducibility issues at certain choices of parameters.

Joint work with Craig Jackson.

We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad T on a topological model category.

This talk will focus on the special case where T is a monad encoding E_infty structure in spectra and H_infty structure in the derived category of spectra. We will present examples from rational homotopy theory illustrating the obstructions to rigidifying homotopy algebra maps to strict algebra maps, and explain in a precise way how the edge homomorphism of this obstruction spectral sequence measures the difference between up-to-homotopy and on-the-nose T-algebra maps.

Given a compact ANR $X$ and $f : X \to \mathbb{C} \setminus 0$ a continuous map, for any $0 \leq r \leq \dim X$, one proposes three monic complex valued polynomials $P_{r,s}(z)$, $P_{r,a}(z)$ and $P_{r,m}(z)$, with $\deg(P_{r,s}(z) = \beta_r(X)$ where $\beta_r(X)$ is the r−th Betti number, $\deg(P_{r,a}(z) = \beta_r^N(X,f)$, where $\beta_r^{N}(X,f)$ is the $r$−th Novikov Betti number, $P_{r,m}(z)$ a homotopy invariant of $f$. The first two are continuous assignments with respect to compact open topology, the last is locally constant (on the space of continuous functions with compact open topology).

A discrete group $\Gamma$ can act freely and properly on $S^n \times R^m$, for some $n, m >0$ if and only if $\Gamma$ is a countable group with periodic Farrell cohomology: Connolly-Prassidis (1989) assuming $vcd(\Gamma)$ finite, Adem-Smith (2001). For free co-compact actions there are additional restrictions, but no general sufficient conditions are known. The talk will survey this problem and its connection to the Farrell-Jones assembly maps in K-theory and L-theory.

P.A. Smith, in the first half of the 20th century, developed homological tools to study actions of finite p-groups on topological spaces. The standard applications of Smith theory are that if a p-group acts on a {disk, sphere, manifold, finite-dimensional space} then the fixed set of the action is a {mod p homology disk, mod p homology sphere, mod p homology manifold, finite dimensional space}.

This talk will review the classical theory, give applications to actions on aspherical manifolds, and extend the theory to give restrictions on periodic knots.

We discuss Yu’s property A as a generalization of amenability for countable groups. A few characterizations of amenability and property A are given, including Johnson's cohomological characterization of amenability and the recent work of Brodzki, Nowak, Niblo, and Wright which characterizes property A in a similar manner. These characterizations play a major role in the relative versions of these properties.

We define the notion of a group having relative property A with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a cohomological characterization shows that if $G$ has property A relative to a family of subgroups $\mathcal{H}$, and if each $H \in \mathcal{H}$ has property A, then $G$ has property A. This result leads to new classes of groups that have property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.

Waldhausen’s A-theory of a space $X$ is sometimes described as the “universal Euler class”. Along these lines, the ‘`universal generalized Lefschetz class'' of an endomorphism would be the reduced algebraic K-theory of an associated ``brave new'' tensor algebra. Recent joint work with Ayelet Lindenstrauss describing this spectrum, when one is working in an analytic range (in the sense of Goodwillie's calculus of functors) will be discussed.

For $\pi_0$, these results go back to Almkvist, Rincki and Lück. More generally these results are related to the theses of Lydakis and Iwachita which built upon the computation of the $A$-theory of the suspension of a space by Carlsson, Cohen, Goodwillie and Hsiang.

The random flag complex is a natural combinatorial model of random topological space. In this talk I will survey some results about the expected topology of these objects, focusing on recent work which gives a sharp vanishing threshold for kth cohomology with rational coefficients.

This recent work provides a generalization of the Erdos-Renyi theorem which characterizes how many random edges one must add to an empty set of n vertices before it becomes connected. As a corollary, almost all d-dimensional flag complexes have rational homology only in middle degree (d/2).

This is topology seminar, so I will assume that people know what homology and cohomology are, but I will strive to make the talk self contained and define all the necessary probability as we go.

For each discrete group G one can find a universal G-space with stabilizers in a prescribed family of subgroups of G. These spaces play a prominent role in the so-called Isomorphism Conjectures, namely the Baum-Connes and the Farrell-Jones conjectures. We will discuss the former conjecture in more detail and describe its topological side: the equivariant K-homology of the universal space for proper actions. Finally, we will report on joint work with Jean-François Lafont and Ivonne Ortiz on the rationalized topological side for some low dimensional groups.

In this joint work with Daniel Farley, we compute the lower algebraic $K$-groups of all split three-dimensional crystallographic groups $G$. These groups account for 73 isomorphism types of three-dimensional crystallographic groups, out of 219 types in all. Alves and Ontaneda in 2006, gave a simple formula for the Whitehead group of a 3-dimensional crystallographic group $G$ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of $G$. The main goal in this work in progress is to obtain explicit computations for $K_0(ZG)$ and $K_{-1}(ZG)$ for these groups.

A sums-of-squares formula over a field $k$ is a polynomial identity of the form $\left( x_1^2 + \cdots + x_r^2 \right) \left( y_1^2 + \cdots + y_s^2 \right) = z_1^2 + \cdots z_t^2,$ where the $z$’s are bilinear in the $x$'s and $y$'s over $k$. If a sums-of-squares formula exists over $\mathbb{R}$, then a theorem of Hopf from 1940 gives numerical restrictions on $r$, $s$, and $t$. This result was one of the earliest uses of the cup product in singular cohomology.

I will describe some joint work with D. Dugger on generalizing Hopf's result to arbitrary fields of characteristic not $2$. The basic idea is to use motivic cohomology instead of singular cohomology.

This leads into the broader subject of computations in motivic homotopy theory.

This is a joint talk with algebraic geometry seminar.

We review some old problems in algebraic topology–namely, the classification of finite-dimensional modules over subalgebras of the Steenrod algebra, and related classification problems in representation theory and finite CW complexes--and some old techniques in deformation theory--namely, the use of Hochschild 1- and 2-cocycles with appropriate coefficients to classify first-order deformations of modules and algebras, respectively. Then we work out how one has to adapt these old methods to solve these old problems, ultimately using some modern technology: a deformation-theoretic interpretation of twisted nonabelian higher-order Hochschild cohomology.

I’ll show that many finite covers of $\mathrm{SL}(m,Z)\backslash \mathrm{SL}(m,R)/\mathrm{SO}(m)$ have non-trivial homology classes generated by totally geodesic flat $(m-1)$-tori. This is joint work with Tam Nguyen Phan.

Suppose $M$ is a smooth manifold and let $G$ denote the connected component of the identity in the group of all compactly supported diffeomorphisms of $M$. It has been known for quite some time that the group $G$ is simple, i.e. has no non-trivial normal subgroups. Consequently, $G$ is a perfect group, i.e. each element $g$ of $G$ can be written as a product of commutators, $$g=[h_1,k_1]\circ\cdots\circ[h_N,k_N].$$ Actually, all available proofs (Herman, Mather, Epstein, Thurston) for the simplicity of $G$ first establish perfectness; it is then rather easy to conclude that $G$ has to be simple.

In the talk I will discuss a new, more elementary, proof for the perfectness of the group $G$. This approach also shows that the factors $h_i$ and $k_i$ in the presentation above can be chosen to depend smoothly on $g$. Moreover, it leads to new estimates for the number of commutators necessary. If $g$ is sufficiently close to the identity, then $N=4$ commutators are sufficient; for certain manifolds (e.g. mapping tori) even $N=3$ will do.

This talk is based on joint work with T. Rybicki and J. Teichmann.

The solvable filtration of the knot concordance group has been studied closely since its definition by Cochran, Orr and Teichner in 2003. Recently Cochran, Harvey and Leidy have shown that the successive quotients in this filtration contain infinite rank free abelian groups and even exhibit a kind of primary decomposition. Unfortunately, their construction relies on an assumption of non-vanishing of certain rho-invariants. By relating these rho-invariants to the signature function defined by Cimasoni and Florens in 2007, we remove this ambiguity from the construction of Cochran-Harvey-Leidy.

We present results concerning the existence of nontrivial homotopy spheres and also discuss the determination of the smallest dimensional Euclidean spaces in which they smoothly embed.

The ecological niche of a species is the set of environmental conditions under which a population of that species persists. This is often thought of as a subset of "environment space" – a Euclidean space with axes labeled by environmental parameters. This talk will explore mathematical models for the niche concept, focusing on the relationship between topological and ecological ideas. We also describe applications of machine learning to develop empirical models from data in the field. These lead to novel questions in computational topology, and we will discuss recent progress in that direction. This is joint with John Drake in ecology and Edward Azoff in mathematics.

We compute the motivic slices of hermitian K-theory and higher Witt-theory. The corresponding slice spectral sequences relate motivic cohomology to Hermitian K-groups and Witt groups, respectively. Using this we compute the Hermitian K-groups of number fields, and (re)prove Milnor’s conjecture on quadratic forms for fields of characteristic different from 2. Joint work with Oliver Röndigs.

This talk centers around the Rank Rigidity Conjecture for groups acting properly and cocompactly on CAT(0) spaces. After discussing some generalities on the conjecture and some of its consequences, I will focus on the two special cases alluded to in the title.

It follows from work of Crowley-Loeh (d>3) and Barge-Ghys (d=2) that in all degrees distinct from d=3, the l^1-seminorm and the manifold semi-norm coincide on homology of degree d. We show that when d=3, the two semi-norms are bi-Lipschitz to each other, with an explicitly computable constant. This was joint work with Christophe Pittet (Univ. Marseille).

Often the answer to a topological question is, by its own admission, nonconstructive, but even when the answer is constructive, serious difficulties can arise in carrying out that construction. We will consider a couple cases like this. As an approachable, low-dimensional example, we decompose surfaces as a square complex with a fixed number of squares meeting at a vertex. As a high-dimensional example, we consider the possible Pontrjagin numbers of highly connected 32-manifolds. To address this latter case, we will be confronted with needing to compute the coefficients of the Hirzebruch $L$-polynomial; some topology provides a recursive method faster than naive symmetric reduction.

For knots invariant under a finite order cyclic symmetry, Seifert, Murasugi and others developed relations constraining the Alexander polynomial of such knots.

We develop similar constraints using the transfer method of generating functions is applied to the ribbon graph rank polynomial. This polynomial, denoted $R(D;X,Y,Z)$, is due to Bollobás, Riordan, Whitney and Tutte. Given a sequence of ribbon graphs, $D_n$, constructed by successive amalgamation of a fixed pattern ribbon graph, we prove by the transfer method that the associated sequence of rank polynomials is recursive: that is, the polynomials $R(D_n;X,Y,Z)$ satisfy a linear recurrence relation with coefficients in $Z[X,Y,Z]$.

We develop conditions for the Jones polynomial of links which admit a periodic homeomorphism, by applying the above result and the work of Dasbach et al showing that the Jones polynomial is a specialization of the ribbon graph rank polynomial.

This is joint work with Jordan Keller and Murphy-Kate Montee)

The original Hennings TQFT is defined for quasitriangular Hopf algebras satisfying various nondegeneracy requirements. We extend this construction to quasitriangular quasi-Hopf algebras with related nondegeneracy conditions and prove that this new “quasi-Hennings” algorithm is well-defined and gives rise to TQFTs. The ultimate goal is to apply this construction to the Dijkgraaf-Pasquier-Roche twisted double of the group algebra, and then show that the resulting TQFT is equivalent to a more geometric one, described by Freed and Quinn.

Aspherical manifolds are manifolds that have contractible universal covers. I will explain how to construct closed aspherical manifolds by gluing the Borel-Serre compactifications of locally symmetric spaces using the reflection group trick. I will also discuss rigidity aspects of these manifolds, such as whether a homotopy equivalence of such a manifold is homotopic to a homeomorphism.

We consider the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot. Using skein theory, we show that the conjecture holds true for some classes of two-bridge knots and pretzel knots.

Kauffman gives a state sum formula for the Alexander polynomial of a knot using states in a lattice that are connected by his clock moves. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by overlaying the two dual Tait graphs of the knot diagram.

Using a partition of the vertices of the bipartite graph, we give a direct computation for the height of Kauffman’s clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified.

We prove structural properties of the bipartite graph in general and mention applications to Chebyshev or harmonic knots (obtaining the popular grid graph) and to discrete Morse functions.

This talk is accessible to those without a background in knot theory. Basic graph theory is assumed.

I will give a historical overview of completions in topology and homotopy theory starting with the work of D. Sullivan, together with motivation and applications of these constructions, including H.R. Miller’s proof of the Sullivan conjecture and Mandell's "homotopical double dual" result for algebraically characterizing p-adic homotopy types. I will then describe a variation of these completion ideas for the enriched algebraic-topological context of homotopy theoretic commutative rings that arises naturally in algebraic K-theory, derived algebraic geometry, and algebraic topology. I will finish by describing some recent results on completion in this new context, which are joint with M. Ching.

Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.

Morava E-theory is an important cohomology theory in chromatic homotopy theory. Rezk described the algebraic structure found in the homotopy of $K(n)$-local commutative E-algebras, via a monad on $E_\ast$-modules that encodes all power operations. However, the construction does not see that the homotopy of a $K(n)$-local spectrum is L-complete (in the sense of Greenlees-May and Hovey-Strickland). We show that the construction can be improved to a monad on $L$-complete $E_\ast$-modules, and discuss some applications. Joint with Tobias Barthel.

To study the exponential law for function spaces with the compact-open topology, R. Brown introduced a topology for product set, which is finer than the product topology, and showed the exponential law for any Hausdorff spaces. The method was improved by P. Booth and J. Tillotson, making use of test maps, and they removed the Hausdorff condition for spaces. The product space they used is called the BBT-product. If we use any class of exponentiable spaces, then we can define a topology for function spaces which enables us to prove the exponential law with the BBT-product for any spaces. We can apply the result to based spaces and we get various good results for homotopy theory. For example, we can prove a theorem of pairings of function spaces without imposing conditions on spaces and base points. If we look at the techniques carefully, we find that the results can also be applied to study group actions on function spaces.

The BBT-product is asymmetric in general and we can define the ‘centralizer’ of the BBT-product, which contains the class of k-spaces defined by the class. The centralizer of the BBT-product has good properties for homotopy theory.

This talk is based on joint work with Yasumasa Hirashima.

In our paper "Polynomial Invariants of Graphs on Surfaces" we found a relationship between two polynomials cellularly embedded in a surface, the Krushkal polynomial, based on the Tutte polynomial of a graph and using data from the algebraic topology of the graph and the surface, and the Las Vergnas polynomial for the matroid perspective from the bond matroid of the dual graph to the circuit matroid of the graph, $\mathcal{B}(G^\ast) \to \mathcal{C}(G)$.

With Vyacheslav Krushkal having (with D. Renardy) expanded his polynomial to the $n$th dimension of a simplicial or CW decomposition of a $2n$-dimensional manifold, a matroid perspective was found whose Las Vergnas polynomial would play a similar role to that in the 2-dimensional case.

We hope that these matroids and the perspective will prove useful in the study of complexes.

I will discuss work arising from Raciel Valle’s thesis concerning complexes built from triangles that are highly symmetrical and have vertex links the join of $n$ pairs of points (equivalently the $1$-skeleton of the $n$-dimensional analogue of the octahedron).

In joint work with Andrew Blumberg, we construct a category of cyclotomic spectra that is (something like) a closed model category and which has well-behaved mapping spectra. We show that topological cyclic homology (TC) is the corepresentable functor on this category given by maps out of the sphere spectrum, verifying a conjecture of Kaledin.

We will discuss a few reduced forms for homotopy types of 1-dim Peano continua. "Deforested" continua contain no attached strongly contractile subsets (dendrites). For 1-dim continua this always gives a minimal deformation retract, or core. In a core 1-dim continuum, the points which are not homotopically fixed form a graph. Furthermore, this can be homotoped to an "arc reduced" continuum, where the non-homotopically fixed points are in fact a union of arcs.

Let $P$ be a knot in a solid torus, $K$ be a knot in $3$-space and $P(K)$ be the satellite knot of $K$ with pattern $P$. This correspondence defines an operator, the satellite operator, on the set of knot types and induces a satellite operator $P:C\to C$ on the set of smooth concordance classes of knots. In a recent paper with Tim Cochran and Arunima Ray, we show that for many patterns this map is injective. I will approach this result from a different perspective, namely by showing that satellite operators really come from a group action. In 2001, Levine studied homology cylinders over a surface modulo the relation of homology cobordism as a group containing the mapping class group. We show that this group also contains satellite operators and acts on an enlargement of knot concordance. In doing so we recover the injectivity result. I will also present some preliminary results on the surjectivity of satellite operators on knot concordance.This is joint work with Arunima Ray of Rice University.

The chromatic convergence theorem of Ravenel and Hopkins asserts that, if $X$ is a $p$-local finite spectrum, then the homotopy limit $\text{holim}_n L_{E(n)}(X)$ of the localizations of $X$ at each of the Johnson-Wilson $E$-theories $E(n)$ is homotopy-equivalent to $X$ itself. One way of seeing the chromatic convergence theorem is by regarding the functor sending a spectrum $X$ to $\text{holim}_n L_{E(n)}(X)$ as a kind of completion, "chromatic completion," which has the agreeable property that $p$-local finite spectra are all already chromatic complete. Then there are two natural questions:

1. Given a (not necessarily finite) spectrum $X$, is there a criterion that lets us decide easily whether $X$ is chromatically complete or not?

2. Given a nonclassical setting for homotopy theory, such as equivariant spectra or motivic spectra, what analogue of the chromatic convergence theorem might hold?

We give an answers to each of these two questions. For a symmetric monoidal stable model category $C$ satisfying some reasonable hypotheses, we produce a natural notion of "chromatic completion," as well as the notion of a "chromatic cover," a commutative monoid object which shares important properties with the complex cobordism spectrum $MU$ from classical stable homotopy theory. We show that, if a chromatic cover exists in $C$, then any object $X$ satisfying Serre’s condition $S_n$ for any $n$ is chromatically complete if and only if the microlocal cohomology of $X$ vanishes. (Of course we have to define Serre's condition $S_n$ as well as microlocal cohomology in this context!)

We get two important corollaries: first, by computing some microlocal cohomology groups, we find that large classes of non-finite classical spectra are not chromatically complete, such as the connective spectra $ku$ and $BP\langle n \rangle$ for all finite $n$. We also get some non-chromatic-completeness results for $\text{ko}$, $\text{tmf}$, and $\text{taf}$. Second, we get conditions under which a chromatic completion theorem can hold for motivic and equivariant spectra: one needs a chromatic cover to exist in those categories of spectra. We identify a candidate for such a chromatic cover for motivic spectra over $\text{Spec}\, C$, assuming the Dugger-Isaksen nilpotence conjecture.

One knows from Artin reciprocity that, for any abelian Galois extension $L/K$ of $p$-adic number fields, there is an isomorphism $K^{\times} / N_{L/K} L^{\times} \to Gal(L/K)$ from the units in $K$ modulo the norms of units in $L$ to the Galois group of $L/K$; this isomorphism is called the "norm residue symbol." Computing the norm residue symbol explicitly on specific elements of its domain is quite difficult and is an open area of research in algebraic number theory.

Given an abelian Galois extension $L$ of $Q_p$ and a finite CW-complex $X$, we use Lubin-Tate theory and the Goerss-Hopkins-Miller theorem to produce a particular subgroup of the $K(1)$-local stable homotopy groups of $X$. We show that this construction provides a filtration, indexed by the abelian extensions of $Q_p$, of the $K(1)$-local stable homotopy groups of finite CW-complexes, and we use Dwork’s computation of the norm residue symbol on the maximal abelian extension of $Q_p$ to compute this filtration explicitly on some interesting finite CW-complexes, such as mod $p$ Moore spaces. We then use the nilpotence and localization theorems of Ravenel-Devinatz-Hopkins-Smith to produce a "dictionary" that lets us pass between norm residue symbols computations from explicit class field theory, and families of nilpotent elements in the $K(1)$-local stable homotopy groups of finite ring spectra.

Time allowing, we will discuss what versions of a (so far only conjectural) $p$-adic Langlands correspondence would permit these methods to be extended to higher heights, i.e., $K(n)$-local stable homotopy groups and nonabelian Galois extensions of $Q_p$.

We present an alternative to Morse-Novikov theory which works for a considerably larger class of spaces and maps rather than smooth manifolds and Morse maps. One explains what Morse-Novikov theory does for dynamics and topology and indicates how our theory does almost the same for a considerably larger class of situations as well as its additional features.

Two knots in the 3-sphere are said to be concordant if they cobound a locally flat, properly embedded annulus in the product of the 3-sphere and the unit interval. The notion of concordance originates from Fox and Milnor, and it is related with other 3- and 4-dimensional topological properties such as homology cobordism and topological surgery theory. In this talk, I will discuss various relationships between concordance and Seifert forms (or the Alexander polynomial) of knots. In particular, I will explain Cha-Orr’s extension of Cochran-Orr-Teichner's concordance invariants, which are von Neumann rho-invariants, and show its application to this subject.

The Farrell-Jones isomorphism conjecture (FJIC) plays an important role in manifold topology as well as computations in algebraic $K$- and $L$-theory. It implies, for example, the Borel conjecture of topological rigidity of closed aspherical manifolds and the Novikov conjecture of homotopy invariance of higher signatures. By the work of A. Bartels, W. Lueck and C. Wegner, it’s now known that FJIC holds for $\mathrm{CAT}(0)$-groups, i.e. groups admitting proper, cocompact actions on finite dimensional proper $\mathrm{CAT}(0)$-spaces. This includes for example fundamental groups of nonpositively curved closed Riemannian manifolds. It's a natural question that if a group admits a "nice" but not necessary proper action on a $\mathrm{CAT}(0)$-space and if the point stabilizers satisfy FJIC, whether the original group satisfies FJIC. In this talk, after outlining the general strategy for proving FJIC, I will talk about the progress that I have made concerning the above question.

We investigate a discrete analogue of Gromov’s systolic estimate and use it to prove facts about triangulations of surfaces. We also discribe a procedure for obtaining Gromov's result from the discrete version.

In this talk I will discuss the problem of classifying all closed Riemannian manifolds whose universal cover has nondiscrete isometry group. Farb and Weinberger solved this under the assumption that $M$ is aspherical. Roughly, they proved that any such $M$ is a fiber bundle with locally homogeneous fibers. However, if $M$ is not aspherical, many new examples and phenomena appear. I will exhibit some of these, and discuss progress towards a classification. As an application, I will characterize simply-connected manifolds with both a compact and a noncompact finite volume quotient.

The Baumslag-Solitar groups are a particular class of two-generator one-relation groups which have played a surprisingly useful role in combinatorial and geometric group theory. They have provided examples which mark boundaries between different classes of groups and they often provide a test-cases for theories and techniques. In this talk, I will illustrate the proof of the Farrell-Jones conjecture for them. This is a joint work with my advisor Tom Farrell.

We will explain why a generic metric on a smooth four manifold (with second Betti number at least three) has the exponential growth property, i.e., the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. Time permitting, we shall explain related topological consequences.

I will provide alternative definitions and methods of calculations for the Alexander Polynomial of a knot and ultimately a generalization of this invariant to all odd dimensional manifolds with large fundamental group. The generalization is a "rational function" on the variety of complex rank K representations of the fundamental group.

We show how recent results of Dundas-Goodwillie-McCarthy can be used to give efficient proofs of i) a Fundamental Theorem for the K-theory of connective S-algebras, ii) an integral localization theorem for the relative K-theory of a 1-connected map of connective S-algebras, iii) a generalized localization theorem for the p-complete relative K-theory of a 1-connected map of connective S-algebras. Following Weibel, we define homotopy K-theory for general S-algebras, and prove that the corresponding NK-groups of the sphere spectrum are non-trivial.

Much of this work arose in an attempt to apply recent results and methods from topological cyclic homology to update Waldhausen’s original program for studying the effect of Ravenel's chromatic tower on the algebraic K-theory of the sphere spectrum. We will give a brief summary of this program, along with recent results of Blumberg-Mandell and how they fit into some deep conjectures of Rognes. As time permits, we will add some conjectures to the list.

We show that quasi-toric manifolds are topologically equivariant rigid with the natural torus action. The proof of the rigidity is done in three steps. First we show that for the manifold equivariantly homotopy equivalent to the quasi-toric manifold the action of the torus is locally standard (it resembles the standard action of the torus on the complex space). The second step is that the manifold is equivariantly homeomorphic to the standard model of such actions. The final step is based on the topological rigidity of the quotient space which is a manifold with corners. This is joint work with Vassilis Metaftsis.

Anti de-Sitter manifolds are Lorentzian manifolds with constant curvature $-1$. In a loose analogy with Teichmuller space, there is a moduli space of AdS 3-manifolds with a given fundamental group. This space is not entirely understood—for instance, we do not know how many connected components it has–-but we do know a fair amount. We know much less about how the geometry of the manifolds varies across the moduli space. I’ll present the some preliminary results on how volume varies across the moduli space and state a few questions the results so far raise.

One can define what it means for a compact manifold with corners to be a “contractible manifold with contractible faces.” Two combinatorially equivalent, contractible manifolds with contractible faces are diffeomorphic if and only if their $4$-dimensional faces are diffeomorphic. It follows that two simple convex polytopes are combinatorially equivalent if and only if they are diffeomorphic as manifolds with corners. On the other hand, by a result of Akbulut, for each n greater than 3, there are smooth, contractible n-manifolds with contractible faces which are combinatorially equivalent but not diffeomorphic. Applications are given to rigidity questions for reflection groups and smooth torus actions.

Haken $n$-manifolds have recently been defined and studied by B. Foozwell and H. Rubinstein in analogy with the classical Haken manifolds of dimension 3, using the the theory of boundary patterns developed by K. Johannson. They can be systematically cut apart along essential codimension-one hypersurfaces until one obtains a system of $n$-cells with a boundary pattern recording some of the information carried by the original manifold and the cutting hypersurfaces. Haken manfolds in all dimensions are aspherical and, in general are amenable to proofs by induction on the length of a hierarchy (and on dimension). As such they provide a a context to explore the classical Euler characteristic conjecture for closed aspherical manifolds, which we are doing in some joint work with M. Davis.

We will discuss a way to “re-package" the colored Jones polynomial knot invariants that allows to read some of the geometric properties of knot complements they detect.

In this talk, we will see the definition of high dimensional graph manifolds and see that there are examples of graph manifolds with quasi-isometric fundamental groups, but where one supports a locally CAT(0) metric while the other cannot. We will use properties of the Euler class as well as various results on bounded cohomology.

We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. A new version of lattice homology is defined: the lattice corresponds to the normalization of the singular germ, and the Hilbert function serves as the weight function. The main result of the paper identifies four homologies: (a) the lattice homology associated with the Hilbert function, (b) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra by valuations given by the normalizations of irreducible components, (c) a certain variant of the Orlik–Solomon algebra of these local arrangements, and (d) the Heegaard--Floer link homology of the local embedded link of the germ. In particular, the Poincaré polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic Poincar\’e series of the singularity.

In this talk I will propose a refinement of the Betti numbers provided by a continuous real valued map. These refinements consist of monic polynomials in one variable with complex coefficients, of degree the Betti numbers. A number of remarkable properties of these polynomials will be discussed.

In case X is a Riemannian manifold these refinements can be even "more refined"; One can assign to the map and each nonnegative integer a collection of mutually orthogonal subspaces of the Harmonic forms = deRham cohomology in degree labelled by the zeros of the above mentioned polynomials and of dimension the multiplicity of the corresponding zero.

If the map is a Morse function the polynomials can be calculated in terms of critical values of the map and the number of trajectories of the gradient of the Morse function between critical points.

While in the homotopy theory of simplicial algebras, the homotopy of “spheres” is known, the unstable Adams spectral sequence is very far from degenerate. I’ll explain how various instances of Koszul duality create various unstable operations on the Adams spectral sequence, and on the composite functor spectral sequences one might use to calculate it.

Farrell Nil-groups are generalizations of Bass Nil-groups to the twisted case. They mainly play role in (1) The twisted version of the Fundamental theorem of algebraic K-Theory (2) Algebraic K-theory of group rings of virtually cyclic groups (3) as the obstruction to reduce the family of virtually cyclic groups used in the Farrell-Jones conjecture to the family of finite groups. These groups are quite mysterious. Farrell proved in 1977 that Bass Nil-groups are either trivial or infinitely generated in lower dimensions. Recently, we extended Farrell’s result to the twisted case in all dimensions. We indeed derived some structural results for general Farrell Nil-groups. As a consequence, a structure theorem for an important class of Farrell Nil-groups is obtained. This is a joint work with Jean Lafont and Stratos Prassidis.

K-theory is the machine that turns symmetric monoidal categories into spectra. I will discuss work, joint with Niles Johnson and Angélica Osorno, in which we study a version of K-theory in which the input is a symmetric monoidal 2-category. My ultimate goal will be to discuss our proof that symmetric monoidal 2-categories model connective spectra.

Goodwillie’s homotopy calculus provides a systematic sequence of “polynomial” approximations (which together are referred to as the Taylor tower) to suitable functors in homotopy theory. Of particular interest is the identity functor (on the category of based topological spaces) which, it turns out, has an interesting calculus. While not polynomial of any degree, the identity functor is ‘`analytic'' and its Taylor tower converges on connected nilpotent spaces (those with a nilpotent fundamental group acting nilpotently on the higher homotopy groups).

In this talk, I want to describe how the nth polynomial approximation to the identity functor can be given the structure of a monad. Algebras over this monad can be thought of as ``n-nilpotent'' spaces—in particular they are nilpotent in the above sense. I will also discuss analogous results for the identity functor on other categories. In the case of algebras over an operad (of spectra), the monad structures on the polynomial approximations to the identity are derived from work of Harper and Hess.

Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite $p$-local spectra induce nonzero homomorphisms on $\mathrm{BP}$-homology. Motivically, this theorem fails to hold: we have a motivic analog of $\mathrm{BP}$ and whilst $\eta:S^{1,1}\to S^{0,0}$ induces zero on $\mathrm{BP}$-homology, it is non-nilpotent. Recent work with Haynes Miller has led to a computation of $\eta^{-1}\pi_{*,*}(S^{0,0})$; we found it to have a very simple description.

I’ll introduce the motivic homotopy category and the motivic Adams-Novikov spectral sequence before describing this theorem. Then I'll talk about the fact that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map.

It is well known that every even positive degree cohomology class of a finite dimensional cell complex pulls back to zero in the total space of some fibration over the cell complex, where the fibre is finite dimensional. For odd degree cohomology classes, however, there are obstructions. In this talk, I will talk about how to characterize these obstructions by using the rational homotopy theory. For all finite dimensional connected cell complexes, we give a complete description, in term of Maurer-Cartan higher products, of the subspace of rational cohomology classes that pull back to zero under fibrations with finite dimensional fibre. This talk is based on joint work with A. Gorokhovsky and D. Sullivan.

The overall goal of this talk is to apply the theory of Goodwillie calculus to the category $\mathrm{Alg}_O$ of algebras over a spectral operad. Its first part will deal with generalizing many of the original results of Goodwillie so that they apply to a larger class of model categories and hence be applicable to $\mathrm{Alg}_O$. The second part will apply that generalized theory to the $\mathrm{Alg}_O$ categories. The main results here are: an understanding of finitary homogeneous functors between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary n-excisive functors can not distinguish between $\mathrm{Alg}_O$ and $\mathrm{Alg}_{O \leq n}$, the category of algebras over the truncated $O

eq n$; and a weak form of the chain rule between such algebra categories, analogous to the one studied by Arone and Ching in the case of Spaces and Spectra.

After discovering the relations among Steenrod powers that bear his name, J. Adem proved a theorem on singly generated cohomology rings. His line of reasoning eventually led to J.F. Adams’ resolution of the Hopf invariant one problem. I will discuss a generalization of Adem's theorem and a different application of it to geometry. When combined with a fundamental result of B. Wilking, this result leads to computations of topological invariants of manifolds that admit Riemannian metrics with positive sectional curvature and symmetry.

We prove that the X(n) spectra, used in the proof of Ravenel’s Nilpotence Conjecture, can be constructed as iterated Hopf-Galois extensions of the sphere spectrum by loop spaces of odd dimensional spheres. We hope to leverage this structure to obtain a better understanding of the Nilpotence Theorem as well as develop an obstruction theory for the construction of complex orientations of homotopy commutative ring spectra. The method of proof is easily generalized to show that other Thom spectra can be considered intermediate Hopf-Galois extensions, for instance the fact that MU is a Hopf-Galois extension of MSU by infinite dimensional complex projective space.

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed. The talk is based on joint work with Thomas Kerler.

This is a pre-talk for the main talk on Thursday afternoon, for those interested in further technical background and ideas. In this talk, I will give a survey of a few model categories which are meant to present “the homotopy theory of homotopy theories.” I will particularly emphasize the theory of quasicategories, and highlight some of its convenient technical advantages. My goal is to introduce enough of the theory that Higher Topos Theory stops looking scary and dense and starts looking exciting and readable.

Goerss–Hopkins obstruction theory is a tool for obtaining structured ring spectra from algebraic data. It was originally conceived as the main ingredient in the construction of tmf, although it’s since become useful in a number of other settings, for instance in setting up a tractable theory of spectral algebraic geometry and in Rognes's Galois correspondence for commutative ring spectra. In this talk, I'll give some background, explain in broad strokes how the obstruction theory is built, and then indicate how one might go about generalizing it to an arbitrary (presentable) $\infty$-category. This last part relies on the notion of a model $\infty$-category—that is, of an $\infty$-category equipped with a “model structure” -- which provides a theory of resolutions internal to $\infty$-categories and which will hopefully prove to be of independent interest.

Let R be a finite-dimensional regular local ring with maximal ideal m. The category of m-complete R-modules is not abelian, but it can be enlarged to an abelian category of so-called L-complete modules. This category is an abelian subcategory of the full category of R-modules, but it is not usually a Grothendieck category. It is well known that a Grothendieck category always has a derived category, however, this is much more delicate for arbitrary abelian categories.

The Hermitian K-theory of a ring with an antistructure is a topological group completion for the monoid of Hermitian forms on the ring. Recent work of Hesselholt and Madsen describes Hermitian K-theory as the fixed points of a genuine Z/2-spectrum: the Real K-theory spectrum of the ring. Their construction uses a variant of Waldhausen’s Sdot construction on categories with duality. This categorical approach makes it possible to construct trace maps to Real variants of THH and TC, as maps of Z/2-spectra. The talk will focus on how, via the trace map, “the equivariant Goodwillie derivative of Real K-theory is Real THH.”

I will describe a method for constructing spectral sequences in homotopy type theory, including ones of Leray-Serre-type and Atiyah-Hirzebruch-type and possibly others. This provides a good test case to discuss ways in which homotopy theory in type theory is similar to and different from classical algebraic topology. I will mention some potential applications, both inside type theory and outside of it (by way of categorical semantics). No prior knowledge of homotopy type theory will be necessary.

Given a metric on a finite CW-complex $X$, how can we use geometry to better understand elements $\alpha \in \pi_n(X)$? One way is by measuring distortion, that is, the way the geometric complexity of an optimal representative of $k\alpha$ grows as a function of $k$. In some sense, the "true" measure of complexity is given by the Lipschitz constant, but volume provides an upper bound which is interesting in many cases. Compared with Lipschitz distortion, which is the topic of an as yet unresolved conjecture of Gromov, volume distortion is tractable and satisfies a strong form of topological invariance. I will present examples of three ways that volume distortion can arise: from rational homotopy invariants, from the action of the fundamental group on higher homotopy groups, and from the geometry of the fundamental group. These three sources of distortion turn out to be enough to characterize those spaces which have no distorted elements.

Consider a flavor of structured ring spectra that can be described as algebras over an operad O in spectra. A natural question to ask is when the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated Koszul dual comonad K can be modified to turn it into an equivalence of homotopy theories. In a paper published in 2012, Francis and Gaitsgory conjecture that replacing O-algebras with the full subcategory of homotopy pro-nilpotent O-algebras will do the trick. In joint with Kathryn Hess we show that every 0-connected O-algebra is homotopy pro-nilpotent; i.e. is the homotopy limit of a tower of nilpotent O-algebras.

This talk will describe recent work, joint with Michael Ching, that resolves in the affirmative the 0-connected case of the Francis-Gaitsgory conjecture; that replacing O-algebras with 0-connected O-algebras turns the fundamental adjunction into an equivalence of homotopy theories. This can be thought of as a spectral algebra analog of the fundamental work of Quillen and Sullivan on the rational homotopy theory of spaces, the subsequent work of Goerss and Mandell on the p-adic homotopy theory of spaces, and the work of Mandell on integral cochains and homotopy type. Corollaries include the following: (i) 0-connected O-algebra spectra are weakly equivalent if and only if their TQ-homology spectra are weakly equivalent as derived K-coalgebras, and (ii) if a K-coalgebra spectrum is 0-connected and cofibrant, then it comes from the TQ-homology spectrum of an O-algebra.

We will recall the usual method, introduced by Schwede and Shipley, of transferring a model structure on a monoidal model category M to the category of P-algebras where P is a colored operad. We’ll then discuss what hypotheses are needed on M so that the resulting model structure on P-algebras is left proper. We'll apply this machinery to the situations where P is a cofibrant colored operad, when P is the commutative monoid operad, and when P is the colored operad for non-reduced operads. We introduce the commutative monoid axiom and prove that the latter two situations inherit left proper model structures from M in the presence of this axiom and the hypothesis of h-monoidality. The primary application of this work is a proof due to Michael Batanin of the Baez-Dolan Stabilization Hypothesis.

In this talk, we will expose how to organize Bredon homology computations for a specific class of groups, with the aim of finding a useful decomposition of Bredon homology in the general case. The Bredon homology of a group allows us to obtain the equivariant K-homology of (the classifying space for proper actions of) the latter, via a spectral sequence. The Baum-Connes conjecture, which has been proved for large classes of groups, constructs an isomorphism from the equivariant K-homology of a group to the K-theory of its reduced C*-algebra. For groups like the ones where our computation is carried out, SL_2 matrix groups over rings of imaginary quadratic integers, this yields the isomorphism type of the mentioned operator K-theory, which would be very hard to compute directly.

(This is joint work with Bob MacPherson.) Configurations spaces of points, in $\mathbb{R}^2$ for example, are well studied in algebraic topology. We understand the homology, homotopy groups, etc., extremely well. But if the points have some finite thickness, and if the region of the plane is restricted, then much less is known. However this situation is quite natural from the point of view of physics—such configuration spaces describe the phase space or energy landscape of a hard spheres gas. As a case study, we investigate the configuration space of $n$ disks of unit radius in an infinite strip of width $w$. We are especially interested in the asymptotic behavior of the $k$th Betti number as $n \to \infty$, and we find two regimes. For some choices of $(w,k)$, the $k$th Betti number grows polynomially fast, and for some it grows exponentially fast. The line separating these two regimes might be considered a kind of phase transition. Our main results are asymptotic estimates for the Betti numbers in every case, correct up to a constant factor.

Otal and Croke proved marked length spectrum rigidity for negatively curved, closed surfaces – if you know the length of the geodesic representative for each free homotopy class, then you know the metric up to isometry. This result has been extended in several directions by loosening the requirements on the metric and—separately -- allowing various ’bad' properties. Versions have been proven when 'negative curvature' is replaced by 'no conjugate points', or when cone point singularities are allowed, or when the metric is flat away from cone points. In this talk I'll discuss how ideas from these various arguments can be patched together to prove MLS rigidity for metrics which are non-positively with cone points; under a technical assumption, this can be extended to 'no conjugate points.'

The Reshetikhin-Turaev TQFT is a functor from a category of 3-dimensional cobordisms to the category of vector spaces. To include surfaces with boundaries and cobordisms with corners in the RT TQFT setting, I will discuss a concrete construction of a 2-categorical extension of the RT TQFT. This extension is a 2-functor from a 2-category of cobordisms with corners to the Kapranov-Voevodsky 2-vector spaces.

I will explain how to construct generic rank-2 Poisson structures on closed oriented smooth 4-manifolds. These are associated to broken Lefschetz fibrations, and also to wrinkled fibrations and their deformations. This talk is based on joint papers with L. García-Naranjo and R. Vera, and with J. Torres Orozco.

Local cohomology was introduced by Grothendieck as a substitute in algebraic geometry for the relative cohomology of a pair of spaces. His studies culminate in his local duality theorem, a result that has proven to be important both computationally and conceptually. Subsequently, this theory has been extended in many directions, most notably by Greenlees and May with the introduction of local homology functors and their interpretation of local duality as special case of what is now know as Greenlees-May duality.

However, local duality type results appear in other instances not covered by the existent theory. In this talk, we will introduce a common categorical framework that exhibits local duality as a fundamental phenomenon in different contexts. We will explain how this can be used to develop the theory for comodules over certain Hopf algebroids and their corresponding geometric interpretation as quasi-coherent sheaves over stacks. If time permits, we will show an application of this theory to the computation of certain cohomology theories in chromatic homotopy theory. This is joint work with Tobias Barthel and Drew Heard.

A large family of theorems all state that if a space is topologically complex, then the functions on that space must express that complexity, for instance by having many singularities. For the theorem in this talk, our preferred measure of topological complexity is the hyperbolic volume of a closed manifold admitting a hyperbolic metric (or more generally, the Gromov simplicial volume of any space). A Morse function on a manifold with large hyperbolic volume may still not have many critical points, but we show that there must be many flow lines connecting those few critical points. Specifically, given a closed n-dimensional manifold and a Morse-Smale function, the number of n-part broken trajectories is at least the Gromov simplicial volume. To prove this we adapt lemmas of Gromov that bound the simplicial volume of a stratified space in terms of the complexity of the stratification.

This talk will mostly be a survey on what is known about the isometric embedding problem for various types of metric spaces. We will compare and contrast the known results for manifolds and polyhedra, and we will classify these results into two types: those that are "flexible" vs. those that are "rigid". If time permits, we will discuss what is known for more general types of metric spaces.

To any pair $(X,f)$, $X$ compact ANR and $f$ a real (angle) valued map defined on $X$ and any $r$, a non-negative integer, we assign: (1) a finite configuration of points “$z$” with multiplicities $\delta_{rf}(z)$ located in the complex plane and (2) a finite configuration of vector spaces $\delta^{f} (z)$ indexed by the same $z$′s in analogy with (1) the configuration of eigenvalues and of (2) generalized eigenspaces of a linear operator in a finite dimensional complex vector space.

The analogy goes quite far as long as the formal properties are concerned and becomes particularly subtle in the case of an angle valued map (involving L-2 topology). The basic properties/implications are discussed.

In the homotopy calculus of functors, Goodwillie defines a way of assigning a Taylor tower of polynomial functors to a homotopy functor and identifies the homogeneous pieces as being classified by certain spectra, called the derivatives of the functor. Michael Ching showed that the derivatives of the identity functor of spaces form an operad, and Arone and Ching developed a chain rule for composable functors. We will review these results and show that through a slight modification to the definition of derivative, we have found a more straight forward chain rule for endofunctors of spaces.

We review the notion of a Hopf-Galois extension of ring spectra and construct a number of new intermediate Hopf-Galois extensions. We use this to introduce a new construction of MU which bears a striking resemblance to Lazard’s construction of the Lazard ring.

We study the problem that whether the Farrell-Jones conjecture in algebraic $K$- and $L$-theories is closed under group extensions. This problem is indeed equivalent to the following two problems. The first one is that whether the conjecture is closed under taking semi-direct product with the infinite cyclic group and the second one is that whether the conjecture is closed under passage to over-groups of finite indices. We show that the $L$-theoretic Farrell-Jones conjecture is closed under taking semi-direct products of torsion free groups with the infinite cyclic group. In particular, this implies the $L$-theoretic Farrell-Jones conjecture holds for many nearly crystallographic groups, all poly-free groups, all poly-surface groups and all groups that are extensions of torsion free non-positively curved groups and so on. Meanwhile, we obtain an obstruction for the same result to hold for the $K$-theoretic Farrell-Jones conjecture. Our study of the second problem leads us to the proposal of a general problem in $K$- and $L$- theory. Our study of this problem enables us to reduce the problem that whether the Farrell-Jones conjecture is closed under passage to over-groups of finite indices to some special cases.

In 1933, van Kampen developed a homological obstruction to embedding simplicial complexes into Euclidean space. Bestvina, Kapovich and Kleiner used this obstruction to give lower bounds on the dimension of a contractible manifold that a group can act on properly discontinuously. I will discuss some examples of groups where this obstructor theory has proven successful, including right-angled Artin groups and lattices in Euclidean buildings. This is based on joint work with Grigori Avramidi, Michael Davis, and Boris Okun.

We consider hyperbolic groups with boundaries homeomorphic to the boundaries of certain types of Kleinian groups. We show that a hyperbolic group quasi-isometric to a mixed Kleinian surface group is virtually a mixed Kleinian surface group. Furthermore, in this case, the rigid pieces of the two groups are commensurable. This is joint work with Peter Haissinsky and Luisa Paoluzzi.

In this talk I will show that the space of complete, pinched negatively curved Riemannian metrics with finite volume on a smooth manifold, is either empty or non-connected, provided one has control at infinity and the dimension of the manifold is large. Time permitting, I will discuss the related problem of endowing a smooth fiber bundle with extra geometric structure.

I give the construction of a "THH-May spectral sequence" which takes as input the topological Hochschild homology of the "associated graded ring spectrum" of a filtration of a ring spectrum by "ring spectrum ideals," and which outputs THH of the original ring spectrum. The construction of this spectral sequence turns out to be quite nontrivial and involves some new ideas.

I describe in general terms how this spectral sequence looks for THH(K(F_q)), the topological Hochschild homology of the K-theory spectrum of certain finite fields; the detailed computation of the spectral sequence (computation of the differentials, etc.) is quite nontrivial, and is the topic of G. Angelini-Knoll’s talk this same week (on Thursday). I also describe, in general terms, how this spectral sequence looks for THH of a connective model for the K(2)-local sphere, which involves new work of Bruner and Rognes on THH of the spectrum of topological modular forms.

These two THH computations are the input one needs for a trace method computation of the topological cyclic homology of connective models for the K(1)-local and K(2)-local sphere spectrum, which by a theorem of C. Ogle on the Goodwillie derivative of algebraic K-theory of ring spectra, computes the (relative) algebraic K-groups of these connective models for the K(1)-local and K(2)-local sphere spectrum. These are the first two presently-unknown lines in the Gersten spectral sequence E_1-term for the sphere spectrum; the computation of this Gersten spectral sequence is an old problem posed by Waldhausen. I will describe this point of view and its relationship to higher Chow groups and algebraic cycles in ring spectra.

This project is joint work with C. Ogle and G. Angelini-Knoll.

It is well known in low-dimensional topology that ’Boroczky's Theorem', which bounds the local density of ball packings of hyperbolic space, also yields sharp density bounds for such packings of hyperbolic manifolds in certain circumstances. Using this fact, in 1996 C. Bavard characterized the 'extremal' closed hyperbolic surfaces: those which admit an embedded disk of largest possible radius. I will discuss the characterization of non-compact extremal surfaces of finite volume, which required new ideas. Then I'll report what I know (not much) about the case of several-disk packings of hyperbolic surfaces.

There have been a host of prime geodesic theorems over the past several decades displaying a surprising analogy between the behavior of primitive, closed geodesics on hyperbolic manifolds and the behavior of the prime numbers in the integers. For instance, just as the prime number theorem dictates the asymptotic growth of the number of primes less than n, there is an analogous asymptotic growth for primitive, closed geodesics of length less than n. In this talk, I will give a brief review of the relevant definitions and go on to give the history of this analogy. I will then discuss some recent work extending this relationship to give the geodesic analogue of the Green–Tao theorem on arithmetic progressions in the prime numbers.

The complement of an arrangement, $A$, of a finite number of affine hyperplanes in $\mathbf{C}^n$ is homotopy equivalent to a poset of CW complexes indexed by the intersection poset, $L(A)$. The subcomplex corresponding to $G\in L(A)$ is homotopy equivalent to the complement of the central arrangement $A_G$ normal to $G$. Similarly, toric hyperplane arrangements have the structure of a diagram of spaces. These structure can be used repair a spectral sequence argument in earlier papers by my collaborators and me on the cohomology of such complements with certain local coefficients. This is joint work with Boris Okun.

Koseleff and Pecker show that all knots can be parametrized by Chebyshev polynomials in three dimensions. These long knots can be realized as trajectories on billiard table diagrams. We use this knot diagram model to study random knot diagrams by flipping a coin at each 4-valent vertex of the trajectory.

We truncate this model to study 2-bridge knots together with the unknot. We give the exact probability of a knot arising in this model. Furthermore, we give the exact probability of obtaining a knot with crossing number c.

This is joint work with Sunder Ram Krishnan and Chaim Even-Zohar.

In this talk, I will report on joint work with Jean Lafont. We show that, for an $n$-dimensional irreducible symmetric space of rank $r\geq 2$ (excluding $SL(3,\mathbb R)/SO(3)$ and $SL(4, \mathbb R)/SO(4)$), the $p$-Jacobian of barycentrically straightened simplices has uniformly bounded norm, provided $p\geq n-r+2$. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group $G$, every degree $p$ cohomology class has a bounded representative. This answers Dupont’s problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension $n-r$ have unbounded volume, showing that the range in which we obtain boundedness of the $p$-Jacobian is very close to optimal. I will also discuss some of my recent work on its generalization.

A Picard category is a symmetric monoidal category in which the morphisms are all isomorphisms and the objects are invertible with respect to the tensor product. These arise from certain kinds of algebraic or geometric invariants, but also in stable homotopy theory as models for stable homotopy 1-types. The computation of a specific universal Picard category shows that there is a canonical action of the cyclic group of order 2 on any invertible object in any symmetric monoidal category, and moreover this computation is an algebraic shadow of the process whereby one obtains the sphere spectrum from the symmetric groups. Exchanging the symmetric groups for the braid groups, one might ask what the corresponding canonical group action will be, and I will explain some topologically-inspired machinery that gives the answer for braids as well as many other possible input structures.

At the prime 2, Behrens-Hill-Hopkins-Mahowald showed $M(1,4)$ admits 32-periodic $v_2$-self-map and more recently B-Egger-Mahowald showed $A_1$ also admits 32-periodic $v_2$-self-map. This leads to the question, whether there exists a finite 2-local complex with periodicity less than 32. This talk will answer the question by producing a finite 2-local complex $Z$ which admits 1-periodic $v_2$-self-map. Apart from admitting 1-periodic $v_2$-self-map, Z has other remarkable properties such as $tmf_\ast Z =k(2)_\ast$ and $E \wedge Z = K(2) \wedge (G_48)_+$, where $E$ is the height 2 Morava $E$-theory and $G_{48}$ is the maximal profinite subgroup of Morava stabilizer group. This work is joint with P. Egger.

The colored Jones polynomial is a sequence of knot invariants. For alternating and adequate knots, it can be shown that the sequence of leading coefficients will stabilize, which allows us to define a power series invariant called the tail of the colored Jones polynomial. We will discuss the definition of this power series as well as techniques to compute it for certain large classes of knots, and it’s relation to the all-A state graph of a particular diagram of the knot.

A Lot of topological obstructions live in the K-theory of Group rings. The Farrell-Jones conjecture makes predictions about that K-theory. Bartels and Lueck showed to Farrell-Jones conjecture for Groups Acting geometrically on CAT0-spaces. In Joint work with D.Kasprowski we generalized their work to Groups that act geometrically on spaces with certain bicombings. There are non-CAT0 groups which satisfy our assumptions. I will give some examples and finish by mentioning some open Problems.

The Yamabe problem consists of finding a complete metric with constant scalar curvature in a prescribed conformal class. A landmark result in Geometric Analysis is that a solution always exists on closed manifolds, but the situation is much more delicate in the noncompact case: there are complete noncompact manifolds where no solutions exist. In this talk, I will discuss how certain topological techniques can be used to obtain infinitely many solutions on some special noncompact manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type. As a consequence, one also obtains infinitely many new solutions to the so-called Singular Yamabe problem on spheres. (This is based on joint work with P. Piccione)

The action dimension of a discrete group $G$ is the minimum dimension of a contractible manifold, on which group $G$ acts properly discontinuously. We will prove that the action dimension for an Artin group with the nerve $L$ of dimension $n$ is less than or equal to $2n + 1$ if the Artin group satisfies the $K(\pi, 1)$ conjecture and the top cohomology group of $L$ with $\mathbb{Z}$-coefficients is trivial.

I will discuss some features of the topology of smooth bundles whose fiber is a closed smooth manifold that supports a nonpositively curved Riemannian metric. Specifically, I will show (topological) rigidity results for the associated vertical tangent bundle and a vanishing theorem for the generalized Miller-Morita-Mumford classes. This is joint work with Tom Farrell and Yi Jiang.

In his landmark 1969 Annals paper, Quillen showed that the rational homotopy type of a simply connected space could be detected at the level of its singular rational chains, and furthermore, that rational chains fit into a derived equivalence with cocommutative dg coalgebras over the rationals, after restricting to 1-connected objects. In 1977 Sullivan subsequently proved the analogous result in the case of rational cochains and commutative dg algebras over the rationals. Since then topologists have worked on attempting to establish analogous results for finite fields (Kriz, Goerss, Mandell), and more recently some partial results have been established in the integral chains case (Mandell, Karoubi). Nevertheless, establishing that integral chains fit into a derived equivalence has proved resistant to all attacks. In this talk I will outline how we recently resolved, in the affirmative, the integral chains problem. Our approach exploits a mixture of (co)simplicial techniques and together with some ideas related to the framework developed in Ching-Harper’s recent resolution of the 0-connected Francis-Gaitsgory conjecture. If time permits, I will also describe how we recently resolved the problem of establishing a recognition principle for iterated suspension spaces, dual to the celebrated iterated loop spaces work of May and Beck. This is joint work with M. Ching and J.E. Harper.

The main result of MorseTtheory is the description of the "Morse Complex", which calculates the homology of a compact Riemannian manifold, whose components are given by the critical points of a smooth generic function and the boundary maps by the isolated trajectories (between critical points) of the gradient of the function with respect to a generic Riemannian metric.

We have recently shown that this complex (when tensored by a field) is completely determined (and determines) up to isomorphism by the “bar codes” of the function. The bar codes are finite collections of pairs of real numbers (ends of intervals) effectively computable by “computer implementable algorithms”.

In addition to computational applications (calculation of number of critical points, and estimates on isolated trajectories) there are pleasant implications both in topology and in dynamics. But more important, the Morse complex we propose can be defined for a considerably larger class of functions defined on a considerably larger class of compact spaces providing informations about the dynamics of flows admitting the function as Lyapunov.

The main result of MorseTtheory is the description of the "Morse Complex", which calculates the homology of a compact Riemannian manifold, whose components are given by the critical points of a smooth generic function and the boundary maps by the isolated trajectories (between critical points) of the gradient of the function with respect to a generic Riemannian metric.

We have recently shown that this complex (when tensored by a field) is completely determined (and determines) up to isomorphism by the “bar codes” of the function. The bar codes are finite collections of pairs of real numbers (ends of intervals) effectively computable by “computer implementable algorithms”.

In addition to computational applications (calculation of number of critical points, and estimates on isolated trajectories) there are pleasant implications both in topology and in dynamics. But more important, the Morse complex we propose can be defined for a considerably larger class of functions defined on a considerably larger class of compact spaces providing informations about the dynamics of flows admitting the function as Lyapunov.

I will discuss the geometry of the class of second-order partial differential equations of parabolic type. In particular, any scalar parabolic differential equation is equivalent to a certain smooth manifold with extra geometric structure. This geometric structure has local invariants that control the behavior of solutions, as well as various properties of the equation itself. I will also explain what a good understanding of these local invariants tells us about the conservation laws of parabolic differential equations.

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. As a consequence, we verify the Chern Conjecture for complete affine manifolds whose holonomy contains a pure translation. This is joint work with Michelle Bucher (Geneva) and Chris Connell (Indiana).

Let $G$ be a finite dimensional Lie group and $G^\delta$ be the same group with discrete topology. The natural homomorphism from $G^\delta$ to $G$ induces a continuous map from $BG^\delta$ to $BG$. Milnor conjectured that this map induces a $p$-adic equivalence. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we show that the map from $BG^\delta$ to $BG$ induces split surjection on cohomology with finite coefficients in "the stable range". If time permits, I will discuss applications of these results in foliation theory, in particular flat surface bundles.

In the late 80’s Gromov and Thurston constructed examples of manifolds which do not admit a hyperbolic metric but do admit metrics whose sectional curvature is pinched arbitrarily close to -1. Their construction involves taking cyclic branched covers of hyperbolic manifolds over "nice" codimension two totally geodesic submanifolds. In a joint project with J.F. Lafont, J. Meyer, and B. Tshishiku we have extended this construction to 4-manifolds whose metric is modeled on the complex hyperbolic plane. This will be the content of the first half of the talk.

For this group project, I needed to understand the metric in the complex hyperbolic plane expressed in polar coordinates about a copy of the real hyperbolic plane. This research will make up the second half of the talk.

Consider a $\pi_{1}$-injective immersion $f:S\to M$ from a compact surface $S$ to a hyperbolic 3-manifold $(M,h)$. Let $\Gamma$ denote the copy of $\pi_{1}S$ in $\mathrm{Isom}(\mathbb{H}^{3})$ induced by the immersion. In this talk, I will discuss relations between two dynamics quantities: the critical exponent $\delta(\Gamma)$ and the topological entropy $h_{top}(S)$ of the geodesic flow for the immersed surface $(S,f^{*}h)$. These dynamics relations lead us to geometry results: through these relations one can characterize certain hyperbolic 3-manifolds such as Fuchsian manifolds, quasi-Fuchsian manifolds, and almost-Fuchsian manifolds. If time permits, I will also discuss applications of these relations to the moduli space of $S$ introduced by C. Taubes.

We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least three do not support Anosov diffeomorphisms. This is joint work with Andrey Gogolev (Binghamton).

In this talk I will discuss obstructions to Riemannian smoothings of a locally CAT(0) manifold. I will focus on obstructions in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their methods can be extended to construct more examples of locally CAT(0) 4-manifolds $M$ that do not support Riemannian metric with nonpositive section curvature. Further, the universal cover of such a manifold, $\tilde M$, satisfies the isolated flats condition and contains a collection of 2-dimensional flats with the property that their boundaries at infinity form non-trivial link in $\partial^\infty \tilde M$.

Algebraic K-theory brings together classical invariants of rings with homotopy groups of topological spaces. In general algebraic K-theory groups are difficult to compute, but in recent years methodsin equivariant stable homotopy theory have led to many important K-theory computations. I will introduce this approach to K-theory computations, and discuss why it is particularly useful in studying the algebraic K-theory of pointed monoid algebras. I will also present some of my recent joint work with Angeltveit on the algebraic K-theory of the group ring $\mathbb{Z}[C_2]$.

Given an injective amalgam at the level of fundamental groups and a specific 3-manifold, is there a corresponding geometric-topological decomposition of a given 4-manifold, in a stable sense? We find an algebraic-topological splitting criterion in terms of the orientation classes and universal covers. Also, we equivariantly generalize the Lickorish–Wallace theorem to regular covers. This is joint work-in-progress with my PhD student, Gerrit Smith.

Hochschild homology is a classical invariant of algebras. A "topological" version, called THH, has important connections to algebraic K-theory, Waldhausen\’s A-theory, and free loop spaces. For coalgebras, there is a dual invariant called "coHochschild homology" and Hess and Shipley have recently defined a topological version called "coTHH," which also has connections to K-theory, A-theory and free loops spaces. In this talk, I\'ll talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH.

I will define and discuss Genus One Fibered Knots (GOF-knots), and a beautiful method to analyze them that touches on automorphisms of trees, representations of SL(2, Z), and hyperbolic geometry. I will then discuss how this analysis can be used to classify all monodromies of GOF-knots, the ambient manifolds in which they sit, and all generalized crossing changes from one GOF-knot to another within a manifold.

This will be a survey-style talk describing ways that categorical algebra helps us understand calculations in homotopy theory. Our main result is the 2-dimensional Stable Homotopy Hypothesis; we will explain what this means and how it relates to basic questions in topology and basic tools in algebra. We will close with some indications of the crucial technical results and what they have taught us about higher-dimensional categories. Much of the talk is based on joint work with Nick Gurski, Angélica Osorno, and Marc Stephan.

A common theme in mathematics is to try to understand an object by breaking it down into smaller pieces via a filtration and then study how those pieces fit together. In (stable) homotopy theory, we use the Postnikov tower to filter a space (or spectrum) by its homotopy groups. In equivariant stable homotopy theory, the slice filtration is one object that plays the role of the Postnikov tower. In this talk, I will first recall the construction of the Postnikov tower, we will then discuss some basics of equivariant stable homotopy theory, and I will describe the construction of the slice tower, highlighting a new way to better understand how this object filters G-spectra.

The dynamics of a geodesic flow on a Riemannian manifold are generated by the metric. In this talk, we are interested in metrics on surfaces that produce uniformly hyperbolic (Anosov) dynamics. There are canonical examples of such metrics, such as those of constant negative curvature and, more generally, of strictly negative curvature. On the other hand, there are also restrictions on which metrics can support such dynamics. Conjugate points were shown by Klingenberg to be an obstruction to Anosov geodesic flows–this implies that neither the sphere nor the torus can support an Anosov geodesic flow, regardless of the metric. However, it is possible for a surface to have a fair amount of positive curvature and still generate an Anosov geodesic flow. In this talk, I will present a new proof of a result of Donnay and Pugh that shows the existence of surfaces with Anosov geodesic flows whose metrics come from embeddings in $\mathbb{R}^3$, and I will discuss an estimate for the genus of such a surface. This is work with V. Donnay.

In this talk, I will discuss the notion of simplicial volume introduced by Gromov and Thurston. For nonpositively curved manifolds, Gromov conjectured that negative Ricci curvature implies postivitity of simplicial volume. I will talk about some recent work joint with Chris Connell. We show that, under a stronger curvature condition, the simplicial volume is always positive, this answers a special case of Gromov’s problem.​

Martin-Löf type theory is an alternative foundation of mathematics invented in the 1970s. It is the mathematical language used by computer proof checkers such as Coq and Agda. One peculiar feature of this language is that its notion of equality is much weaker than the normal mathematical notion of equality.

In the early 2000s, it was realized that the rules for this equality describe some kind of homotopy theory. In particular, these rules look a lot like the axioms for a weak factorization system. These observations have grown into a research program called Homotopy Type Theory.

In this talk, I will describe the basics of Martin-Löf type theory, various models of this type theory, and how these models give rise to weak factorization systems.

If two finite complexes or compact manifolds are homotopy equivalent, then they are Lipschitz homotopy equivalent (in the obvious sense.) Therefore, in this context, Lipschitz homotopy invariants are a natural object of study. More than 20 years ago, Gromov made a number of conjectures about such invariants; today we are finally equipped to tackle some of them. The main technique is a kind of inverse to the well-known results of rational homotopy theory: it is possible to produce maps which are "close" to arbitrary DGA homomorphisms. Combined with some quantitative results about DGAs, this proves a number of results about maps from a finite complex $X$ to a finite simply connected complex $Y$, such as:

The number of homotopy classes in $[X,Y]$ which have an $L$-Lipschitz representative grows at most polynomially in $L$. However, pace Gromov, this is not always exactly a polynomial; in one example the growth is asymptotic to $L^8 \log L$.

Any two homotopic $L$-Lipschitz maps $f, g:X \to Y$ are homotopic via a $C(L+1)^p$-Lipschitz map $X \times I \to Y$, where $C$ and $p$ are constants depending on $X$ and $Y$.

Many questions remain open. Some of this is joint work with Shmuel Weinberger.

A triangle group $T(p,q,r)$ is the group of rotational symmetries of a tiling of the hyperbolic plane by geodesic triangles. We will begin by discussing a component of the representation variety of $T(p,q,r)$ in $\mathrm{PSL}(n,R)$ called the Hitchin component, noteworthy in part because representations inside are all discrete and faithful.

For $n = 3$, the dimension of the Hitchin component for hyperbolic triangle groups follows from a special case of work by Choi and Goldman. More recently, Long and Thistlethwaite determined its dimension for general $n \geq 3$. Our results retain this broader n-dimensional context, but focus in on those representations contained in the subgroup $G = \mathrm{SO}(m,m+1)$ or $G = \mathrm{Sp}(2m)$. In particular, we will give a formula for the dimension of this "restricted" Hitchin component for hyperbolic $T(p,q,r)$ within $G$ for all $n \geq 3$.

Basmajian’s identity expresses the length of the boundary of a compact hyperbolic surface as a summation over the orthogeodesics on the surface. In this talk, I will introduce Basmajian-type series identities on limit sets associated to familiar complex dynamical systems. In particular, I will show how to extend Basmajian's identity to certain Schottky groups via analytic continuation. I will also discuss Basmajian-type identities for quadratic polynomials and its applications to Hausdorff dimension of Julia sets.

The goal of this talk will be to describe some of the methods used in my recent results in quantitative homotopy theory. In particular, I’d like to outline the proof of the following theorem: If $X$ and $Y$ are finite metric simplicial complexes with $Y$ simply connected, then every nullhomotopic $L$-Lipschitz map $f:X \to Y$ has a constant time nullhomotopy through $O(L^2)$-Lipschitz maps. I will start with an introduction to the relevant aspects of rational homotopy theory and no prior knowledge in that area will be required. The talk will be nominally independent but the context and motivation come from my Topology Seminar and Welcome Seminar talks.

Topological complexity is a homotopy invariant introduced by Michael Farber in the early 2000s. Denoted $TC(X)$, it counts the smallest size of a continuous motion planning algorithm on X. In this sense, it solves optimally the problem of continuous motion planning in a given topological space. In topological robotics, a part of applied algebraic topology, several variants of $TC$ are studied. In a recent paper, I introduced the relative topological complexity of a pair of spaces $(X, Y )$ where $Y \subset X$. Denoted $TC(X, Y )$, this counts the smallest size of motion planning algorithms that plan from $X$ to $Y$.

Right-angled Artin groups have grown in importance lately with their connection to braid groups and their connection to real-world robotics problems. In this talk, we will present the background needed to compute the relative topological complexity of pairs of right-angled Artin groups and hopefully discuss the details of the optimal motion planner involved.

In the first lecture I will recall some background in hyperbolic dynamics and discuss several well-known open classification problems. In the second lecture I will explain certain cut-and-paste constructions which produce new higher dimensional examples of partially hyperbolic diffeomorphisms and flows. I plan to make both talks very accessible, especially the first one. In the first lecture I will recall some background in hyperbolic dynamics and discuss several well-known open classification problems. In the second lecture I will explain certain cut-and-paste constructions which produce new higher dimensional examples of partially hyperbolic diffeomorphisms and flows. I plan to make both talks very accessible, especially the first one.

We describe a new algorithm which determines if the intersection of a quasiconvex subgroup of a negatively curved group with any of its conjugates is infinite. The algorithm is based on the concepts of a coset graph and a weakly Nielsen generating set of a subgroup. We also give a new proof of decidability of a membership problem for quasiconvex subgroups of negatively curved groups.

I will explain why our understanding about manifolds is much further advanced than our understanding of their symmetry. With some luck, I will then try to convince you that there’s a reason for some optimism.

If $C$ is the categorical representation of a finite partially ordered set, then a $C$-module refers to a covariant functor from $C$ to the category $(\mbox{vect}/k)$ of finite-dimensional vector spaces over a field $k$. When $C$ represents a totally ordered set, one simply recovers the notion of finite persistence module, so this framework represents a natural generalization of persistence modules, and includes all finite multi-dimensional persistence modules.

For a fixed $C$ the moduli space of isomorphism classes of $C$-modules is in general non-discrete, and a fundamental problem - for multi-dimensional persistence modules in particular - has been to find an appropriate generalization of the classification theorem for finite 1-dimensional persistence modules which is nevertheless consistent with the consequences of Gabriel’s Theorem.

In these two talks we present such a generalization in the case the $C$-module admits an inner product (is an IPC-module). This is additional structure which occurs naturally in applications; specifically when the module is geometrically injective - it arises as the homology of a $C$-diagram of spaces where the morphisms are all closed cofibrations.

We show that any such IPC-module $M$ admits a weakly tame cover $T(M) \twoheadrightarrow M$, functorial in $M$, from which one can recover the block codes of the module even when the module $M$ itself is not weakly tame. This cover $T(M)$ is the closest approximation to $M$ by a weakly tame $C$-module, in the sense that $M$ is weakly tame iff the projection $T(M) \twoheadrightarrow M$ is an isomorphism. Finally, in the event the indexing category $C$ is holonomy-free, the block summands of $T(M)$ may be further decomposed as a direct sum of generalized bar codes (GBCs). In particular, this is the case when $C$ indexes a finite multi-dimensional persistence diagram.

The paper forming the basis for this talk is available at http://arxiv.org/abs/1803.08108