Department of Mathematics

The University of Western Ontario

The question motivating the first part of this talk is the following: What information can one deduce about ordinary (de Rham) cohomology of a manifold using the theory of equivariant cohomology, if the manifold admits a special type of Lie group action? The class of group actions we will consider is that of cohomogeneity one actions (i.e., those that admit an orbit of codimension one). Among other things, one can derive the following topological obstruction: if a compact manifold with positive Euler characteristic admits an action of cohomogeneity one, then all of its odd Betti numbers vanish. (A result that was previously shown by Grove and Halperin using rational homotopy theory.) In the second part we will go into a completely different geometric situation and show how one can use similar techniques to derive a link between the basic cohomology of certain Riemannian foliations and the number of closed leaves of the foliation. The main example here will be the Reeb foliation of a K-contact manifold. (The first part is a joint work with Augustin-Liviu Mare, and the second one with Hiraku Nozawa and Dirk Töben)

It is a classical result that groupoids model homotopy 1-types, in the sense that there is an equivalence between the homotopy categories, via the classifying space and fundamental groupoid functors. We extend this to stable homotopy 1-types and Picard groupoids. Using an algebraic description of Picard groupoids, we give a model for the Postnikov invariant of a stable 1-type and describe the action of the truncated sphere spectrum in these terms. We relate this data to exact sequences of Picard groupoids developed by Vitale, constructing a model for the homotopy cofiber of a map of stable 1-types. Joint with Angélica Osorno.

We given an elementary proof of Waldhausen Additivity using key ideas from earlier proofs. Then we discuss how to prove the quasicategorical version. Model category arguments do not play a role, nor do any technical results about quasicategories. This is joint work with David Gepner and Wolfgang Lueck.

While a symplectic quotient coming from a Hamiltonian action of a compact Lie group is generally not a manifold (it is a stratified space), one can still define a notion of differential form on it. Indeed, one can obtain a de Rham Theorem, Poincaré Lemma, and a version of Stokes' Theorem using this de Rham complex of forms. I will show how these forms are defined, and then explore the question of intrinsicality of the complex. This question leads into a discussion of different definitions of a smooth structure on the quotient, and the pros and cons of each

Manifold calculus of functors was introduced and developed by T. Goodwillie and M. Weiss in order to study spaces of embeddings. In a few words the goal of their method is to understand how from the spaces Emb(U,N) of smaller open subsets U of M we can describe the space Emb(M,N) of embeddings of the entire manifold M into N. Naively it is sometimes called "patching method". I will describe briefly the ideas of this theory and also explain some recent advances which gives a connection with the theory of operads.

Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. In this talk we will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.

Homology is easy to compute, thanks to excision, but it isn't very sensitive. It only detects homotopy types. In this talk I'd like to give one answer to the question: Is there a notion of homology theory for manifolds that's sensitive to more? I will present the definition of factorization homology, which Lurie has also called topological chiral homology. Factorization homology generalizes usual Eilenberg-Steenrod homology, and is and invariant of manifolds and stratifications on them. The main result will be a classification of all homology theories, namely by giving an equivalence between the category of homology theories and the category of certain kinds of algebras. I will explain how the theorem in turn gives candidates for new sources of invariants of embedding spaces (and in particular, link invariants). If time allows, I can discuss connections to topological field theories and to Koszul duality. This is joint work with David Ayala and John Francis.

Given a graded Hopf algebra $A$, one wants to compute the stable representation ring $Stab(A)$. By work of Margolis, computing all possible Adams spectral sequence $E_2$-terms for finite module spectra over certain commutative ring spectra amounts to computing the cohomology of A with coefficients in each generator for Stab(A), when is a subalgebra of the Steenrod algebra. However, actually computing $Stab(A)$ is (in Margolis' words) "a very difficult problem in general." In this talk we describe this relationship between Stab(A) and Adams spectral sequences, and we describe a new approach to the computation of Stab(A) which uses a twisted version of the deformation theory of modules. While untwisted first-order deformations of an A-module M are classified by the Hochschild cohomology group $HH^1(A, End(M))$, our twisted deformations instead are classified by a nonabelian (that is, with coefficients in a nonsymmetric module) version of the "higher-order Hochschild cohomology" of Pirashvili. We discuss existence and uniqueness results for these nonabelian higher-order Hochschild cohomologies, and the relative difficulty of actually making these computations (in particular, when they do and do not run up against of the unsolvability of the word problem!).

Ryan Budney recently constructed an operad that encodes splicing of knots and extends his little 2-cubes action on the space of (long) knots. He further showed that the space of knots is freely generated over the splicing operad by the subspace of torus and hyperbolic knots. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we construct a colored operad that encodes this infection operation.

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l-adic cohomology groups of the quiver moduli space is strongly pure. This is joint work with Michel Brion.

Dendroidal simplicial sets satisfying an analogue of the Segal condition are a model for ($\infty$, 1)-colored operads, as shown by Cisinski and Moerdijk. We consider weak group actions on such a ``Segal operad'' and prove a rigidification theorem. This is joint work with Julie Bergner.

Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, there are no non-zero operations on the $E_2$ page of the spectral sequence. Despite the negative results we are able to use old computations of Steinbergers with our current work to compute operations in the homotopy of some relative smash products.

Operads, and the more general multicategories, are combinatorial devices originally used in algebraic topology as a ``bookkeeping'' devices that described the internal operations of iterated loop spaces. The basic idea of an operad, however, is quite flexible and can be adapted to problems in algebra, mathematical physics, and computer science. The goal of this talk is to give a quick introduction to the Grothendieck-Teichm\"{u}ller group, as introduced by Drinfeld and Ihara, describe some of the conjectures relating this group to quantized deformations, and explain how this conjecture is being understood through the machinery of operads (up to homotopy).

In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss, yielding new computations of algebraic K-theory groups. Techniques from equivariant stable homotopy theory are often key to algebraic K-theory computations. In this case we use n-cubes of cyclotomic spectra to compute the topological cyclic homology, and hence K-theory, of truncated polynomial algebras in several variables.

We discuss the equivalence of perfect obstruction theories (extensively used in formation of Gromov-Witten invariants) and quasi-smooth derived schemes. The latter can be thought of as a derived zero locus. When the zero locus is locally given by a complete intersection, one gets a classical scheme.

I will describe some categories of "smooth spaces" which generalize the notion of manifold. The generalizations allow us to form smooth spaces consisting of subsets and quotients of manifolds, as well as loop spaces and other function spaces. In more technical language, these categories of smooth spaces are complete, cocomplete and cartesian closed. I will give examples, discuss possible applications and explain what we have learned about the homotopy theory of these categories. This is work in progress with Enxin Wu.

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds. In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description. This is joint work with Chris Allday and Volker Puppe.

In 1995, de Concini and Procesi investigated certain iterative blowups of affine space along intersections of linear subspaces, their wonderful models, a fundamental example being the Fulton-Macpherson configuration space compactification. In doing so, they developed suitable combinatorics to describe, among other things, the cohomology of the wonderful models. In 2006, Feichtner and Yuzvinsky constructed smooth toric varieties from de Concini and Procesi's combinatorial data, and found that, for any arrangement of hyperplanes, the cohomology ring of the de Concini-Procesi wonderful model is isomorphic to the Chow ring of their toric variety. Their argument is indirect, via the combinatorics defining the rings in question. I will outline a toric construction of de Concini and Procesi's wonderful models for hyperplane arrangements. This is an example of Tevelev's notion of a tropical compactification. One advantage is that it provides a geometric explanation of Feichtner and Yuzvinsky's isomorphism.

We define a classifying stack for the bounded derived category associated to any scheme X. When X is projective, we show that this stack is locally geometric, i.e., we can treat it as a slight abstraction of a scheme. We will also provide some applications of this result.

Quillen's derived functor notion of homology provides interesting and useful invariants in a variety of homotopical contexts, and includes as special cases (i) singular homology of spaces, (ii) homology of groups, and (iii) Andre-Quillen homology of commutative rings. Working in the topological context of symmetric spectra, we study topological Quillen homology of commutative ring spectra, E_n ring spectra, and more generally, algebras over any operad O in spectra. Using a QH-completion construction---analogous to the Bousfield-Kan R-completion of spaces---we prove under appropriate conditions (a) strong convergence of the associated homotopy spectral sequence, and (b) that connected O-algebras are QH-complete---thus recovering the O-algebra from its topological Quillen homology plus extra structure. A key problem in usefully describing this extra structure was solved recently using homotopical ideas in joint work with Kathryn Hess that describes a rigidification of the derived comonad that coacts on the object underlying topological Quillen homology, and plays the analogous role (in symmetric spectra) of the Koszul cooperad associated to a Koszul operad in chain complexes. This talk is an introduction to these results with an emphasis on proving (a) and (b) which is joint work with Michael Ching.

When a symplectic manifold M carries a Hamiltonian torus R action, the injectivity theorem states that the R-equivariant cohomology of M is a subring of the one of the fixed points and the GKM theorem allows us to compute this subring by only using the data of 1-dimensional orbits. The results in the first part of this talk are a generalization of this technique to Hamiltonian R actions on orbifolds and an application to the computation of the equivariant cohomology of compact toric orbifolds. In the second part, we will introduce the equivariant Chen-Ruan cohomology ring which is a symplectic invariant of the action on the orbifold and explain the injectivity/GKM theorem for this ring.

Several models for $(\infty, 1)$-categories have been defined and shown to be equivalent, and they are all being used in different areas of algebra and topology. More recently, there has been interest in more general $(\infty, n)$-categories, especially with Lurie's recent work on the Cobordism Hypothesis. Comparison of different definitions is still work in progress by several authors. In this talk, we will go over some of the models for $(\infty, 1)$-categories and discuss some of the methods for inductively generalizing them to models for $(\infty, n)$-categories.

In the 1980's (at UWO) I gave a local construction of the Deligne-Langlands epsilon factors attached to representations of Galois groups of local field extensions. The method was to resolve an arbitrary representation by monomial representations, for which the construction was straightforward. At the time my idea was to attack the Langlands programme by making a similar resolution of an arbitrary admissible representation of $GL_{n}K$ where $K$ is a local field. Returning to this with a bit more knowledge, I now more or less have the correct definition and the outline of the construction to the extent that I can handle $GL_{2}K$! The entire Langlands programme has many features which were suggested by properties of representations of finite groups such as $GL_{n}{\mathbb F}_{q}$. So I shall spend a lot of the time illustrating the constructions and properties in the case of finite groups - looking at some or all of: (i) Weil representations, (ii) cuspidality and monomial resolutions, (iii) local L-functions and (iv) Shintani descent.

Let $p$ be a prime. A 1989 theorem of Ossa calculates the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$ and purports to calculate the corresponding orthogonal connective K-theory when $p=2$. Sadly the latter is wildly wrong! Using a simple K\"{u}nneth formula short exact sequence I shall derive Ossa's unitary connective K-theory result in an elementary manner. As a corollary, I shall derive the correct version of Ossa's orthogonal theorem. As an application of this result I shall show that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ gives a stable homotopy element having Arf-Kervaire invariant one.

We will recall Voevodsky's definition of the presheaves of rigid motivic homotopy groups and show that they admit a canonical structure of presheaves with transfers when we consider rational coefficients and quasi-excellent base schemes.

In joint work with John Rognes, we show how to transfer the technologies and results of Quillen and Waldhausen in higher algebraic $K$-theory to the context of $\infty$-categories. Analogues of the $S_{\bullet}$ and $Q$-constructions — as well as versions of the additivity, localization, and d evissage theorems — are among the results we find in this new context. As a motivation for this work, we discuss a conjecture of Hopkins, Waldhausen, and Rognes on the algebraic $K$-theory of $BP\langle n\rangle$.

Equivariant spectra have received a good deal of attention lately due to their central role in the Hill-Hopklins-Ravenel proof of the Kervaire invariant one problem. I will describe joint work with Peter May that provides an alternative model for equivariant spectra (indexed on a complete G-universe).

A Gerstenhaber algebra is a special kind of graded Poisson algebra. A homotopy Gerstenhaber algebra is a specific "up to homotopy" version of the former. Important examples of homotopy Gerstenhaber algebras are the Hochschild cochains of an associative algebra and the cochain complex of a simplicial set. In this talk I will address the following problem: What structure does the tensor product of two homotopy Gerstenhaber algebras have? If time permits, I will also talk about formality results for homotopy Gerstenhaber algebras.

(joint with Ausoni, Baas, Richter and Rognes) Two vector bundles give rise to a geometrically defined cohomology theory extrapolating past the theory of vector bundles (K-theory) and differential forms (de Rham cohomology), capturing information related to cobordisms of manifolds beyond K-theory and deRham cohomology's reach. The analytic and differential geometric understanding of two vector bundles is still very much in its infancy. There was a hope that an "integration of determinants through loops" construction would give an integral functor from two vector bundles to quantum field theories. However, the fact that the commutative ring spectrum representing complex K-theory does not support a determinant rules this out. The first obstruction has a geometric interpretation: the one-dimensional two vector bundle represented by the Dirac monopole over the three sphere splits virtually.

Topological spaces - such as classifying spaces of small categories and spacetimes - often admit extra temporal structure. Such "directed spaces" often arise as geometric realizations of simplicial sets and cubical sets; the temporal structure encodes orientations of simplices and 1-cubes. Directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Nevertheless, we present simplicial and cubical approximation theorems for a homotopy theory of directed spaces. In our directed setting, ordinal subdivision plays the role of barycentric subdivision and cubical sets equipped with coherent compositions of higher cubes serve as analogues of Kan complexes. We consequently show that geometric realization induces an equivalence between certain weak homotopy diagram categories of cubical sets and directed spaces. As applications, we show that directed analogues of homotopy groups of spheres are uninteresting, sketch constructions of a (more interesting) cubical singular cohomology theory for directed spaces, and calculate such "directed cohomology" monoids for various directed spaces of interest.

Let $T$ be a compact torus. Goresky, Kottwitz and Macpherson showed that for a large and interesting class of $T$-equivariant projective varieties $X$, the equivariant cohomology ring $H_T^*(X)$ may be encoded in a graph, now called the GKM-graph, with vertices corresponding to the fixed points of $X$ and edges labeled by the weights, $Hom(T, U(1))$. In this lecture, we explain how the GKM construction can be generalized to any finite $T$-CW complex. This generalization gives rise to new mathematical objects: GKM-hypergraphs and GKM-sheaves. If time permits, we will show how these methods were used to resolve a conjecture concerning the moduli space of flat connections over a non-orientable surface.

In the chromatic take on stable homotopy theory, the homotopy type of a finite $p$-local spectrum $X$ is reassembled from its localizations with respect to the various Morava $K$-theories. In the early 1990s, Hopkins proprosed a brash conjecture for how the reassembly process works. I'll review the conjecture and the state of the art -- including a verfication of the conjecture at $p=3$ and chromatic level $2$, where the question is not simply algebraic and where there has been a proposed counterexample. This is joint work with Hans-Werner Henn.

We examine the equivariant birational linearisation problem for algebraic tori equipped with a finite group action. We also study bounds on degree of linearisability, a measure of the obstruction for such an algebraic torus to be linearisable. We connect these problems to earlier work with Vladimir Popov and Zinovy Reichstein on the classification of the simple algebraic groups which are Cayley and on determining bounds on the Cayley degree of an algebraic group, a measure of the obstruction for an algebraic group to be Cayley.

A projective toric scheme is specified by combinatorial data, viz., a polytope with integral vertex coordinates. I will show how the geometry of the polytope leads to a simple splitting result in the algebraic K-theory of the scheme. In the special case of projective space (given by a standard simplex) this reduces to the well-known splitting of K(P^n) into n+1 copies of the K-theory of the ground ring. - The combinatorial approach is flexible enough to include the case of schemes defined over an arbitrary (possibly non-commutative) ring.

A smooth proper variety with a zero cycle of degree one (that is, closed points of relatively prime degrees) need not have a rational point. In this talk we aim to explain how this phenomenon relates to the difference between unstable and stable $A^1$-homtopy theory.

The notion of a partial product space is a relatively recent unification of various combinatorial constructions in topology. This construction is variously known as the generalized moment-angle complex, or (more euphoniously) as the polyhedral product functor. Some instances of it are closely related to Davis and Januszkiewicz's quasitoric manifolds: these include the moment-angle complexes (Buchstaber and Panov) and homotopy orbit spaces for quasitoric manifolds. By making suitable choices, one also obtains classifying spaces for right-angled Artin groups and Coxeter groups, as well as certain real and complex subspace arrangements. One advantage to this generality is that some topological information about such spaces can sometimes be expressed directly in combinatorial terms: presentations of cohomology rings; a homotopy-theoretic decomposition of the suspension of a partial product space; descriptions of rational homotopy Lie algebras and the Pontryagin algebra. I will give an introductory overview of some remarkable results along these lines.

In 1983 Jeanne Duflot published some results showing that certain long exact sequences were, in fact, short exact. Her methods confined her to smooth actions. Recently, Matthias Franz, Volker Puppe and I have needed to reconsider Duflot's results, and, in the process, we arrived at new proofs that also work for continuous actions on topological manifolds. The main tool in the proof is a spectral sequence that is an equivariant version of Poincare - Alexander - Lefschetz duality. The proof also makes use of some basic properties of Cohen - Macaulay modules and a little bit of local cohomology.

Let k be a positive integer. A k-differential on a Riemann surface C is a section of the k-th tensor power of the canonical bundle of C. I will review what is known about the space of holomorphic k-differentials in the case when C is compact. I will state some new results for the case when C is non-compact.

From the topological fact that the circle is the representing space for the functor $X \to H1(X,\mathbb Z)$ follows that by computing degree 1 cohomology and picking cocycle representatives corresponds to computing equivalence classes of continuous maps $X\to S1$. In a research project with Vin de Silva and Dmitriy Morozov, we use this in a data analysis context to produce intrinsic coordinate functions with values on the circle. We shall discuss the derivation of circle-valued coordinates for point clouds using persistent cohomology to distinguish useful coordinate functions from functions appearing from noise in the data set, and discuss applications to the analysis of periodic signals and periodic dynamical systems.

A dynamical system is a map of spaces $X \times S \to X$, and a map of dynamical systems $X \to Y$ over $S$ is an $S$-equivariant map. There is both an injective and a projective model structure for this category. These model structures are special cases of injective and projective model structures for space-valued diagrams $X$ defined on a fixed category $A$ enriched in simplicial sets. Simultaneously varying the parameter category $A$ (or parameter space $S$) along with the diagrams $X$ up to weak equivalence is more interesting, and requires new model structures for $A$-diagrams having weak equivalences defined by homotopy colimits, as well as a generalization of Thomason's model structure for small categories to a model structure for simplicial categories.

In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad O will also provide interesting and useful invariants. Working in the context of symmetric spectra, we prove a Whitehead theorem for topological Quillen homology of algebras and modules over operads. This is part of a larger goal to attack the problem: how much of an O-algebra can be recovered from its topological Quillen homology? We also prove analogous results for algebras and modules over operads in unbounded chain complexes. This talk is an introduction to these results (joint with K. Hess) with an emphasis on several of the motivating ideas.

Operads, multicategories, and their representations (also called operadic/multicategorical algebras) play a key role in organizing hierarchies of higher homotopies in any category with a good notion of homotopy theory. In this talk we show how one can generalize work of Toen and Rezk to provide a description of the derived category of any multicategorical algebra. Time permitting, we will discuss applications of this theory to problems in combinatorial representation theory. We do not assume prior knowledge of the theory of operads and multicategories.

Classically, in algebraic geometry, descent for quasi-coherent sheaves is valid for maps between schemes satisfying certain flatness and finiteness conditions. With the machinery developed in stable infinity categories, one can extend decent to the stable infinity category $QCoh(X)$ (whose homotopy category is the derived category of the scheme $X$) and to maps that are not necessarily flat. We will give some examples and discuss plausible conditions on maps to satisfy descent.