I will give a survey of the recently discovered connection between constructive logic and homotopy theory. This forms the basis of Voevodsky's Univalent Foundations program, a new approach to foundations with intrinsic geometric content and a computational implementation. Time permitting, I will explain the Univalence axiom.

An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Γ-space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Peter May and Mona Merling.

The Picard scheme Pic⁰ representing invertible sheaves can be compactified by a moduli space J-bar of rank 1, torsion-free sheaves called the compactified Jacobian. For a smooth algebraic curve X over a field k with boundary ∂X, applying H₁ to the Abel-Jacobi map X → Pic⁰ (X/ ∂X) gives the Poincaré duality isomorphism H₁(X, Z/l) → H¹_c(X, Z/l(1)) = H¹(X, ∂X, Z/l(1)). We show the analogous result for the compactified Jacobian that applying H₁ to the Abel-Jacobi map X/∂X → J-bar gives the Poincaré duality isomorphism H₁(X, ∂X, Z/l) → H¹(X, Z/l(1)). In particular, H₁(X/ ∂X → J-bar) is an isomorphism. This is joint work with Jesse Kass.

Quasi-categories (aka ∞-categories) are convenient models of categories weakly enriched in spaces. Analogs of the standard categorical theorems involving limits and colimits, adjunctions, equivalences, monads and so forth have been proven by Joyal, Lurie and others. The goal of this talk is to describe a new ground-level approach that allows for 'formal' re-proofs of these facts that requires only very mild model category prerequisites and hence generalizes. A highlight will be the construction and characterization of the quasi-category of algebras associated to a homotopy coherent monad. This is a progress report on ongoing joint work with Dominic Verity.

Given a fibration f: X → S of CW-complexes one can use Eilenberg obstruction theory to study the spaces of sections of f. These obstruction theory give rise to obstructions to the existence of a section lying in the groups H^s+1(S, π_s(F)) where F is the fibre of f. A topos is a generalization of the concept of topological space which is ubiquitous in algebraic geometry. In the talk I shall present joint work with I. Barnea generalizing Eilenberg obstruction theory for sections of maps of topoi f: X → S. If time permits I will describe applications to Galois theory of number fields.

For G a finite group with Sylow subgroup S, the conjugation action of G on the subgroups of S gives rise to the data of a saturated fusion system F_S(G) on S. On the other hand, S acts on G by left and right multiplication. The resulting (S,S)-biset _SG_S turns out to contain much of the same information as F_S(G), in that the biset determines the fusion system, but not conversely. These notions can be abstracted to make no reference to the ambient group G, resulting in an abstract saturated fusion system F on S and a characteristic biset Ω for F. Again, Ω determines F, but each F has many associated characteristic bisets. This talk will focus on the failure of a saturated fusion system to uniquely determine a characteristic biset. We will show that there is a parametrization of all characteristic bisets for a fixed F, which will have as a consequence the surprising result that each saturated fusion system has a unique minimal associated characteristic biset.

We combine three strategies to studying cooperations in connective topological modular forms: the Adams spectral sequence and its relation to Brown-Gitler modules following Mahowald's approach to cooperations in connective real K theory, Laures's theory of q-expansions of multi-variable modular forms, as well as level structure approximations. As a result, we obtain an algorithmic procedure for determining the structure of the smash product of tmf with itself. This is a report on joint work in progress with Behrens, Ormsby, and Stapleton.

The generalized character theory of Hopkins, Kuhn and Ravenel has proved to be a very useful tool in the study of Morava E_n. In this talk, I will outline a compact construction of the transchromatic generalized character maps. The Morava E-theory of cyclic groups and symmetric groups have well known algebro-geometric interpretations. Using the relationship between the character maps and the transfer maps for Morava E-theory, I will provide algebro-geometric interpretations of the cohomology of some finite groups other than symmetric groups and cyclic groups.

The motivic truncated Brown-Peterson spectra BP<n> interpolate between motivic cohomology (BP<0>), algebraic K-theory (BP<1>), and the motivic Brown-Peterson spectrum itself, a close relative of algebraic cobordism. We use the motivic Adams spectral sequence and global-to-local comparison maps to compute the BP<n>-homology of the rational numbers. Along the way, we prove a Hasse principle for the motivic BP<n> and deduce several classical and recent theorems about the algebraic K-theory of particular fields. This is joint work with Paul Arne Østvær.

I'm going to talk about connections between the geometry of a map and its homotopy type. Suppose we have a maps from the unit m-sphere to the unit n-sphere. We say that the k-dilation of the map is < L if each k-dimensional surface with k-dim volume V is mapped to an image with k-dim volume at most LV. Informally, if the k-dilation of a map is less than a small ε, it means that the map strongly shrinks each k-dimensional surface. Our main question is: can a map with very small k-dilation still be homotopically non-trivial? Here are the main results. If k > (m+1)/2, then there are homotopically non-trivial maps from S^m to S^m−1 with arbitrarily small k-dilation. But if k ≤ (m + 1)/2, then every homotopically non-trivial map from S^m to S^{m − 1} has k-dilation at least c(m) > 0.

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.

I will discuss recent joint work with Oscar Randal-Williams concerning the manifolds W_g²ⁿ obtained as the connected sum of g copies of Sⁿ ×Sⁿ. For n=1 this is a genus g surface, and there is a moduli space M_g parametrizing smooth surface bundles with genus g fibers. For higher n there is an analogous moduli space M_gⁿ parametrizing smooth fiber bundles with fibers W_g (although for n > 1 it is no longer finite dimensional). We prove that for n > 2 the cohomology groups H^k(M_gⁿ) are independent of g as long as g >> k, generalizing a result of John Harer and others for n=1.

It was shown by Greenlees-Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality. This duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks. In this talk I will define differentiable stacks and explain this construction. I will also discuss the generalization of the above result to a more general class of stacks and give some examples.

A multicategory, also known as a colored operad, is simply a generalized non-commutative algebra. In this talk we focus on studying maps between multicategories enriched in simplicial sets. We show that the homotopy function complex of maps between any two multicategories can be computed as the moduli space of a certain small category of (operatic) bimodules. As an application, we show how this description leads to several important decompositions which allow one to compute various geometric invariants.

Budney recently constructed an operad that encodes splicing of knots and extends his little 2-cubes operad action on the space of (long) knots. He further decomposed the space of knots as the space 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 have constructed a colored operad that encodes this infection operation. This suggests looking for other ways to decomposes spaces of knots and links, which is a main direction of our work in progress.

I will report on a work in progress in collaboration with Pantev-Vaquié-Vezzosi. The central notion will be of a shifted symplectic structure on a derived scheme or derived stack, which I will explain in details. The main result is an existence statement of shifted symplectic structures on derived mapping spaces toward a symplectic target, that will be used to construct many examples (moduli of sheaves on CY varieties, moduli of representation of π₁ of a compact oriented manifold, Lagrangian intersections ...). Finally, I will explain what quantization means in this context as well the general strategy to prove existence of canonical quantizations.

Symonds (2010) showed that the cohomology ring of a finite group G with a faithful complex representation of dimension n is generated by elements of degree at most n². This was a remarkable advance, since no bound was known before. Symonds's proof combined equivariant cohomology with commutative algebra (Castelnuovo-Mumford regularity). We give better bounds for the cohomology ring of a p-group. The methods also apply to the Chow ring of algebraic cycles on BG.

Many proposed higher categories come from geometric situations. This talk will demonstrate a constructive connection between a homotopy theory of local invariants of n-manifolds and that of weak n-categories in the sense of Rezk. Connections to specific topological field theories will be discussed. This is a report on joint work with Nick Rozenblyum.

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. 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.

I will discuss topological Hermitian cobordism, an RO(G)-graded Z/2 x Z/2- spectrum constructed in a joint paper with Igor Kriz. I will also talk about the method for calculating its RO(G)-graded coefficients completely.

I plan to discuss my recent joint work with Po Hu and Daniel Kriz on a lifting of Khovanov homology to connective k-theory. I will also talk about its relation to recent work by Lipshitz and Sarkar. As a technical tool, our approach exhibits a curious link between modular functors and the Elmendorf-Mandell approach to multiplicative infinite loop space theory.

The motivic perspective on algebraic K-theory is a useful means of constructing trace maps as well as other additive or localizing invariants which are often easier to compute. For instance, a homotopy invariant form of K-theory is obtained by forming the A¹-localization of noncommutative motives before passing to homotopy groups. In order to relate this to the usual notion of homotopy K-theory it is useful to have a direct construction of this localization, and this involves understanding the additive group A¹ in the noncommutative context. This is joint work with Andrew Blumberg and Goncalo Tabuada.

The aim of this talk is to discuss the homotopy coherence properties of adjunctions between quasi-categories. Taking as our lead the theory of the "walking adjunction" A of 2-category theory, we generalise to categories enriched in quasicategories and show that this same 2-category plays a similar role in this new context. Specifically, using insights drawn from the calculus of string diagrams we give an explicit presentation of A as a simplicially enriched category. We then use this to show that if C is any quasicategory enriched category and u is a right adjont 0-arrow in C, in some suitable sense to be discussed, then this data may be completed to give a simplicially enriched functor A->C. Furthermore, we show that the space of all such exensions is contractible. That adjunctions of quasicategories may be completed up to enriched functors on A in this way contains, in its very essence, the adjunction data discussed by Jacob Lurie. Such enriched functors encapsulate both the coherent monad and the coherent comonad generated by such an adjunction and provide the building blocks upon which to found a formal theory of such things along the lines established by Street in the 2-categorical context.

The filtration on the infinite symmetric product of spheres by number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. In this talk I will discuss the equivariant stable homotopy types, for finite groups, obtained from this filtration for the infinite symmetric product of representation spheres. The filtration is more complicated than in the non-equivariant case, and already on the zeroth homotopy groups an interesting filtration of the augmentation ideal of the Burnside ring functor arises. Our method is by `global' homotopy theory, i.e., we study the simultaneous behaviour for all finite groups at once. The equivariant subquotients are no longer rationally trivial, nor even concentrated in dimension 0. 2012/02/13

Algebraic model categories are a variant of Quillen's classical notion in which the (co)fibrations are equipped with extra structure witnessing their defining lifting properties. Many ordinary model categories admit this extra structure, giving rise to a plethora of examples. In this talk we present several theorems illustrating various features of this theory. In particular we focus on a series of results that guarantee the existence of algebraic Quillen adjunctions and algebraic monoidal model structures just when particular cofibrations are cellular: eg relative cell complexes not mere retracts of such. On account of these results the algebraic theory places great emphasis on a distinction that is also present in expository accounts of the classical theory where its role is less transparent.

The algebraic K-theory of the complex cobordism spectrum is an object of basic interest, both because it provides an interesting example of K-theory of a non-classical ring and because it should shed light on K(S). There is reason to believe that K(MU) should be approachable via trace methods, which focuses attention on understanding THH(MU) and TC(MU). This talk describes work in progress to describe the equivariant homotopy type of THH of a Thom spectrum as an equivariant Thom spectrum. The ingredients for this description include the Hill-Hopkins-Ravenel norm and a modernized view of equivariant infinite loop space theory. This is joint work with Angeltveit, Gerhardt, and Hill.

Factorization homology, or the topological chiral homology of Lurie, is a homology theory for manifolds conceived as a topological analogue of the chiral homology of Beilinson and Drinfeld. I'll describe an axiomatic characterization of factorization homology, generalizing the Eilenberg-Steenrod axioms for usual homology. The use of excision for factorization homology facilitates a short proof of the nonabelian Poincare duality of Salvatore and Lurie; this proof generalizes to give a nonabelian Poincare duality for stratified manifolds, joint work with David Ayala and Hiro Tanaka. Work in progress with Kevin Costello aims to express quantum invariants of knots and 3-manifolds in factorization terms, which, time permitting, I'll outline. 2011/11/21

The Milnor conjecture identifies the cohomology ring H^*(Gal([ˉk]/k) \mathbbZ/2) with the tensor algebra of k^* mod the ideal generated by x ⊗1−x for x in k − {0,1} mod 2. In particular x ∪1−x vanishes where x in k^* is identified with an element of H¹. We show that order n Massey products of n−1 factors of x and one factor of 1−x vanish by embedding \mathbbP¹ − {0,1,∞} into its Picard variety and constructing Gal([ˉk]/k)-equivariant maps from π₁^et applied to this embedding to unipotent matrix groups. This also identifies Massey products of the form 〈1−x, x ... x, 1−x〉 with f ∪1−x where f is a certain cohomology class which arises in the description of the action of Gal([ˉk]/k) on π₁^et(\mathbbP¹ − {0,1,∞}).

The K(2)-localization of the sphere spectrum admits a conjectural small resolution built from TMF and "TMF with level structures" - the evaluation of the TMF sheaf on the stack of elliptic curves equipped with an order l subgroup. In this talk, I will use variations on Tate normal form to describe several Hopf algebroids that stackify to elliptic curves with level structure. These Hopf algebroids lead to computations of the Behrens-Lawson spectrum Q(l). This is current work with Mark Behrens, Nat Stapleton, and Vesna Stojanoska.

In the 80's Hopkins, Kuhn, and Ravenel developed a way to study cohomology rings of the form E^*(BG) in terms of a character map. Their map can be interpreted as a map of cohomology theories beginning with a height n cohomology theory E and landing in a height 0 cohomology theory with a rational algebra of coefficients that they construct out of E. In this talk we will use the language of p-divisible groups to discuss various ways of generalizing their map to every height between 0 and n.

In this talk I will start by describing what Hamiltonian Floer homology is and how it relates to 1-periodic orbits of a Hamiltonian flow. Then I will consider the case \mathbbR²ⁿ and describe how finite dimensional approximations lead to considering periodic cobordism theories (complex and real) as "coefficient rings". I will then in the more general case of a Liouville domain sketch how to define a spectrum-module over these coefficient rings with a set of generators in 1-1 correspondence with periodic orbits. 2011/10/10

In this talk we report on joint work with Clark Barwick. We give a short list of axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of (\inftyn)-categories. We show that the space of such quasicategories is homotopy equivalent to B(\mathbbZ/2)ⁿ generalizing a theorem of Toen when n=1 and verifying two conjectures of Simpson. In particular any two such quasicategories are equivalent. We also provide a large class of examples satisfying our axioms including those of Joyal, Kan, Lurie, Simpson and Rezk. 2011/09/26

It has been observed that certain localizations of the spectrum of topological modular forms tmf are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M_ell yet is only true in the derived setting. When p is inverted, choice of level-p-structure for an elliptic curve provides a geometrically well-behaved cover of M_ell, which allows one to consider tmf as the homotopy fixed points of tmf(p), topological modular forms with level-p-structure, under a natural action by GL₂(\mathbbZ/p). Specializing to p=2 or p=3 we obtain that as a result of Grothendieck-Serre duality, tmf(p) is self dual. The vanishing of the associated Tate spectra then makes tmf itself Anderson self-dual.

Fusion systems are an algebraic description of the p-local structure of a finite group. A centric linking system is the algebraic data needed to construct a topological classifying space for a fusion system; recent work of Andy Chermak shows that in fact, every saturated fusion system has a unique associated centric linking system. The collection of these data is what is known as a p-local finite group. In this talk I will try to give some insight about the structure of p-local finite groups. I will also give a brief discussion of the more topological notion of retractive transfer triple which appears to be equivalent to the data of a p-local finite group.

Let f: X → R be a Morse function on a manifold X and v its gradient-like vector field. Classically, the topology of a closed X can be described in terms of the spaces of v-trajectories that link the singular points of f. On manifolds with boundary, the situation is somewhat different: there, a massive set of nonsingular functions is available. For such Morse data (f, v), the interactions of the gradient flow with the boundary dX take central stage. We will introduce and measure the convexity and concavity of a v-flow relative to dX. E€ESome manifolds are intrinsically more concave than others with respect to any gradient flowE€E is the main slogan of the talk. Stated differently, the intrinsic concavity of X is a reflection of its complexity. We will explain how this approach leads to new topological invariants, both of the flow v and of the manifold X. In 3D, we have a good grasp of these invariants and their connection to the classification of 3-folds.

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.

Greg Arone and I have been trying to understand the structure that exists on the derivatives of a functor in the sense of Goodwillie calculus. Previously we have shown that these derivatives possess the structure of a coalgebra over a certain comonad on the category of symmetric sequences. In this talk I'll try to describe in more depth what this structure amounts to for functors from based spaces to spectra. Specifically I'll relate these coalgebras to right modules over the (Koszul duals of) the little disc operads. This is all joint work with Greg with substantial input also from Bill Dwyer.

Abstract: The L(k) spectra arise in a variety of ways in homotopy theory: the Whitehead conjecture, splittings of classifying spaces, and as the layers of the Goodwillie tower of the identity evaluated on S¹. I will give a description of the Morava E-homology of these spectra in terms the Morava E-theory Dyer-Lashof algebra, and the modular isogeny complex. This is joint with Charles Rezk.

Some vanishing results associated with the Costello construction of TCFT's on Hochschild homology.

For an algebraic group G and a projective curve X, we study the category of D-modules on the moduli space Bun_G of principal G-bundles on X using ideas from conformal field theory. We describe this category in terms of the action of infinitesimal Hecke functors on the category of quasi-coherent sheaves on Bun_G. This family of functors, parametrized by the Ran space of X, act by averaging a quasi-coherent sheaf over infinitesimal modifications of G-bundles at prescribed points of X. We show that sheaves which are, in a certain sense, equivariant with respect to infinitesimal Hecke functors are exactly D-modules, i.e. quasi-coherent sheaves with a flat connection. This gives a description of flat connections on a quasi-coherent sheaf on Bun_G which is local on the Ran space.

We will discuss connectedness in unstable A¹-homotopy theory, focusing on some examples. We will also discuss the classification of low dimensional A¹-connected smooth proper varieties and its some similarities to and differences from the corresponding topological classifications.

We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces RPⁿ for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. We also present several questions about tmf.

We prove a conjecture of Kontsevich which states that if A is an E_d−1 algebra then the Hochschild cohomology object of A is the universal E_d algebra acting on A. The notion of an E_d algebra acting on an E_d−1 algebra was defined by Kontsevich using the swiss cheese operad of Voronov. We prove a homotopical property of the swiss cheese operad from which the conjecture follows.

In this talk I will describe joint work with Vigleik Angeltveit, Mike Hill, and Ayelet Lindenstrauss yielding new computations of algebraic K-theory groups. In particular we consider the K-theory of truncated polynomial algebras in several variables. 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 the rings in question.

Picard and Brauer groupoids, which capture information about invertible modules and central simple algebras, are objects of classical interest in algebra and number theory. The calculation of these objects is often aided by the use of Galois cohomology. We will discuss the generalizations to ring spectra, due to Hopkins-Mahowald-Sadofsky and Baker-Richter respectively. We then discuss how to compute these using Galois cohomology in higher category theory. (This talk based on joint work with David Gepner.)

I'll talk about joint work with Greg Arone to describe the data needed to reconstruct the Taylor tower of a functor from its layers. That data consists of a bimodule over the derivatives of the relevant identity functors together with a coaction of a particular cotriple on the category of bimodules. I'll describe what we know about this cotriple for functors of based spaces and/or spectra.

We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups - this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology and use this to define Quillen homology completion - in the sense of Bousfield-Kan - for algebras over operads in symmetric spectra. We prove that under appropriate connectivity conditions the coaugmentation into Quillen homology completion is a weak equivalence - in particular such algebras over operads can be recovered from their topological Quillen homology. Many of the results described are joint with K. Hess.

This is joint work with Guillermo Cortinas, Christian Hassemeyer and Chuck Weibel. Let A be a commutative monoid and k a field. What part of the algebraic K-theory of the monoid-ring k[A] comes from just the monoid and is independent of the field k? I describe a partial answer to this question one which involves topological cyclic homology toric varieties and Voevodsky's cdh topology. I will also explain how our answer leads to a proof of Gubeladze's "nilpotence" conjecture for the algebraic K-theory of toric varieties in arbitrary characteristic.

Bott and Taubes considered a bundle over the space of knots whose fiber is a compactified configuration space and they constructed knot invariants by performing integration along the fiber of this bundle. Their method was subsequently used to construct real cohomology classes in spaces of knots in Rⁿ n > 3. Replacing integration of differential forms by a Pontrjagin-Thom construction I have constructed cohomology classes with arbitrary coefficients. Motivated by work of Budney and F. Cohen on the homology of the space of long knots in R³ I have also proven a product formula for these classes with respect to connect-sum. If time permits I will mention some progress towards further understanding these classes using a cosimplicial model for knot spaces coming from the Goodwillie-Weiss embedding calculus.

I will propose a simple and combinatorial E_n-operad which is built out of finite posets indexing a stratification of configuration spaces of points in an n-disk. This poset is constructed from a category θ_n which has recently become an important player for modeling weak n-categories (Joyal, Berger, Rezk). The techniques involved use the formalism of quasi-categories (Lurie). This project is joint with Richard Hepworth (Copenhagen) and is a work in progress. One (nearly achieved) goal is to directly and geometrically relate three well-developed methods for recognizing n-fold loop spaces: as certain algebras over the little n-disk operad as certain algebras over the Barratt-Eccles E_n operad (via the Smith filtration) and as certain presheaves on θ_n (Berger). A version of Dunn's additivity theorem becomes a formal consequence of the setup. A farther away goal is to imitate the construction of topological chiral homology using this proposed E_n-operad. This should have the benefit of making topological chiral homology (and possibly other field theories) more prepared for computations.

I will describe recent work with Dave Benson and Henning Krause wherein we classify colocalizing subcategories of StMod(kG) for a finite group G in the spirit of earlier work by Hopkins Neeman and Benson Carlson and Rickard on thick subcategories and localizing subcategories in various contexts. A central result in our work is a criteria for the vanishing of a function object Hom_k(M N) in StMod(kG) in terms of geometric data (to be precise: support and cosupport) associated to M and N via local cohomology and completion functors on the stable module category.

We show a 2-nilpotent section conjecture over R: for a smooth curve X over R with negative Euler characteristic π₀(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action showing a 2-nilpotent birational real section conjecture.

The Lefschetz fixed point theorem gives a sufficient condition for a continuous endomorphism to have a fixed point: If the Lefschetz number of a continuous endomorphism of a closed smooth manifold is nonzero that endomorphism has a fixed point. Usually no conclusions can be drawn if the Lefschetz number is zero but with some (restrictive) hypotheses there is a converse. I will describe an approach to the converse of the Lefschetz fixed point theorem using traces that also gives converses to the equivariant and fiverwise Lefschetz fixed point theorems.

I will discuss the (motivic or A¹) homotopy theory of G-equivariant schemes G a finite group. Stabilizing with respect to regular representation of spheres produces a good stable theory which in the case G=Z/2 contains motivic analogues of Atiyah's Real K-theory and Araki's Real cobordism over arbitrary characteristic 0 base fields. The algebraic Real K-theory spectrum is closely related to Hermitian K-theory (a.k.a. Higher Grothendieck-Witt theory). Tools from stable equivariant topology like the Tate diagram and slice spectral sequence allow us to resolve the completion (or homotopy limit) problem for the Hermitian K-theory of fields.

Lie groupoids (and Lie algebroids) play an increasingly important role in foliation theory symplectic and Poisson geometry and non-commutative geometry. In this lecture I will explain how some basic properties of Lie groups extend to groupoids and how some other properties don't.

In this talk I will report on recent work joint with Christopher Douglas and Noah Snyder on understanding the nature of fully extended (a.k.a. Local) 3-dimensional topological quantum field theories. Specifically we show that fusion categories are fully-dualizable objects in the 3-category of tensor categories, a natural categorification of the bicategory of algebras bimodules and bimodule maps. Fusion categories themselves are well-known are arise in several areas of mathematics and physics - conformal field theory, operator algebras representation theory of quantum groups and others. In light of Hopkins and Lurie's work on the cobordism hypothesis this provides a fully local TQFT for arbitrary fusion categories. Moreover we will discuss how many familiar structures from the theory of fusion categories are given a natural explanation from this point of view.

In this talk I will give a brief survey of what I call "categorical informatics which is the study of information and communication from a category-theoretic perspective. In order to begin such a study one must ground it in something concrete. To that end I'll explain a simple categorical model of databases which are real-world store-houses of organized information. I will then discuss information transfer between databases and move on to define a more general notion of communication networks. Briefly, a communication network is a simplicial complex (of interacting groups) equipped with a sheaf of databases (common languages) on its simplices. I will show how this same structure can be pared down to give a categorical model for combining information obtained from various sources. I'll end by showing some interesting consequences of the topological nature of this model.

The first part of this talk is basic definitions and statement of the Nearby Lagrangian Conjecture (NLC). I will then go on describing previous results relating to my talk especially that of Viterbo in 1998 and that of Fukaya Seidel and Smith in 2007 (refined this year by Abouzaid to a very strong result in the case of vanishing Maslov class). I will then describe an approach using fibered spectra which partly unifies the two rather different approaches and then state new results in the general case (non-vanishing Maslov class) following from a product structure on a fibered spectrum combined with the intersection product on the base manifold (probably relatable to the pair of pants product and a twisted version of the Chas-Sullivan product).

The computer software package Sage (www.sagemath.org) understands some basic constructions from algebraic topology: it understands simplicial complexes, cubical complexes, and it seems to be the only major mathematical software package with an implementation of ∆-complexes. It can perform basic operations like joins, products, and connected sums. It can compute homology and cohomology over the integers or over a field. In this talk, we will discuss and demonstrate some of these capabilities, present some related unsolved problems, and discuss future directions.

A Π-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object A, one may ask if it can be realized topologically: Is there a space X such that π_*X is isomorphic to A as a Π-algebra, and if so, can we classify them? Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a Π-algebra A where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of A. What do these groups look like and can we compute them? We will tackle this question from the algebraic side focusing on Quillen cohomology of truncated Π-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types and compare them to what is known from other approaches.

In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the K-theory of a bimonoidal category R, and show that it is equivalent to the algebraic K-theory space of the ring spectrum HR. In this thesis we show that K(R) is the group completion of the classifying space of the 2-category Mod_R of modules over R, and show that Mod_R is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a Γ-(2-category), which gives an infinite loop space structure on K(R). We show that the equivalence mentioned above is an equivalence of infinite loop spaces.

This is one of the main technical step in the proof of the Friedlander-Milnor conjecture, and uses a lot of "classical" topological arguments (lower central series, A¹-derived functors of the free Lie algebra functors, devissage of the Lie algebra functor in terms of other polynomial functors.

Fusion systems, an abstraction of the p-local structure of finite groups, lie in the intersection of algebraic topology and finite group theory. In this talk I detail a "theory of fusion systems with many objects," generalizing fusion system theory to imitate actions of a finite group on a finite set. This generalization will come in three stages: as a simple condition to be put on a finite p-group-set, as algebraic structure added to the fusion system, and finally as an additional layer of structure necessary to construct classifying spaces and do homotopy theory.

In this talk I will describe the construction of the category of non-commutative motives [1,2,3] in Drinfeld-Kontsevich's non-commutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher K-theory since Quillen's foundational work in the 70's. As an application, I will show how these results allow us to obtain for free the higher Chern character from K-theory to cyclic homology. References:

[1] D.-C. Cisinski and G. Tabuada, Symmetric monoidal structure on Non-commutative motives. Available at arXiv:1001.0228.

[2] D.-C. Cisinski and G. Tabuada, Non-connective K-theory via universal invariants. Available at arXiv:0903.3717.

[3] G. Tabuada, Higher K-theory via universal invariants. Duke Math. Journal,145 (2008),no.1, 121-206.

A Π-algebra is a sequence of groups, along with an action of the primary homotopy operations, which are indexed by homotopy groups of wedges of spheres. It is natural to view the homotopy groups of a space as an object in the category of Π-algebras. The v_n-periodic homotopy groups of a space are obtained by inverting non-nilpotent self maps of finite complexes. In this talk I will investigate v_n-periodic Π-algebras. 2010/03/08

Shimomura and Yabe computed the homotopy groups of the E(2)-local sphere at primes greater than or equal to 5. However, their computation is difficult to understand. I will describe a conceptual way to understand the answer, and the computation. The goal is to make the answer as concrete as the image of the J homomorphism. We'll see of this goal is achieved...

Homotopy n-types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than n. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of n-fold categories internal to groups, also called catⁿ-groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of catⁿ-groups is a weak equivalence. In this talk we introduce a new model of connected n-types through a subcategory of catⁿ-groups which we call weakly globular for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model and discuss the relevance of these structures for higher category theory.

Ando constructed power operations for the Lubin-Tate cohomology theories using the theory of finite subgroups of a formal group. Moreover, he was able to produce a necessary and sufficient condition for a complex orientation of these cohomology theories to be compatible with the power operations. This result concerns the stable homotopy category of spectra. However, the Lubin-Tate spectra of Morava are very rigid objects. Using ideas of Ando, Hopkins and Rezk, we can classify those orientations of complex K-Theory that are compatible with Ando's power operations, but on the point set level. In this talk, we will show the equivalence of these two descriptions for complex p-adic K-Theory. To achieve this goal, we use the language of Bernoulli numbers attached to a formal group law and their relationship with distributions on a p-adic Lie group. Many of these tools were developed by N. Katz and J. Tate.

Morava E-theory (the complex oriented cohomology theories associated to deformations of formal groups) are structured commutative ring spectra, and so support a well-behaved theory of power operations. We describe what is know about this theory, and we prove a conjecture of Ando, Hopkins, and Strickland, that the ring of power operations for such theories is Koszul.

Costenoble and Waner showed that grouplike equivariant E_∞- spaces model equivariant infinite loop spaces. Shimakawa gave an equivariant analog of Γ-spaces to model equivariant infinite loop spaces. We describe equivariant Γ spaces as defined by Shimakawa. We show that the categories of equivariant E_∞-spaces and equivariant Γ-spaces are Quillen equivalent with appropriate model categories. Following Segal's work, we give a construction of equivariant Γ- spaces (and hence of equivariant infinite loop spaces) from symmetric monoidal G-categories for finite group G.

The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will try to describe a filtration of the stable chromatic spectral sequence induced by the Hopf rings for the odd spheres. There are natural questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.

Lurie's theorem allows the functorial construction of E_∞ ring spectra associated to certain p-divisible groups. In this talk I will discuss three situations in which we can apply this and attempts to understand the computational results. The first is joint work with Behrens on the relationship between the moduli of elliptic curves and certain moduli of abelian surfaces with complex multiplication. The second is joint work with Hill on Shimura curves that parametrize "false elliptic curves", and in particular trying to obtain computations of the homotopy of the associated spectra without niceties such as q-expansions and Weierstrass equations. The third is on using Zink's work on displays to produce E_∞ ring spectra from purely algebraic input data, in the form of invertible matrices over Witt rings.

Grothendieck's anabelian conjectures say that hyperbolic algebraic curves over number fields should be K(π,1)'s in algebraic geometry. It follows that conjecturally the rational points on such a curve are the sections of étale π₁ of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. We use cohomological obstructions of Jordan Ellenberg coming from nilpotent approximations to the curve to study the sections of étale π₁ of the structure map. We will relate Ellenberg's obstructions to Massey products, and explicitly compute mod 2 versions of the first and second for P¹−{0,1,∞} over Q. Over R, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian.

The playing field of profinite homotopy theory is provided by the homotopy categories of profinite spaces and profinite spectra. A motivating application is the connection to algebraic geometry. For example the etale fundamental group and continuous etale cohomology of a scheme can be defined in a unified way using a profinite etale realization functor. We will discuss this functor and use it to define etale topological cobordism. But it turned out that profinite structures might be useful in other areas such as Lubin-Tate spectra. If time permits we will discuss this idea in progress as well.

This talk will motivate and develop a bordism category consisting of singular manifolds. Applications will be discussed having relevance to 'stable' characteristic classes of families of smooth manifolds and Gromov-Witten theory.

The 6-connected cover of Spin(n), known as the group String(n), has fascinating connections with both abstract homotopy theory (through String Bordism and TMF) and with quantum field theory (through the 2D SUSY non-linear sigma model). A better geometric understanding of String geometry has the potential to offer new interactions between these fields. Unfortunately all previous models of String(n) are infinite dimensional, making a thorough geometric understanding elusive. In this talk we will construct a finite dimensional model of String(n) as a higher categorical version of a group (known as a 2-group). In the process, we will "categorify" the classical notions of group cohomology and derived functor. In particular we will categorify Segal's topological group cohomology, thereby obtaining a classification of extensions of topological 2-groups.

In this talk we will show that the space of almost commuting elements in a compact Lie group splits after one suspension.

Given a smooth manifold M and two submanifolds A and B, their intersection need not be a smooth manifold. By Thom's transversality theorem, one can deform A to be transverse to B and take the intersection: the result, written A∩B, will be a smooth manifold. Moreover, if A and B are compact, then there is a cup product formula in cobordism, integral cohomology, etc. of the form [A]∪[B]=[AB], where [·] denotes the cohomology fundamental class. The problem is that AB is not unique, and there is no functorial way to choose transverse intersections for pairs of submanifolds. The goal of the theory of derived manifolds is to correct this defect. The category of derived manifolds contains the category of manifolds as a full subcategory, is closed under taking intersections of manifolds, and yet has enough structure that every compact derived manifold has a fundamental class. Even if the submanifolds A and B of M are not transverse (in which case their intersection can be arbitrarily singular), their intersection A×_MB will be a derived manifold with [A×_MB] = [AB], and thus satisfy the above cup product formula. To construct the category of derived manifolds, one imitates the constructions of schemes, but in a smooth and homotopical way. I will begin the talk by explaining this construction. Then I will give some examples and discuss some features of the category of derived manifolds. I will end by sketching the Thom-Pontrjagin argument which implies that compact derived manifolds have fundamental classes.

A second subspace of a product is the generalized moment-angle complex first defined in generality by Neil Strickland. Definitions, examples, as well as connections will be addressed. One notable case is given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of Goresky-MacPherson Davis-Januskiewicz Buchstaber-Panov-Ray Denham-Suciu Franz as well as many others encode information ranging from the structure of toric varieties in one guise Stanley-Reisner rings as well as 'motions of certain types of robotic legs' in another guise. What do these spaces have to do with the motions of legs of a cockroach? This feature will be illustrated with slides. Features of these spaces are developed within the context of classical homotopy theory based on joint work with A. Bahri M. Bendersky and S. Gitler.

The first basic example here is the configuration space of unordered k-tuples of distinct points in a space M. When specialized to the case of M given by the complex numbers, these spaces can be identified as the space of classical complex, monic polynomials of degree k which have exactly k distinct roots. Elementary features of these spaces as well as their connections to spaces of knots, links and homotopy groups of spheres will be addressed. These topics are based on joint work with R. Budney as well as J. Berrick, Y. Wong and J. Wu.

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations. These spaces as well as further spaces of representations admit the structure of a simplicial space at the heart of the work here. Features of geometric realizations will be developed. What is the fundamental group or the first homology group of the associated space in case G is a finite discrete group? This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres and J. Gomez.

I will describe the stable (in genus) structure of the universal moduli space of flat connections on riemann surfaces. I will also introduce the category of 1-manifolds and 2-cobordisms endowed with flat connections. Using classical techniques of Atiyah-Bott, and more recent techniques introduced by Madsen-Weiss and coauthors, we will give a complete description of the classifying space of this category. This is joint work with R. Cohen and S. Galatius.

Let C be a model category. In a 2001 paper, Dan Dugger showed that if C is combinatorial, it can be realized as a left Bousfield localization of simplicial presheaves on some small site. I'll describe a variation of this theorem: by replacing simplicial sets with a cubical model for the homotopy category, we can produce a presentation for C when C is symmetric monoidal that retains the monoidal structure of C as the Day convolution product.

For a given compact smooth manifold M we consider the space Emb(M,R^k) of smooth embeddings of M into some large Euclidean space R^k, or rather some geometric variant of it, which is a homotopy invariant of M. I will explain how Goodwillie's cutting method enables us to understand the homotopy type of this space of emeddings. I will then prove that the rational homology of that space is actually an invariant of the rational homotopy type of M. The proof is based on Kontsevich's theorem on the formality of the little cube operad and Arone's description of the layers of Weiss' orthogonal tower for the space of embeddings. This is a joint work with Greg Arone and Ismar Volic.

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 will also provide interesting and useful invariants. Working in the two contexts of symmetric spectra and unbounded chain complexes, we establish a homotopy theory for studying Quillen homology of modules and algebras over operads, and we show that this homology can be calculated using simplicial bar constructions. A key part of the argument is proving that the forgetful functor commutes with certain homotopy colimits. A larger goal is to determine the extra structure that appears on the derived homology and the extent to which the original object can be recovered from its homology when this extra structure is taken into account. This talk is an introduction to these results with an emphasis on several of the motivating ideas.

Let R be an E_∞ ring spectrum. Given a map f:X → BGL₁(R), we can construct a Thom spectrum Xf. If f is a loop map, then there is an A_∞ R module structure on the Thom spectrum. I will consider various examples of these Thom spectra and construct A_∞ structures on them. I will then use this identification to calculate Topological Hochschild Homology.

For smooth manifolds P, Q, and N, let Link(P,Q;N) denote the space of smooth maps of P in N and Q in N such that their images are disjoint. I will discuss the connectivity of a "generalized linking number" from the homotopy fiber of the inclusion of Link(P,Q;N) into Map(P,N)×Map(Q,N) to a certain cobordism space of manifolds over a space which is a homotopy theoretic model for the intersections of P and Q. The proof of the connectivity uses some easy statements about connectivities in the world of smooth manifolds as a guide for obtaining similar estimates in a setting where the tools of differential topolgy do not apply. This is joint work with Tom Goodwillie.

In this talk I will present recent work, joint with Radu Stancu, in which we obtain a bijection between saturated fusion systems on a finite p-group S and idempotents in the double Burnside ring of S satisfying a "Frobenius reciprocity relation". (These terms will all be defined in the talk.) The theorem and its proof are purely algebraic, so I will focus attention on implications in algebraic topology, answering long-standing questions on the stable splitting of classifying space and generalizing a variant of the Adams-Wilderson theorem, as well as the obvious implications for p-local finite groups.

I will discuss connections between the calculus of functors and the Whitehead Conjecture, both for the classical theorem of Kuhn and Priddy for symmetric powers of spheres and for the analogous conjecture in topological K-theory. It turns out that key constructions in Kuhn and Priddy's proof have bu-analogues, and there is a surprising connection to the stable rank filtration of algebraic K-theory.

I will describe some joint work with J.P. May in which we investigate when enriched model categories can be modeled as enriched diagrams on a small (enriched) domain category. As an application, we are able to obtain a new model for the equivariant stable homotopy category of a compact Lie group.

Carefully developing the homology and cohomology of ordered configuration spaces leads to a pretty model for the Lie cooperad. We use this model to unify the Quillen approach to rational homotopy theory with the theory of Hopf invariants. We will also share progress on a new approach to the cohomology of unordered configurations spaces (i.e. symmetric groups), which are of course relevant to homotopy theory at p.

The symmetric groups S_p are considered with the norm induced by the word length (with respect to transpositions as generators). This gives a filtration of their classifying spaces. Furthermore, using certain deletion functions S_p → S_p−1 the family of all symmetric groups can be regarded as filtered simplicial object. we show: in its realization, the stratum for norm equal to h has several components, each being homoemorhic to a vector bundle over the moduli space M_g,₁^m of genus g surfaces with one boundary curve and m punctures (for h = 3D 2g + m).

Let R be an associative ring spectrum. I shall describe several new constructions of the R-module Thom spectrum associated to a map f: X → BGL₁ R. The space BGL₁ R classifies the twists of R-theory, and to a fibration of manifolds g: Y → X I shall associated an Umkehr map g_! from the fg-twisted R-theory of Y to the f-twisted R- theory of X. In the case of K-theory, this twisted Umkehr map appears in the study of D-brane charge. I shall review this story, and then discuss the analogous construction for TMF.

In joint work with Keir Lockridge, we have been developing theories of global and weak dimensions for ring spectra. We have good results for ring spectra of dimension zero, and partial results but good conjectures for the finite dimensional case.

The cohomology jumping loci of a space X come in two basic flavors: the characteristic varieties (the jump loci for cohomology with coefficients in rank 1 local systems), and the resonance varieties (the jump loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X). I will discuss various ways in which the geometry of these varieties is related to the formality, quasi-projectivity, and homological finiteness propoerties of the fundamental group of X.

We will describe how (multivariable) manifold calculus of functors can be used for studying classical knots and links. In particular, this theory yields a classification of finite type invariants and Milnor invariants of knots, links, homotopy links, and braids. Another novelty is that a certain cosimplicial variant of manifold calculus provides a way for studying knots and links in a homotopy-theoretic framework. Higher-dimensional analogs will also be discussed. This is joint work with Brian Munson.

The moduli space of Riemann surfaces M is a classifying space for families of Riemann surfaces. It has a compactification [ˉM], which is a classfying space for families of modal Riemann surfaces. A nodal Riemann surface is allowed to have singularities which look like the solutions to zw=0 in complex 2-space. I will describe how to decompose [ˉM] as a homotopy colimit of spaces which look more like M. Then I will use this to study part of the homology of [ˉM], using what is known about the homology of M.

On every bimonoidal category with anti-involution, R, there is an involution on the associated K-theory. This K-theory is the algebraic K-theory of the spectrum associated to R. In the talk I will construct this involution, discuss examples and indicate why the involution is non-trivial in several examples.