Burt Totaro

Obstruction theory in algebraic geometry

Given a curve on a smooth projective surface, when does it move in a family of disjoint curves? The self-intersection number of the curve must be zero, but there are also higher-order obstructions. We give an obstruction theory in the spirit of homotopy theory to analyze the infinitely many obstructions. As an application, we answer questions by Mumford and Keel about curves on surfaces over finite fields.

Stephan Stolz

Supersymmetric Euclidean field theories and generalized cohomology

This is a report of ongoing joint work with Peter Teichner (Berkeley). Elaborating Segal's axiomatic approach to conformal field theories, we define supersymmetric Euclidean field theories over a manifold X. It turns out that the set of concordance classes of such field theories over X of dimension d isin bijective correspondence to the cohomology of X (with complex coefficients) for d=0 and to the K-theory of X for d=1. We speculate that for d=2 we obtain the "Topological Modular Form theory" of Hopkins-Miller. Evidence is provided by our result that the partition function of a supersymmetric Euclidean field theory of dimension 2 is a weakly holomorphic integral modular form.

Nitu Kitchloo

Kac-Moody groups over the last decade

I will try to give general overview of the developments in the study of Kac-Moody groups from the standpoint of topology. Kac-Moody groups are natural generalizations of compact lie groups. I will stress the analogy between Kac-Moody groups and compact groups in motivating the reason to study them. The talk will be accessible to a general topological audience.

Bertrand Toen

Chern character, loop spaces and derived algebraic geometry

The purpose of this talk is to present a general framework for the Chern character map, based on techniques from derived algebraic geometry and higher category theory. I will explain in particular how it can be useful in order to define a Chern character map for sheaves of categories rather than sheaves of modules.

Natalia Castellana

Mod p Noetherian loop spaces

One of the big achievements in the study of the homotopy theoretical properties characterizing compact Lie groups is the classification of p-compact groups by Andersen, Grodal, Møller and Viruel. p-compact groups were defined by Dwyer and Wilkerson. They are loop spaces with finite mod p cohomology whose classifying space is p-complete. In this joint project with J.A. Crespo and J. Scherer we define a p-Noetherian group to be a loop space with noetherian mod p cohomology and a p-complete classifying space. We show how they are all constructed from p-compact groups and certain Eilenberg-Mac Lane spaces. This topological characterization allows us to understand the cohomological finiteness conditions satisfied by the mod p cohomology of the classifying space.

Birgit Richter

The separable closure of the K_n-local sphere

Let E_n be the n-th Lubin-Tate spectrum at an odd prime p and adjoin all roots of unity whose order is not divisible by p. We show that the resulting spectrum does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the even prime we can show that there are no connected finite Galois extension with solvable Galois group. This is joint work with Andy Baker.

Bob Oliver

Fusion systems of groups of Lie type via homotopy theory

The talk will be centered around the following theorem, shown in joint work with Carles Broto and Jesper M\o{}ller. Fix a prime $p$, and let $q$ and $q'$ be prime powers which are not powers of $p$ and which generate the same closed subgroup of $\mathbb{Z}_p^\times$. Then for any Lie group ``type'' $G$, the $p$-subgroup fusion systems of $G(q)$ and $G(q')$ are equivalent. In other words, the Sylow $p$-subgroups of $G(q)$ and $G(q')$ are isomorphic, via an isomorphism which preserves all conjugacy relations in $G(q)$ and $G(q')$ between their subgroups. A similar result holds for ``twisted'' groups of Lie type. This is a result which is not surprising to people who have studied these groups, but there does not yet seem to be a published proof, and none of the group theorists we asked knew of a proof. Our proof uses the connection between fusion and topology, taking as starting point a theorem of Martino and Priddy that two finite groups $G$ and $G'$ have equivalent fusion systems at $p$ if their $p$-completed classifying spaces $BG^\wedge_p$ and $BG'{}^\wedge_p$ are homotopy equivalent. Other theorems, by Friedlander and Mislin, describe $BG(q)^\wedge_p$ and $BG(q')^\wedge_p$, in the above situation, as being homotopy fixed point sets of Frobenius endomorphisms acting on $BG(\C)^\wedge_p$. We showed that under certain hypotheses on a space $X$, homotopy fixed point sets of two self equivalences of $X$ are homotopy equivalent if they generate the same closed subgroup of the group of all self equivalences of $X$. Together with the other results described above, this proves the theorem, and also gives information about some other equivalences between fusion systems of groups of Lie type.

Jean Lannes

Kervaire Invariant And Manifolds With Corners

I will report on a joint work with Haynes R. Miller showing that the introduction of manifolds with corner sheds some light on the link, uncovered a long time ago in a famous paper by W. Browder, between the Kervaire invariant - defined in terms of framed manifolds - and the Adams spectral sequence for mod 2 cohomology.

Dennis Sullivan

Compactified String Topology

There is a transversality , reconnecting and gluing construction that defines for each manifold of dimension d a chain mapping from (spaces of closed curves in the manifold) cartesian product (the harmonic compactification of moduli space of surfaces) to (spaces of surfaces with nodes in the manifold). The top chains of moduli space lead to a "quantum lie bialgebra" on the string homology which means the following: if [k] denotes the kth tensor power of the equivariant homology of the free loop space mod constant loops  , there are maps F(k,l,g) from [k] to [l] for each g = 0,1,2,...and k,l positive of degree 3d(eulercharacterstic of S) -1 satisfying quadratic identities .Here S is the connected oriented surface of genus g with k input boundaries and l output boundaries and the quadratic identities correspond to all ways of factoring S as a gluing composition.

Tilman Bauer

On the K(n)-based Eilenberg-Moore spectral sequence

The Eilenberg-Moore spectral sequence tries to compute the homology of the fiber of a fibration from the homology of the base space and the total space. For non-connective homology theories like Morava K-theory, convergence is very problematic and fails even in simple examples. In my talk I wish to introduce a new, useful and rather strong notion of convergence for this and similar spectral sequences and show that this kind of convergence is inherited under fibrations; in particular, when the base space is a finite Postnikov system, the Eilenberg-Moore spectral sequence always converges in this strong sense.

Mark Behrens

Congruences amongst modular forms and the divided beta family

The v_1-periodic stable homotopy groups of spheres at an odd prime are generated by the alpha family, and the orders of these groups admit a global description in terms of denominators of Bernoulli numbers. I will describe a similarly global description of the v_2-periodic beta family in terms of explicit congruences of modular forms, for primes greater than 3.

Charles Rezk

The Frobenius congruence for power operations in Morava E-theory

We describe the structure inherent in the Morava E-theory of a commutative S-algebra. Ando, Hopkins, and Strickland have shown how this structure encodes information about isogenies of deformations of a finite height formal group. We will use their work to describe the natural target category C of the functor defined by Morave E-homology whose domain category is commutative S-algebras. The answer is that C is a category of sheaves on a certain generalized stack, with a twist: the objects of C are exactly those sheaves which satisfy a certain congruence condition related to Frobenius isogenies. This answer is a precise analogue to the "Wilkerson criterion" for lambda-rings.

Jeff Smith


Kasper Andersen

Reduced fusion systems over 2-groups

Saturated fusion systems were introduced in group theory by L. Puig and made popular in homotopy theory by works of C. Broto, R. Levi and B. Oliver and others. For odd primes there are a number of exotic examples of saturated fusion systems (i.e. examples not coming from finite groups). For p=2 however, there is only one known family of exotic examples (constructed by R. Levi and B. Oliver based on earlier work of R. Solomon). In the talk we will describe the results of a systematic search for new exotic saturated fusion systems over "small" 2-groups. This is joint work in progress with B. Oliver and J. Ventura.

Jacob Lurie

Algebraic Groups over the Sphere Spectrum

Let G be a compact Lie group. The complexification of G has the structure of an algebraic group: that is, it can be described as the space of solutions to a set of polynomial equations over the complex numbers. This algebraic group admits a canonical (split) form over the ring of integers (in other words, the polynomial equations defining the complexification of G can be taken to have integer coefficients). In this talk, I will discuss the problem of realizing this algebraic group over the sphere spectrum.

Vassily Gorbounov

Divergent series and invariants of projective varieties

We will describe a way of summation of divergent series suggested in physics which goes back to the Gauss cyclotomic identity. It appears natural to present some invariants of projective varieties as such divergent sums. This we believe is a reflection of interesting dualities considered in physics.

Jack Morava

To the left of the sphere spectrum

Both Waldhausen's A-theory spectrum of a point and the topological cyclic homology TC of the sphere spectrum are interesting ringspectra which map naturally to S; in some ways they resemble equicharacteristic local rings with S as residue field. Both specialize bivariant constructions (Williams, Dundas-Ostvaer...) whic can be regarded as the Hom-objects for natural categories of correspondences. The trace homomorphisms from A or TC to S define monoidal functors from those categories to spectra, with S \smash_A S [resp. S \smash_TC S] as candidates for Hopf-Galois objects representing the automorphisms of these forgetful functors. The latter example seems particularly accessible, as a kind of cotensor algebra much like a dual to the ringspectrum S[Omega Sigma CP^\infty_+] studied recently by Baker and Richter. Over the rationals this is the universal enveloping algebra of a free graded Lie algebra, with striking similarities to the Hopf algebra of the Tannakian Galois group appearing in the arithmetic theory of mixed Tate motives over the integers. This suggests that that theory (closely related to the algebraic K-theory of the integers) has a homotopy-theoretic analog, related similarly to A.