We meet (online) every other Thursday at 1pm (Mexico City/Central time). If you'd like to participate, please contact Bernardo Villarreal (villarreal at matem dot unam dot mx) so we can add you to our email list. A link to the meeting will be sent the day of the talk.
Feb 18. Simon Gritschacher, University of Copenhagen
Title: On the second homotopy group of spaces of commuting elements in Lie groups
Abstract: In this talk we will consider the topological space of n-tuples of pairwise commuting elements in a Lie group G. This space arises as a moduli space of flat connections on principal G-bundles over the n-torus. We will describe homotopy and homology calculations. One of our main results is the description of the second homotopy group of the space of commuting pairs in any compact Lie group. The connection with gauge theory will be discussed. This is joint work with Alejandro Adem and José Manuel Gómez.
Mar 4. Sam Nariman, Purdue University
Title: Mather-Thurston’s theory, non abelian Poincare duality and diffeomorphism groups
Abstract: I will discuss a remarkable generalization of Mather’s theorem by Thurston that relates the identity component of diffeomorphism groups to the classifying space of Haefliger structures. The homotopy type of this classifying space plays a fundamental role in foliation theory. However, it is notoriously difficult to determine its homotopy groups. Mather-Thurston theory relates the homology of diffeomorphism groups to these homotopy groups. Hence, this h-principle type theorem has been used as the main tool to get at the homotopy groups of Haefliger spaces. We talk about generalizing Thurston's method to prove analogue of MT for other subgroups of diffeomorphism groups that was conjectured to hold. Most h-principle methods use the local statement about M=R^n to prove a statement about compact manifolds. But Thurston's method is intrinsically compactly supported method and it is suitable when the local statement for M=R^n is hard to prove. As we also shall explain Thurston's point of view on this "local to global" method implies non abelian Poincare duality.
Mar 18. Alexander Kupers, University of Toronto
Title: E_2-algebras and the unstable homology of mapping class groups
Abstract: We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
Apr 15. Kristine Bauer, University of Calgary
Title: Categorical differentiation of homotopy functors
Abstract: The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, B., Johnson, Osborne, Tebbe and Riehl (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute, Cockett and Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure. To address this question, Burke and Ching and I instead use the notion tangent categories, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. The structure of a tangent category is highly reminiscent of the structure of a tangent bundle on a manifold. Indeed, the category of smooth manifolds is a primary and motivating example of a tangent category. In recent work with Burke and Ching, we make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors appears as the associated categorical derivative of a particular tangent infinity category. This ties together Lurie’s tangent bundle construction to the categorical literature on tangent categories. In this talk, I aim to explain the categorical notions of differentiation and tangent categories, and explain their relationship to Goodwillie’s functor calculus.
Apr 29. Rachael Boyd, Max Planck Institute for Mathematics
Title: Homological stability for Temperley-Lieb algebras
Abstract: Many sequences of groups and spaces satisfy a phenomenon called 'homological stability'. I will present joint work with Hepworth, in which we abstract this notion to sequences of algebras, and prove homological stability for the sequence of Temperley-Lieb algebras. The proof uses a new technique of 'inductive resolutions', to overcome the lack of flatness of the Temperley-Lieb algebras. I will also introduce the 'complex of planar injective words' which plays a key role in our work. Time permitting, I will explore some connections to representation theory and combinatorics that arose from our work. I will aim this talk at a broad topological audience, and assume no prior knowledge of homological stability or Temperley-Lieb algebras.
May 13. Jeffrey Carlson, Imperial College London
Title: Biquotients and a product on the two-sided bar construction
Abstract: In 1960s and '70s five separate teams of authors showed that the Eilenberg–Moore spectral sequence computing the cohomology of a homogeneous space collapsed, but there was no word on the ring structure until 2019, when Franz showed it was what one would hope. In this talk we generalize this proof to biquotients K\G/H, an attractive class of smooth manifolds that contains exotic spheres and essentially all known examples of manifolds admitting a Riemannian metric of nonnegative sectional curvature. The key new ingredient is a natural multiplication on the two-sided bar construction B(M,A,N) of differential graded algebras, subject to a homotopy-commutativity condition. Background will be explained and proofs limply gestured at. The mod 2 reduction of the formula for the product was proposed by Franz.
May 27. Mona Merling, University of Pennsylvania
Title: Scissors congruence for manifolds via K-theory
Abstract: The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German "schneiden und kleben," cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. This is joint work with Hoekzema, Semikina, Rovi, and Wells.
Jun 10. Dan Petersen, Stockholm University
Title: Factorization statistics and bug-eyed configuration spaces
Abstract: Fix a class function of the symmetric group S_n, i.e. a function on the set of partitions of n. For any monic polynomial of degree n over a fixed finite field the degrees of its irreducible factors will form such a partition and we can ask about the average value of the class function evaluated on all such polynomials. Trevor Hyde ('18) proved by direct calculation that the answer can be expressed in terms of the S_n-action on the cohomology of the configuration space of n points in R^3 (!), but his argument gave no geometric reason for such a formula to exist. We give a geometric proof of Hyde's formula by applying the Grothendieck--Lefschetz trace formula to the cohomology of a certain highly nonseparated algebraic space obtained by gluing together complements of hyperplanes in the braid arrangement. (Joint with P. Tosteson)
Jun 24. Calista Bernard, Stanford University