Purdue Topology Seminar, Spring 2020

The Purdue Topology Seminar is held Wednesdays 11:30am-12:30pm at MA 731online unless otherwise noted. If announced there is an introductory lecture beforehand 10:30am-11:15am in the same room. How to join will be announced by email on Monday before the seminar. If you want to be informed let Peter Patzt (ppatzt at purdue dot edu) know.

We have begun to record our online seminars. We publish them on our new YouTube-Channel.

January 15: Maximilien Péroux (University of Illinois at Chicago)
no video recording

Introductory Talk: Coalgebraic tools in algebraic topology
Abstract: Coalgebras and comodules are dual concepts that of algebras, rings and modules. In this talk, we will define the notions and present well-known results and examples. For instance, every Thom space (or spectrum) is a particular kind of comodules over the base space. Spectra with a coalgebra structure also appear in chromatic homotopy theory.

Main Talk: Rigidification of higher coalgebras and comodules
Abstract: A classical result of higher algebra relates homotopy coherent rings in spectra with strictly associative (or commutative) ring spectrum in a particular choice of model of spectra (such as symmetric spectra, orthogonal spectra, etc). This process is called rigidification (or rectification). In this talk, we are interested in the dual questions of rigidification for higher coalgebras and comodules. We will show that in full generality, there is no rigidification of coalgebras in spectra. We will instead focus on the discrete case (i.e. differential graded context) and show new results for comodules and coalgebras.

January 22: Yang Mo (Purdue)

Topics Exam

January 29: no seminar
February 5: no seminar
February 12: no seminar
February 19: Wee Liang Gan (University of California at Riverside)
no video recording

Title: The Nakayama functor in representation stability theory
Abstract: I will explain the construction of the Nakayama functor for FI and VI and why it induces an equivalence from a quotient of a category of finitely generated modules to a category of finite dimensional modules. The proof of the equivalence uses some of the key results for FI-modules and VI-modules. This is a joint work with Liping Li and Changchang Xi.

February 26: Tony Feng (MIT)
no video recording

Title:The Galois action on symplectic K-theory
Abstract: The classifying spaces of arithmetic groups are often modeled by moduli spaces in algebraic geometry. A notable example of this phenomenon is the moduli space of complex abelian varieties, which models the classifying space of integral symplectic groups. As a consequence, the group homology and algebraic K-theory of symplectic groups carry a natural Galois action. In joint work with Soren Galatius and Akshay Venkatesh we compute this Galois action, and find it to have an interesting universal characterization.

March 4: Florian Naef (MIT)
no video recording

Title: String topology and the configuration space of two points
Abstract: Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points.

There is an introductory lecture at 10:30.

March 11: Zach Himes (Purdue)
no video recording

Title: Secondary stability and periodicity for unordered configuration spaces
Abstract: Secondary stability is a stability pattern in a range that homological stability does not hold. The first example of secondary stability is Galatius--Kupers--Randal-Williams results on mapping class groups. We prove secondary stability for unordered configuration spaces of manifolds. The main difficulty is the closed case (the open case was previously known by some experts). In the closed case, there are no obvious stabilization maps and the homology does not stabilize but is periodic. We resolve this issue by constructing a chain-level stabilization map for configurations of closed manifolds.

March 18: Spring Break
March 25: no seminar
April 1: Andrey Lazarev (Lancaster University)
[Video recording]

Title: Differential graded Koszul duality: a global approach
Abstract: Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. It exists on various levels: as a correspondence between operads and cooperads, algebras over operads and coalgebras over cooperads and modules and comodules over associative algebras (the latter, simplest, version will be relevant to this talk). Usually some kind of conilpotence is assumed on the `co' side. In this talk I explain what happens if this condition is dropped; the consequences turn out to be quite dramatic. I will show how this non-conilpotent (or global) version of Koszul duality comes up naturally in the study of derived categories of complex algebraic manifolds and infinity local systems on topological spaces. Time permitting, I will also explain how one can construct a global version of deformation theory for certain deformation problems based on this approach. This is joint work with Ai Guan.

April 8: Calista Bernard (Stanford University)
[Video recording]

Title: Twisted Dyer-Lashof Operations
Abstract: In the 70s, Fred Cohen and Peter May gave a description of the mod p homology of a free E_n algebra in terms of certain homology operations, known as Dyer-Lashof operations, and the Browder bracket. These operations capture the failure of the E_n multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen's work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and explain computational results that show the existence of additional operations in the twisted case.

April 15: Yining Zhang (University of Colorado at Boulder)
[Video recording]

Title: Hodge decomposition of string topology
Abstract: Given a closed oriented manifold X, Chas and Sullivan constructed a Lie bracket on the (reduced) S^1-equivariant homology of the free loop space of X, called the string bracket. Furthermore, if the manifold is simply connected, there is a Hodge type decomposition on the S^1-equivariant homology induced by the n-fold coverings of the circle. It is natural to ask whether the string bracket preserve this decomposition. We provide a positive answer under the additional assumption that X is rationally elliptic. Our argument is based on algebraic models of string topology and analogous operations on Hochschild and cyclic homologies. This is a joint work with Yuri Berest and Ajay Ramadoss.

April 22: Nir Gadish (MIT)
[Video recording]

Title: The “generating function” of configuration spaces, as a source for explicit formulas and representation stability
Abstract: As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. In this talk I will apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.

April 29: Sam Nariman (Copenhagen University)
[Video recording]

Title: Diffeomorphisms of reducible three manifolds and bordisms of group actions on torus
Abstract: I first talk about the joint work with K. Mann on certain rigidity results on group actions on torus. In particular, we show that if the torus action on itself extends to a C^0 action on a three manifold M that bounds the torus, then M is homeomorphic to the solid torus.
Motivated by this result, I will talk about certain finiteness results about classifying space of reducible three manifolds which is related to an unfinished project by Allen Hatcher.

May 1:
(Friday)
Søren Galatius (Copenhagen University)

Title: E_\infty algebras and general linear groups of infinite fields
Abstract: Homology of general linear groups of a field F is calculated by the chain complex R(n) = C_*(BGL_n(F)). There are chain maps from the tensor product of R(n) and R(m) to R(n+m), induced by direct sum of F-vector spaces. These products make R into a bigraded differential graded algebra, which is homotopy commutative, and can moreover be given the structure of an E_\infty algebra. In joint work with Alexander Kupers and Oscar Randal-Williams, we study the structure of this E_\infty algebra when F is infinite. We are led to new results about homology of general linear groups, for instance an extension of a relationship between the relative homology of BGL_n relative to BGL_{n-1} and Milnor K-theory, due to Nesterenko and Suslin, as well as new questions.

May 5:
(Tuesday)
Markus Land (Copenhagen University)
[Video Recording]

Title: Hermitian K-theory
Abstract: I will start by introducing the Grothendieck-Witt group associated to a ring and discuss some examples. I will show how it relates to the algebraic K-group and the Witt group. I will then explain that this relation can be refined to fibre sequence of spaces relating the Grothendieck-Witt space to algebraic K-theory and Ranicki’s algebraic L-theory. Finally, I will indicate that the Grothendieck-Witt space is closely related to an algebraic version of the geometric cobordism category, and highlight some similarities between the geometric and algebraic world. All of this is joint work with Calmès, Dotto, Harpaz, Hebstreit, Moi, Nardin, Nikolaus and Steimle.

May 6: Camilo Arias Abad (Universidad Nacional de Colombia- Medellín)
[Video Recording]

Title: Singular chains on Lie groups, the Cartan relations and Chern-Weil theory
Abstract: Let G be a simply connected Lie group. The space C(G) of smooth singular chains on G has de structure of a differential graded algebra, with product induced by the Eilenberg-Zilber map. We will describe how the category of (sufficiently smooth) modules over this algebra can be described infinitesimally. More precisely, we will show that the category of modules over C(G) is equivalent to the category of represenations of the differential graded Lie algebra Tg, which is universal for the Cartan relations. This extends the usual correspondence between representations of G and representations of the Lie algebra g. We also prove that in case G is compact, this equivalence of categories can be extended to an A_∞-equivalence of dg-Categories. The main ingredients in the proof are Gugenheim’s A_∞-version of de Rham’s theorem, the non- commutative Weil algebra of Alekseev-Meinrenken, and the Van-Est map. If time permits, I will describe how this construction relates to a version of Chern-Weil theory for ∞-local systems on classifying spaces. This talk is based on joint work with Alexander Quintero.

May 13: Jeremy Miller (Purdue University)
Joint with the CUNY Topology, Geometry, and Physics seminar

Title: André–Quillen homology and homological stability for general linear groups of Euclidean domains
Abstract: I will discuss joint work in progress with Kupers and Patzt on improved stable ranges for homological stability for general linear groups of Euclidean domains. The main ingredient is a vanishing result for E_infinity André–Quillen homology. By work of Galatius--Kupers--Randal-Williams, these André--Quillen homology groups are isomorphic to the equivariant homology of certain posets.

May 20: Edouard Balzin (Centre de Mathématiques Laurent Schwartz, École Polytechnique, France)
[Video Recording]

Title: Operation-indexing categories and Grothendieck fibrations
Abstract: It has been known since Segal that various small categories can be used as blueprints for algebraic structures in homotopy theory, providing alternatives to operads in such questions as for example delooping. The examples of those categories include finite sets, ordered sets, n- ordinals of Batanin and various exit path categories of configuration spaces, as well as categories of operators of general topological operads.
We would like to offer a definition for such operation-indexing categories, called weak operads or algebraic patterns, and how to describe homotopy-algebraic structures over such things via fibrations of model and higher categories. Depending on time and the interest of the audience, we may attempt to introduce the notion of a weak approximation as a means of establishing certain “Morita”-type equivalences in the world of weak operads.

May 27: No Seminar
June 3: Manuel Krannich (University of Cambridge, UK)

Title: Pseudoisotopies of discs and algebraic K-theory of the integers
Abstract: There is an intimate connection between algebraic K-theory and the space of pseudo-isotopies P(M) of a compact d-manifold M (that is, diffeomorphisms of a cylinder M x I that are the identity on M x 0 and ∂M x I). Classically, the pseudo-isotopy space P(M) is studied in two steps: there is a stabilisation map P(M)-> P(M x I) which is approximately d/3-connected by a result of Igusa, and the colimit has a description in terms of Waldhausen's algebraic K-theory of spaces due to Waldhausen--Jahren--Rognes' stable parametrised h-cobordism theorem. In this talk, I will focus on the case of an even-dimensional disc and explain a new method to relate its space of pseudo-isotopies to the algebraic K-theory of the integers in a range up to roughly the dimension. This approach is independent of the usual route, does not involve stabilising the dimension, and is homological in nature.

June 10: Mark McConnell (Princeton University)
[Video Recording]

Title: Computing Hecke Operators for Cohomology of Arithmetic Groups
Abstract: I will describe three projects. The first, which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups Gamma of G = SL(4,R). We compute the cohomology of the locally symmetric spaces Gamma\G/K, focusing on the cuspidal degree H^5. We compute a range of Hecke operators on this cohomology. We find Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion and torsion classes. The method is to use a cell complex, the well-rounded retract, due to Ash. The second project, joint with Bob MacPherson, is an algorithm for computing the Hecke operators on the cohomology H^d of Gamma in SL(n,R) for all n and all d. For this we introduce a new retract, the well-tempered complex. In the third project, joint with Dylan Galt, we give an algorithm for Hecke operators on H^d of Gamma in Sp(4,R) for all d. The method is to define an appropriate subcomplex of the well-tempered complex for SL(4,R).

June 17: Manuel Rivera (Purdue)
[Video Recording]

Title: The fundamental group and homotopy theory over a field
Abstract: I will explain how the fundamental group of a space, as well as its homology groups with coefficients in all possible local systems of vector spaces, are completely determined by chain level algebraic data. This algebraic data may be packaged as the weak equivalence class of the simplicial cocommutative coalgebra of chains under a notion of weak equivalence which involves adjusting the coalgebra structure through the cobar construction and is stronger than quasi-isomorphism. I will outline a theory, of independent interest, of twisting cochains and twisted tensor products at the level of simplicial coalgebras and algebras, which we used to construct the fundamental bialgebra and the universal cover of an abstract connected simplicial cocommutative coalgebra. This opens up the possibility of using the power of abstract algebraic homotopy theory to study geometric spaces with arbitrary fundamental group.

June 24: no seminar
July 1:
July 8:
July 15: Anibal Medina (EPFL)