## Purdue Topology Seminar

The Purdue Topology Seminar is held Wednesdays 2:30pm - 3:30pm EST (at SCHM 308 if we meet in person) unless otherwise noted. Some of the talks will be online through Zoom. If you want to be added to our email list please contact Arun Debray (adebray@purdue.edu).

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

## Spring 2024

### January 17 (in-person)

Vijay Higgins (Michigan State)

Title: Central elements in the SL(d) skein algebra

Abstract: The skein algebra of a surface is spanned by links in the thickened surface, subject to skein relations which diagrammatically encode the data of a quantum group. The multiplication in the algebra is induced by stacking links in the thickened surface. This product is generally noncommutative. When the quantum parameter q is generic, the center of the skein algebra is essentially trivial. However, when q is a root of unity, interesting central elements arise. When the quantum group is quantum SL(2), the work of Bonahon-Wong shows that these central elements can be obtained by a topological operation of threading Chebyshev polynomials along knots. In this talk, I will discuss joint work with F. Bonahon in which we use analogous multi-variable 'threading' polynomials to obtain central elements in higher rank SL(d) skein algebras. Time permitting, I will discuss how a finer version of the skein algebra, called the stated skein algebra, can be used to show that the threading operation yields a well-defined algebra embedding of the coordinate ring of the character variety into the root-of-unity skein algebra in the case of SL(3).

### February 7 (in-person - different location: STON 215)

Morgan Opie (UCLA)

Title: Enumerating stably trivial topological vector bundles with higher real K-theories

Pre-talk 1:30-2:20pm

Abstract: The goal of the pre-talk will be to set up some basic notions and ideas for the talk "Enumerating stably trivial topological vector bundles with higher real K-theories", to make the main talk more accessible. I will discuss vector bundles (and how I view them as a homotopy theorist); will talk about some historical methods for enumerating vector bundles; and will also talk a bit about "higher real K-theories" and why these generalized cohomology theories come into play. I aim for this can be an interactive talk, so if you have questions about specific things in the abstract please feel free to ask!

Main talk 2:30-3:30pm

Abstract: The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking questions about topological vector bundles. But, in general, there are many non-equivalent vector bundles with the same K-theory class. Bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest CW complexes.

Building on work of Hu, we use Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate vector bundle enumeration questions to tractable stable homotopy theory computations. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. The talk will include a gentle discussion of the tools involved. This is joint work with Hood Chatham and Yang Hu.

### February 13, 14, and 15 (in-person)

Ajay Ramadoss (Indiana)

This week, we will have a special three-lecture series on representation homology. Please note the unusual times and locations.

First Lecture: Tuesday, February 13, 2:30-4:00. Location: MATH 731

Title: Representation homology of associative algebras

Abstract: Classical representation varieties of associative algebras, deriving the representation functor (and definition of representation homology), cyclic homology and derived character maps, the stabilization theorem, combinatorial applications.

Second Lecture: Wednesday, February 14, 2:30-3:30. Location: SCHM 308

Title: Representation homology of Lie algebras

Abstract: Classical representation varieties of Lie algebras, derived representation schemes of Lie algebras and their representation homology, Hodge decomposition of cyclic homology (of universal enveloping algebras), derived character maps and the Drinfeld homomorphism, the derived Harish-Chandra homomorphism, an (open) conjecture on derived commuting schemes.

Third Lecture: Thursday, February 15, 2:30-4:00. Location: MATH 731

Title: Representation homology of spaces

Abstract: Representation varieties of groups, derived representation schemes of simplicial groups (and hence, spaces), basic properties of representation homology of spaces, analogy with and relation to Pirashvili-Hochschild homology, computations for Riemann surfaces (time permitting, knot complements and Lens spaces), representation homology of simply connected spaces, topologically interpreting the Strong Macdonald Conjectures via representation homology.

### February 21 (in-person)

Jake Rasmussen (Illinois)

Title: Heegaard Floer homology, sutures, and functoriality

Abstract: Heegaard Floer homology is a powerful invariant of 3 and 4-manifolds, which fits (roughly) into the framework of a 3+1 dimensional TQFT. The "hat" version of this theory should fit into an extended TQFT in dimensions 2,3, and 4; in dimensions 2 and 3 this theory is described by the bordered Floer homology of Lipshitz, Ozsvath, and Thurston. To make the bordered theory into a 2-functor, one needs an appropriate 2-category of sutured manifolds, I'll discuss this category and what the 2-functor should look like.

### February 28 (in-person)

Soren Galatius (Copenhagen)

Title: The action of Aut(C) on symplectic K-theory of the integers

Abstract: I will discuss aspects of the groups Sp_{2g}(Z), the integral symplectic groups, and their homology. On the one hand, Charney proved homological stability for these groups and Karoubi defined a flavor of K-theory based on symplectic groups instead of general linear groups, and also studied its relationship to ordinary algebraic K-theory. On the other hand, the relationship to principally polarized abelian varieties gives rise to actions of certain Galois groups on this flavor of K-theory, at least after completing at a prime p. I will discuss joint work with Feng and Venkatesh, in which we study the resulting Galois representations.

### March 6 (online)

### March 13 (online)

Carlos De La Cruz Mengual

Title: The Degree-Three Bounded Cohomology of Complex Lie Groups of Classical Type