# BilkentMath SHCS-2021

This is the homepage of the workshop Spaces of Homomorphisms and Classifying Spaces (2021).

Location: Zoom (Register here to get the link.)
Time: 6:00 pm - 11:30 pm UTC+3, Sept 7-8-9 2021

To view the schedule in your local time visit researchseminars.org. Recordings of talks are available on YouTube.

# Day 1 - Sept 7

• 5:50 pm UTC+3: Opening - Welcome to SHCS 2021

• 6:00 pm UTC+3: $E$-polynomials of character varieties for real curves by Tom Baird
Abstract: Given a Riemann surface $\Sigma$ denote by $M_n(\mathbb{F}) := Hom_{\xi}( \pi_1(\Sigma), GL_n(\mathbb{F}))/GL_n(\mathbb{F})$ the $\xi$-twisted character variety for $\xi \in \mathbb{F}$ a $n$-th root of unity. An anti-holomorphic involution $\tau$ on $\Sigma$ induces an involution on $M_n(\mathbb{F})$ such that the fixed point variety $M_n^{\tau}(\mathbb{F})$ can be identified with the character variety of “real representations” for the orbifold fundamental group $\pi_1(\Sigma, \tau)$. When $\mathbb{F} = \mathbb{C}$, $M_n(\mathbb{C})$ is a complex symplectic manifold and $M_n^{\tau}(\mathbb{C})$ embeds as a complex Lagrangian submanifold (or ABA-brane). By counting points of $M_n(\mathbb{F}_q)$ for finite fields $\mathbb{F}_q$, Hausel and Rodriguez-Villegas determined the E-polynomial of $M_n(\mathbb{C})$ (a specialization of the mixed Hodge polynomial). I will show how similar methods can be used to calculate the E-polynomial of $M_n^\tau(\mathbb{F}_q)$ using the representation theory of $GL_n(\mathbb{F}_q)$. We express our formula as a generating function identity involving the plethystic logarithm of a product of sums over Young diagrams. The Pieri’s formula for multiplying Schur polynomials arises in an interesting way.
This is joint work with Michael Lennox Wong.

• 7:00 pm UTC+3: An extension theory for partial groups by Carles Broto
Abstract: A partial group [Chermak, Fusion systems and localities, Acta Math. 211 (2013), 47–139] might be seen as a simplicial set subject to a certain conditions. With this point of view we study the self-equivalences of partial groups. Then, we use the classical theory of fibre bundles in order to develop an extension theory of partial groups. We show how extensions are classified by low dimensional cohomology, as in the case of groups. (Joint work with Alex Gonzalez)

• 8:00 pm - 8:30 pm UTC+3: Break

• 8:30 pm UTC+3: Stability properties of spaces of commuting elements by Mentor Stafa
Abstract: In this talk we explore the homological stability properties of spaces of commuting elements in a Lie group $G$, denoted by $Hom(\mathbb{Z}^n,G)$. We will discuss the different types of sequences of spaces to consider and the stability results for each case. This work is done in collaboration with Daniel Ramras.

• 9:30 pm UTC+3: Commutator Equations in Finite Groups by Enrique Torres
Abstract: The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. Frobenius considered the case of ordered pairs and obtained a formula to count them in terms of the irreducible characters of the group. In this talk we will consider the case of ordered triples and will show how to obtain formulas that extend Frobenius’ character formula in this case. We will also discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and will talk about some open problems.

• 10:30 pm - 11:30 pm UTC+3: Discussion

# Day 2 - Sept 8

• 6:00 pm UTC+3: Poisson maps between character varieties: gluing and capping by Lisa Jeffrey
Abstract: (joint with Indranil Biswas, Jacques Hurtubise, Sean Lawton, arXiv:2104.05589) Let $G$ be either a compact Lie group or a reductive Lie group. Let $\pi$ be the fundamental group of a $2$-manifold (possibly with boundary). We can define a character variety by $Hom(\pi, G)/G$, where $G$ acts by conjugation. We explore the mappings between character varieties that are induced by mappings between surfaces. It is shown that these mappings are generally Poisson. In some cases, we explicitly calculate the Poisson bi-vector.

• 7:00 pm UTC+3: Singularities in free-group character varieties by Dan Ramras
Abstract: I’ll discuss joint work with Florentino, Guérin, and Lawton regarding the varieties $Hom(F_r, G)//G = G^r//G$, which parametrize r-tuples in a (connected, linearly reductive) Lie group G modulo conjugation. Extending work of Richardson from the 1980s, we show that for $r\gt 2$ the algebraic singularities in these spaces come in just two forms: reducible representations and bad representations (irreducible but with non-trivial, finite stabilizer). I’ll discuss the methods behind this result, as well as related results about the topology of these moduli spaces. For instance, it turns out that not only are reducible and bad representations algebraic singularities of the character variety, they are in fact always topological singularities.

• 8:00 pm - 8:30 pm UTC+3: Break

• 8:30 pm UTC+3: Commutative simplicial bundles by Pal Zsamboki
Abstract: Joint work with Cihan Okay, see https://arxiv.org/abs/2001.04052. Let $G$ be a topological group. Then if $G$ is $\mathbb{Z}$-good, for example if $G$ is a compact Lie group, then the topological space $B(\mathbb{Z},G)$ classifies transitionally commutative principal $G$-bundles. We study an analogue of this result in the realm of simplicial sets. We generalize transitionally commutative structure to $\tau$-structure where $\tau$ is an endofunctor of the category of groups of quotient type. This includes for example p-torsion. We show that via geometric realization, our construction corresponds to the topological one. We introduce the notion of $\tau$-structure for simplicial principal bundles. We show that our space classifies simplicial principal bundles with $\tau$-structure, provided that a conjecture holds, which is of independent interest.

• 9:30 pm UTC+3: Classifying Space for Commutativity in $U(3)$ by Santanil Jana
Abstract: The classifying space for commutativity in Lie groups has been a major focus of research in the study of Spaces of Homomorphism. In this talk, I will present a computation of the cohomology of the total space of the principal bundle associated to the classifying space for commutativity in $U(3)$ using the spectral sequence associated to a homotopy colimit. I will also talk about some interesting computations such as the integral cohomology of $U(3)/N(T)$, where $T$ is the maximal torus of $U(3)$ with normalizer $N(T)$.

• 10:30 pm - 11:30 pm UTC+3: Discussion

# Day 3 - Sept 9

• 6:00 pm UTC+3: Torsion in the space of commuting elements in a Lie group by Daisuke Kishimoto
Abstract: The (co)homology of the space of commuting elements in a Lie group is a recent subject of interest. There are nice results on rational cohomology, but there are few results on torsion in homology. I will present a method to compute torsion in homology, which is based on a new homotopy decomposition and the combinatorics of the Weyl group.
This is a joint work with Masahiro Takeda.

• 7:00 pm UTC+3: On the second homotopy group of spaces of commuting elements in Lie groups by José Manuel Gómez
Abstract: In this talk we will explore the problem of computing the second homotopy group of the spaces of the form $Hom(\mathbb{Z}^n, G)$, where $G$ is a compact Lie group. We will show in particular, that if $G$ is compact, simply connected and simple, then $\pi_{2}( Hom(\mathbb{Z }^2, G))=\mathbb{Z}$ and furthermore one can obtain geometric generators.
This talk is based on a joint work with Alejandro Adem and Simon Gritschacher

• 8:00 pm - 8:30 pm UTC+3: Break

• 8:30 pm UTC+3: Combinatorial models of $E(2,S_n)$ by Omar Antolín Camarena
Abstract: I will review several simplicial complexes and posets associated to a discrete group $G$ that have the homotopy type of $E(2,G)$, the total space of the universal transitionally commutative principal $G$-bundle. The homotopy equivalence of these models is given by some underappreciated results of Abels and Holz. Then I will focus on the symmetric groups and describe some very early work in progress to understand the homotopy type of $E(2,S_n)$.

• 9:30 pm UTC+3: Commuting unitary matrices and Hochschild homology by Simon Gritschacher
Abstract: I will explain how spaces of commuting unitary matrices relate to Hochschild homology, and how this can be used in homology calculations.
This is based on work in progress with M. Hausmann.

• 10:30 pm - 11:30 pm UTC+3: Discussion