Schedule for: 21w5011 - Cohomology of Arithmetic Groups: Duality, Stability, and Computations

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I give a few examples of how Galois representations can help in the understanding and computation of the homology of congruence subgroups of $\mathrm{GL}_n(\mathbb{Z})$. Then I sketch a current project of mine with Darrin Doud in which we hope to prove the following: If $\rho=\sigma_1 \oplus \sigma_2$ is an $n$-dimensional odd mod $p$ Galois representation, with $\sigma_1$ and $\sigma_2$ irreducible odd Galois representations that are attached to Hecke eigenclasses in the homology of the predicted congruence subgroups, with predicted weights, then $\rho$ is attached to a Hecke eigenclasses in the homology of the predicted congruence subgroup of $\mathrm{GL}_n(\mathbb{Z})$, with predicted weight. Here, "predicted" refers to the Serre-type conjecture of Ash–Doud–Pollack–Sinnott. We assume that $p$ is greater than $n+1$ and that the Serre conductor of $\rho$ is square-free.

In this talk, I will explain how a homotopy equivalence between certain $E_k$-buildings both proves Rognes' connectivity conjecture for fields and computes the Koszul dual of Steinberg. Rognes' connectivity conjecture states that the common basis complex is highly connected. This is relevant as the equivariant homology of this complex appears in a rank filtration spectral sequence computing the homology of the $K$-theory spectrum. The Steinberg modules appear in various contexts, importantly as the dualizing modules of special linear groups of number rings. They can be put together to form a ring. When considered equivariantly over the general linear groups of fields, one can show that this ring is Koszul and we compute its Koszul dual. Results in this talk include joint work with Jeremy Miller, Rohit Nagpal, and Jennifer Wilson.

I will explain what is known about homological stability for the general linear groups of the integers. In particular, I will discuss a recent result, joint work with Jeremy Miller and Peter Patzt, that improves the homological stability range to slope 1. It builds on machinery developed with Soren Galatius and Oscar Randal-Williams, and is closely related to homology with coefficients in the Steinberg module.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear. I’ll discuss this question from the point of view of automorphic forms, and outline how the phenomenon of endoscopy can be used to explain the slow rates of growth and to compute upper bounds.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

General linear groups, symplectic groups, unitary groups and orthogonal groups have long been known to satisfy homological stability under appropriate conditions. In joint work with David Sprehn, we improved the earlier known homological stability ranges for $\mathrm{Sp}_{2n}(\mathbb{F})$, $\mathrm{O}_{n,n}(\mathbb{F})$ and $\mathrm{U}_{2n}(\mathbb{F})$ over any field $\mathbb{F}$ other than $\mathbb{F}_2$, following a strategy of Quillen for general linear groups $\mathrm{GL}_n(\mathbb{F})$. Under more restricted assumptions, we deduce a stability theorem for the orthogonal group $\mathrm{O}_n(\mathbb{F})$. I'll present these results, focussing on what these groups have in common, and presenting this maybe less well-known strategy of Quillen that gives a slope 1 stability range for $\mathrm{GL}_n(\mathbb{F})$.

We prove that the Steinberg representation of $\mathrm{GL}_n$ (or, more generally, a connected reductive group) over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis. This is joint work with Andrew Snowden.

The moduli space $\mathcal{M}$ of complex curves of fixed topology is an orbifold classifying space for surface bundles. As such the cohomology rings of $\mathcal{M}$ and its various orbifold covers give characteristic classes for surface bundles. I will discuss the Steinberg module which is central to the duality present in these cohomology rings. I will then explain current joint work with T. Brendle and A. Putman on surfaces with marked points which expands on results of N. Fullarton and A. Putman for surfaces without marked points. We show that certain finite-sheeted orbifold covers $\mathcal{M}[l]$ of $\mathcal{M}$ have large nontrivial $Q$-cohomology in their cohomological dimension.

Let $p$ be a prime. One can make sense of various “compatible” algebraic representations of $\mathrm{SL}_N(\mathbb{F}_p)$ as $p$ is fixed and as $N$ varies (for example, the standard representation, or the adjoint representation, or the trivial representation). It turns out that the cohomology groups of these representations are stable as $N$ gets large. So what are they? We discuss a conjectural answer to this. We also discuss how this relates to a conjectural computation of $H^i(\mathrm{SL}_N(\mathbb{F}_p),\mathbb{F}_p)$ for $i$ fixed and $N$ going off to infinity which should be true for “almost all $p$”.

Given a real quadratic field, there is a naturally defined Hecke-stable subspace of the cohomology of a congruence subgroup of $\mathrm{SL}_3(\mathbb{Z})$. We investigate this subspace and make conjectures about its dependence on the real quadratic field and the relationship to boundary cohomology. This is joint work with Avner Ash.

In joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe, we recently identified new top-weight rational cohomology classes for moduli spaces $\mathcal{A}_g$ of abelian varieties, by using computations of Voronoi complexes for $\mathrm{GL}(g,\mathbb{Z})$ of Elbaz-Vincent--Gangl--Soulé. In this talk, I will try to explain these results from the beginning, surveying some of the main techniques and ingredients.

Modular symbols, due to Birch and Manin, provide a very concrete way to compute with classical holomorphic modular forms. Later modular symbols were extended to $\mathrm{GL}(n)$ by Ash and Rudolph, and since then such symbols and variations have played a central role in computational investigation of the cohomology of arithmetic groups over number fields, and in particular in explicitly computing the Hecke action on cohomology. $$ \qquad \\[-2em]$$ A theory of modular symbols for $\mathrm{GL}(2)$ over the rational function field was developed by Teitelbaum and later by Armana. In this talk we extend this construction to $\mathrm{GL}(n)$ and show how it can be used to compute Hecke operators on cohomology. This is joint work with Dan Yasaki.

Robert MacPherson and I developed an algorithm for computing the Hecke operators on the cohomology $H^d$ of arithmetic subgroups of $\mathrm{SL}(n)$ defined over any division algebra, for all $d$ and all $n$. It extends Voronoi's notion of perfect forms by introducing tempered perfect forms. To find the tempered perfect forms, our code must compute the facets of a large convex polytope of $n(n+1)/2$ dimensions, which is slow even for $n = 3$ or $4$. The talk will report on recent work, in the classical case of $\mathrm{SL}(2,\mathbb{Z})$, where we have succeeded in identifying the tempered perfect forms directly. The story comes down to binary quadratic forms in the spirit of Lagrange and Gauss, together with some modern class field theory. This is joint work with Erik Bahnson and Kyrie McIntosh.

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.

We construct unstable cohomology classes of nonuniform arithmetic subgroups of $\mathrm{SO}(p,q)$ using ideas of Millson-Raghunathan and more recent work of Avramidi and Nguyen-Phan. The classes we construct are dual to maximal periodic flats in the locally symmetric space. One motivation for this result is to produce characteristic classes for certain manifold bundles that are not in the algebra generated by the stable (MMM) classes.

Algebraic geometry contains an abundance of miraculous constructions. Examples include ``resolving the quartic''; the existence of 9 flex points on a smooth plane cubic; the Jacobian of a genus $g$ curve; and the 27 lines on a smooth cubic surface. In this talk I will explain some ways to systematize and formalize the idea that such constructions are special: conjecturally, they should be the only ones of their kind. I will state a few of these many (mostly open) conjectures. They can be viewed as forms of rigidity (a la Mostow and Margulis) for various moduli spaces and maps between them.

``Categorical Langlands'' refers to a perspective on the local Langlands correspondence in which the *category* of smooth representations of a $p$-adic groups is related to the *category* of coherent sheaves on an appropriate moduli stacks of Galois representations. This perspective probably arose first in the geometric Langlands program, but it also relates to the number-theoretic technique known as ``Taylor--Wiles--Kisin'' patching (which is used to prove modularity theorems). In this talk I will explain a conjectural formula for the (singular, or etale) cohomology of Shimura varieties in terms of categorical Langlands. I hope the talk will be accessible to non-experts, and give a sense of one contemporary number-theoretic perspective on the problem of describing this cohomology.

I will talk about some connections between the cohomology of arithmetic groups, $K$-theory, and number theory. One reason for these connections is the fact that there is a natural Galois action on the cohomology of symplectic groups of integers, which turns out to provide Galois representations important in the Langlands correspondence. The same mechanism leads to a Galois action on a symplectic variant of K-theory of the integers. In joint work with Soren Galatius and Akshay Venkatesh, we compute this Galois action and find that it also enjoys a certain universality.

By results of Lee–Szczarba and Church–Putman, the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in "codimensions zero and one", i.e. $H^{{n \choose 2} -i}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $i\in \{0,1\}$ and $n \geq i+2$, where ${n \choose 2}$ is the virtual cohomological dimension of $\operatorname{SL}_n(\mathbb{Z})$. I will talk about work in progress on two generalisations of these results: The first project is joint work with Miller–Patzt–Sroka–Wilson. We show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in codimension two, i.e. $H^{{n \choose 2} -2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 4$. The second project is joint with Patzt–Sroka. Its aim is to study whether the rational cohomology of the symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes in codimension one, i.e. whether $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 2$.

In this talk, we will discuss the geometry of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ and its compactifications. As is well known, in degree $k < g$ the rational cohomology of $\mathcal{A}_g$, which coincides with the cohomology of the symplectic group, is freely generated by the odd Chern classes of the Hodge bundle by a classical result of Borel. Work of Charney and Lee provides an analogous result for the stable cohomology of the minimal compactification of $\mathcal{A}_g$, the Satake compactification. However, for most geometric applications it is more natural to work with the toroidal compactifications of $\mathcal{A}_g$. We will report on joint work with Sam Grushevsky and Klaus Hulek on the toroidal compactifications of $\mathcal{A}_g$, and describe stability results for the perfect cone compactification and the matroidal partial compactification and their combinatorial features.

I will report on joint work with M. Land and T. Nikolaus in which we compute the stable part of the cohomology of both symplectic groups and orthogonal groups with vanishing signature over the integers at regular primes, in particular at the prime 2. Our approach is by identifying the stable cohomology with that of a certain Grothendieck-Witt space, whose homotopy type can be analysed using recent advances in hermitian $K$-theory.

This talk is really a problem proposal.  Mark Shusterman and I have been talking about the problem of controlling sums of Legendre symbols $\left(\frac{f}{g}\right)$ as $f$ and $g$ range over squarefree polynomials of degree $m$ and $n$ over $\mathbb{F}_q$, with $m$ and $n$ growing while the finite field $\mathbb{F}_q$ stays the same.  This can be expressed as a problem about the trace of Frobenius acting on the etal cohomology of a space whose complex points are a $K(\pi,1)$ for a certain finite-index subgroup of a colored braid group; it seems to me that the behavior we expect to see for these averages would follow from a good result on secondary homological stability for these subgroups.  The question is whether the assembled topological might of this workshop can help figure out whether such a statement is true and provable with current methods.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.