Polylogarithms, homology of linear groups, and Steinberg modules

Contribution List

1. Polylogarithms, K-theory, and motives I

Clement Dupont

6/9/25, 9:00 AM

In this mini-course I will try and paint a general landscape where polylogarithms interact with algebraic K-theory via the formalism of motives. The goal will be to explain the idea behind Goncharov’s program, which aims at computing the rational K-theory of fields via explicit complexes involving formal versions of multiple polylogarithms. The emphasis will be put on concrete computations...

10. $E_\infty$-algebras of general linear groups I

Oscar Randal-Williams

6/9/25, 10:00 AM

For a field $F$, the collection of all group chains $C_*(GL_n(F) ; k)$ of general linear groups over $F$ assemble into an $E_\infty$-algebra $\mathbf{BGL}$, which is equipped with an $\mathbb{N}$-grading by the rank $n$: the multiplicative structure is induced by block-sum of matrices. (Something similar can be done with $F$ replaced by more general rings.) One may and should treat this object...

2. Polylogarithms, K-theory, and motives II

Clement Dupont

6/9/25, 11:30 AM

In this mini-course I will try and paint a general landscape where polylogarithms interact with algebraic K-theory via the formalism of motives. The goal will be to explain the idea behind Goncharov’s program, which aims at computing the rational K-theory of fields via explicit complexes involving formal versions of multiple polylogarithms. The emphasis will be put on concrete computations...

16. The Steinberg module and the Church--Farb--Putman conjecture

Jennifer Wilson

6/9/25, 5:00 PM

In this talk, I will introduce the Steinberg module, and describe its relationship to a conjecture of Church--Farb--Putman on the high-degree rational cohomology of SL_n(Z). The talk will feature work joint with Benjamin Brück, Jeremy Miller, Peter Patzt, and Robin Sroka.

14. Cocycles for GL_n and graph complexes from differential forms

Erik Panzer

6/9/25, 6:00 PM

We study invariant differential forms on the space of positive definite matrices. Integration of these forms gives rise to functionals that satisfy relations due to Stokes’ theorem. In certain degrees, these relations allow us to interpret the integrals as cocycles for the general linear group or graph complexes. This construction explains many new cohomology classes. Perhaps surprisingly, for...

11. $E_\infty$-algebras of general linear groups II

Oscar Randal-Williams

6/10/25, 9:00 AM

For a field $F$, the collection of all group chains $C_*(GL_n(F) ; k)$ of general linear groups over $F$ assemble into an $E_\infty$-algebra $\mathbf{BGL}$, which is equipped with an $\mathbb{N}$-grading by the rank $n$: the multiplicative structure is induced by block-sum of matrices. (Something similar can be done with $F$ replaced by more general rings.) One may and should treat this object...

3. Polylogarithms, K-theory, and motives III

Clement Dupont

6/10/25, 10:00 AM

In this mini-course I will try and paint a general landscape where polylogarithms interact with algebraic K-theory via the formalism of motives. The goal will be to explain the idea behind Goncharov’s program, which aims at computing the rational K-theory of fields via explicit complexes involving formal versions of multiple polylogarithms. The emphasis will be put on concrete computations...

12. $E_\infty$-algebras of general linear groups III

Oscar Randal-Williams

6/10/25, 11:30 AM

For a field $F$, the collection of all group chains $C_*(GL_n(F) ; k)$ of general linear groups over $F$ assemble into an $E_\infty$-algebra $\mathbf{BGL}$, which is equipped with an $\mathbb{N}$-grading by the rank $n$: the multiplicative structure is induced by block-sum of matrices. (Something similar can be done with $F$ replaced by more general rings.) One may and should treat this object...

20. Tropicalizations of locally symmetric varieties

Juliette Bruce

6/10/25, 5:00 PM

I will discuss recent work constructing tropicalizations of locally symmetric varieties. Beyond being of interest just in tropical geometry, I will discuss how such tropicalizations have applications to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups.

17. The common basis complex and the partial decomposition poset

Benjamin Brück

6/10/25, 6:00 PM

With Piterman-Welker, we showed that Rognes' common basis complex is homotopy equivalent to a certain poset of partial decompositions. I will give an idea of our mostly combinatorial argument and also mention other contexts to which it applies, such as symplectic groups, automorphisms of free groups and matroids.

7. Multiple polylogarithms and the Steinberg module I

Danylo Radchenko

6/11/25, 9:00 AM

In this minicourse I will explain a surprising connection between multiple polylogarithms and the Steinberg module. In the first lecture I will outline Goncharov's programme relating multiple polylogarithms and algebraic K-theory of a field, describe the Hopf algebra structure for multiple polylogarithms and the basic properties of its different families of generators (multiple polylogarithms,...

4. Algebraic structures of Steinberg modules 1: Ring structure

Jeremy Miller

6/11/25, 10:00 AM

I will begin by reviewing the definition of Steinberg modules and apartments. By concatenating apartments, one can endow the direct sum of all Steinberg modules with the structure of an associative ring. I will also describe a space-level construction of this product due to Galatius—Kupers—Randal-Williams. This ring is not graded-commutative. However, it is commutative in an equivariant sense....

13. Cusp forms and the depth filtration

Richard Hain

6/11/25, 11:30 AM

The goal of this talk is to explain the mechanism by which cusp forms of certain congruence subgroups of SL₂(Z) impose relations in the depth filtration of the motivic fundamental group of the category of mixed Tate motives unramified over the ring Z[mu_N,1/N]. The known depth 2 relations in the case N=1, first observed by Ihara and Takao, were proved by Francis Brown and...

5. Algebraic structures of Steinberg modules 2: Indecomposables

Jeremy Miller

6/12/25, 9:00 AM

I will begin by describing a conjecture of Rognes regarding high connectivity of a certain simplicial complex called the common basis complex. This complex appears when studying rank filtrations of algebraic K-theory. I will describe the relationship between this simplicial complex and Steinberg modules. In particular, the homology of the common basis complex computes the derived commutative...

8. Multiple polylogarithms and the Steinberg module II

Danylo Radchenko

6/12/25, 10:00 AM

In the second lecture, following a joint work with Charlton and Rudenko, I will define certain spaces of multiple polylogarithms on algebraic tori, and show that they are isomorphic to the tensor square of the Steinberg module of rationals. I will discuss some implications of this isomorphism both for polylogarithms and for the Steinberg module.

9. Multiple polylogarithms and the Steinberg module III

Danylo Radchenko

6/12/25, 11:30 AM

Finally, in the third lecture, following a work-in-progress by Kupers-Rudenko-Sierra, I will talk about a conjectural relation between the first homology group of GL with coefficients in (the space of indecomposables of) the tensor square of the Steinberg module and the conjectural motivic Lie coalgebra from Goncharov's programme.

19. Goncharov’s conjecture and higher Chow group

Vasily Bolbochan

6/12/25, 5:00 PM

I will talk about my recent result (arXiv:2404.06271) which states that for any n, the cohomology of polylogarithmic complex in degree (n-1) and weight n is isomorphic to the appropriate graded piece of algebraic K-theory. This gives a new case of Goncharov’s conjecture stating that graded pieces of algebraic K-theory should be isomorphic to the cohomology of the polylogarithmic...

21. The Rognes rank spectral sequence and Goncharov program via $E_\infty$-homology.

Ismael Sierra

6/12/25, 6:00 PM

In this talk I will explain ongoing work with Kupers and Rudenko on a new approach to understanding the Rognes rank spectral sequence and the Goncharov program using the $E_\infty$-homology of the $E_\infty$-algebra associated to symmetric monoidal category of vector spaces over a field. One of the new ideas is to compute the Koszul dual Lie cobracket on the indecompodables of this algebra and...

18. Goncharov's programme, and symmetries of weight 6 multiple polylogarithms

Steven Charlton

6/13/25, 9:05 AM

In his programme to investigate Zagier's Polylogarithm Conjecture on the values of $\zeta_F(m)$, Goncharov gave a conjectural criterion -- the Depth Conjecture -- to determine the depth (number of variables) of a linear combination of multiple polylogarithms using the motivic coproduct. I will give an overview of this conjecture and its implications; in particular this Conjecture explains why...

6. Algebraic structures of Steinberg modules 3: Hopf algebras

Jeremy Miller

6/13/25, 10:00 AM

I will describe how Steinberg modules not only form a ring but in fact form a bi-algebra in a “duoidal” sense. This endows the homology of general linear groups with Steinberg module coefficients with the structure of a Hopf algebra (work joint with Ash and Patzt, independently work of Brown—Chan—Galatius—Payne). I will describe applications to the unstable cohomology of SL_n(Z) and...

15. Fun with Tits buildings

Cary Malkiewich

6/13/25, 11:30 AM

I'm going to talk about a couple of cool facts about affine and linear Tits buildings for the real numbers. One is that you can get a Solomon-Tits theorem if you take any collection of hyperplanes and their intersections, rather than taking all subspaces. The other is that if you suspend the Tits building twice, then the apartments become cubes, and this leads to a beautiful geometric picture...