All talks will take place in David Rittenhouse Laboratory (DRL) room A1 on the first floor. Abstracts will appear below the schedule. The address for the math building, DRL is 209 South 33rd St, Philadelphia.
Monday
9:30-10 Donuts and coffee + Registration
10-11 Featured talk 1: Inna Zakharevich
Cutting apart, putting together
11-11:30 Coffee break
11:30-12:30 Featured talk 2: Jeremy Miller
Scissors congruence automorphism groups
12:30 - 2:30 Lunch break
2:30-3:30 Collaborative Workshop talk 1: Josefien Kuijper, David Mehrle, Thor Wittich
Spherical scissors congruence
3:45-5 Wine and cheese
Tuesday
9:30-10 Donuts and coffee
10-11 Featured Talk 3: Daniil Rudenko
Multiple polylogarithms, algebraic K-theory, and the Steinberg module
11-11:30 Coffee break
11:30-12:30 Collaborative Workshop talk 2: Sanjana Agarwal, Diego Manco, Zhonghui Sun
Witt vectors, bicategorical traces and K-theory
12:30 - 2:30 Lunch break
2:30-3:30 Featured Talk 4: Jenny Wilson
The Steinberg module and the rational cohomology of SL_n(Z)
3:30-4 Coffee and dessert break
4-5 Collaborative Workshop talk 3: Alba Sendón Blanco, Ming Ng, Lucas Williams
Scissors congruence K-theory for equivariant manifolds
7 Conference Dinner
Wednesday
9:30-10 Donuts and coffee
10-11 Featured talk 5: Jesse Wolfson
Derived motivic integration
11-11:30 Coffee break
11:30-12:30 Collaborative Workshop talk 4: Tatiana Abdelnaim, David Chan, Matt Scalamandre
Double Steinberg coinvariants
Free afternoon
Thursday
9:30-10 Donuts and coffee
10-11 Collaborative Workshop talk 5: Myungsin Cho, Liam Keenan, Juan Moreno
Homological trace methods for Real topological Hochschild homology
11-11:30 Coffee break
11:30-12:30 Featured Talk 6: Cary Malkiewich
Open problems in scissors congruence
Cary Malkiewich
Open problems in scissors congruence
Abstract: It is a longstanding open problem to classify all five-dimensional Euclidean polyhedra up to scissors congruence. Although I don't know how to solve that problem at the moment, I can tell you about several related problems whose solutions we do know, and a few more that I think someone could solve in the next few years. The audience should feel free to suggest problems as well, and if there is time at the end we can have a more open-ended discussion.
Jeremy Miller
Scissors congruence automorphism groups
Abstract: Two polytopes are called scissors congruent if they can be cut up into isometric pieces. The question of classifying polytopes up to scissors congruence dates back to Hilbert. In joint work with Kupers, Lemann, Malkiewich, and Sroka, we study an orthogonal question, investigating the group of scissors congruence automorphisms of a fixed polytope. We prove a homological stability result for these groups. This yields a new model of Zakharevich’s scissors congruence spectrum. Combining our result with work of Malkiewich on Tits building models of the scissors congruence spectra yields an approach to calculating the homology of these groups and their variants.
Daniil Rudenko
Multiple polylogarithms, algebraic K-theory, and the Steinberg module
Abstract: Multiple polylogarithms appear to be central to many seemingly unrelated areas of mathematics, including the volumes of hyperbolic polytopes, scissors congruence, algebraic K-theory, and special values of zeta functions. Despite this broad network of connections, the most fundamental properties of these functions, as predicted by the Goncharov program, remain conjectural. I will discuss recent progress in the Goncharov program, which is based on the connection between multiple polylogarithms and the Steinberg module. The talks are based on joint work with Steven Charlton and Danylo Radchenko, as well as on an ongoing work with Alexander Kupers and Ismael Sierra.
Jenny Wilson
The Steinberg module and the rational cohomology of SL_n(Z)
Abstract: In this talk, I will survey the role of the Steinberg module in some recent work: on a conjecture due to Church--Farb--Putman on the cohomology of the special linear group, and on a conjecture of Rognes related to algebraic K-theory. I will describe these conjectures and some partial progress, which includes work joint with B. Brueck, J. Miller, P. Patzt, and R. Sroka.
Jesse Wolfson
Derived motivic integration
Abstract: Motivic integration was introduced by Kontsevich in order to prove that the Hodge numbers of birational smooth Calabi-Yaus are equal, and it remains one of the most powerful tools to analyze the class of a variety in the Grothendieck ring of varieties. In this talk, I will discuss joint work-in-progress to lift motivic integration to a map of spectra taking values in Zakharevich's K-theory spectrum of varieties, with the goal of constructing new invariants of birational automorphisms of smooth Calabi-Yaus. This began in 2015 as joint work with Oliver Braunling, Michael Groechenig and Inna Zakharevich, ground to a premature halt for life reasons (too many babies, too few jobs . . . pandemic), and is now being revived with Anubhav Nanvaty.
Inna Zakharevich
Cutting apart, putting together
Abstract: In this talk we give a historical overview of scissors congruence, starting with classical Euclidean scissors congruence and Hilbert's third problem, continuing to more involved example such as McMullen's polytope algebra and total scissors congruence, and finishing with more modern and abstract ideas about cutting and pasting.
Tatiana Abdelnaim, David Chan, and Matthew Scalamandre
Double Steinberg Coinvariants
Abstract: The Steinberg module $St$ is a representation of the general linear group of a field, encoding important information about the structure of that group. In recent work, Galatius—Kupers--Randal-Williams study the GL-coinvariants of $St\otimes St$. Using the theory of $E_k$ cells, they apply this computation to study the homology of GL. In this talk we discuss work in progress (joint with Kupers and Sroka), on extending these ideas and computations to the special linear group SL. In particular, we will show that in many cases the coinvariants are significantly larger than in the GL case, and discuss complications arising from differences in monoidal structures.
Sanjana Agarwal, Diego Manco, and Zhonghui Sun
Frobenius and Verschiebung on K-theory of endomorphisms
Abstract: The ring of Witt vectors associated to a commutative ring R, denoted by W(R), is a highly structured object that crucially shows up in many different fields. Yet, this structure can feel mysterious and opaque. The K-theory of endomorphisms of R provide a natural home for understanding the Witt vector structure in simple linear algebra terms. In this talk, we explain this perspective and extend it to the K-theory of twisted endomorphisms over noncommutative rings, where structures like Frobenius and Verschiebung still arise naturally.
Myungsin Cho, Liam Keenan, and Juan Moreno
Homological trace methods for Real topological Hochschild homology
Abstract: In their foundational work, Bruner and Rognes developed a homological approach to study topological Hochschild homology (THH) and related invariants, where the input for calculations is now homology rather than homotopy (in practice, we often have access to the former rather than the latter). We will discuss a C₂-equivariant analogue of this homological program, with a focus on real forms of THH and topological negative cyclic homology (TC⁻). Central to our approach is a “real” analogue of the homotopy fixed point spectral sequence (HFPSS) for the circle group, which has multiplicative structure similar to that of the usual HFPSS. However, equivariant multiplicativity is rather subtle, leading to complications in mirroring the program of Bruner and Rognes; for example, equivariant extended power constructions are more difficult to analyze than their nonequivariant counterparts.
Josefien Kuijper, David Mehrle, and Thor Wittich
Higher Spherical Scissors Congruence
Abstract: A central question in Euclidean, spherical, or hyperbolic geometry is scissors congruence of polytopes: given two polytopes, can they be cut into finitely many pieces and rearranged into each other? If so, such polytopes must have the same volume, but does the converse hold as well? In 3-dimensional Euclidean space, this is Hilbert’s famous third problem, which was swiftly resolved by Dehn in the negative.
In spherical geometry, classical work of Sah shows that polytopes can be assembled into a commutative Hopf algebra, now called the Sah algebra. A modern point of view on scissors congruence, due to Zakharevich, recasts scissors congruence in terms of K-theory spectra. We assemble the spherical scissors congruence K-theory spectra into a spectral version of the Sah algebra, and lift the Hopf algebra structure to the spectral level.
Alba Sendón Blanco, Ming Ng, and Lucas Williams
Scissors congruence K-theory for equivariant manifolds
Abstract: Suppose you have a manifold M, a pair of scissors and some glue. When can you cut M into pieces and reglue them together to get another manifold N? The answer: this is possible so long as the manifolds M and N have the same boundary and Euler characteristic. But in the presence of a group action on the manifold, the picture becomes more complicated. Even for finite abelian G, we lack a full set of invariants that determine equivariant cut-and-paste of G-manifolds. Progress on this question has largely stalled since the 1970s — it seems that new ideas are needed.
In this talk, we revisit the equivariant cut-and-paste problem through the lens of homotopy theory. We describe how the basic features of the problem lead naturally to a K-theoretic framework. In our language: there exists a K-theory spectrum whose pi_0 recovers the equivariant cut-and-paste groups. In fact, we can lift the Burnside ring-valued equivariant Euler characteristic to the level of spectra. Moreover, as we vary over subgroups of G, the resulting cut-and-paste groups assemble into a Mackey functor — this provides evidence for a conjectural higher genuine equivariant structure.