William Chen
TITLE: When can we describe a branched cover of an elliptic curve in terms of torsion points?
ABSTRACT: Let G be a finite group. For a G-Galois cover f of an elliptic curve ramified at most over the origin, the moduli space of G-covers of elliptic curves of the same topological type as f is a quotient of the upper half plane by a finite index subgroup $\Gamma_f$ of SL(2,Z). When this subgroup is a congruence subgroup, covers of this topological type can be encoded in terms of torsion data. If G is abelian, f must be unramified, and hence $\Gamma_f$ is congruence. From empirical data, highly nonabelian groups G typically give rise to highly noncongruence subgroups $\Gamma_f$. However, the line between congruence and noncongruence seems to be remarkably subtle. In this talk we will describe some recent progress on this front. In particular, we will give an infinite family of nonabelian simple covers with congruence moduli spaces. This uses an exceptional relationship between SL(2,Z) and the Artin braid group B_4, first due to Dyer, Formanek, and Grossman, which we then apply to the Burau representation to obtain covers with Galois group PSU(3,q) and PSL(3,q). This is joint work with Alexander Lubotzky, Pham Huu Tiep, and Nick Salter.
Wanlin Li
TITLE: Non-vanishing of Ceresa and Gross–Kudla–Schoen cycles
ABSTRACT: The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle are algebraic 1-cycles associated to a smooth algebraic curve. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus >2. Given an algebraic curve, it is an interesting question to study whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem. Moreover, I will discuss some recent exciting developments on the study of algebraic cycles stemmed from the interest on these two cycles.
Eduard Looijenga
TITLE: The cohomology of the configuration space of an open surface
ABSTRACT: By an open surface we mean the complement of a finite nonempty subset of a closed, oriented, connected surface. The cohomology of the configuration space of n (numbered) points of such a surface is very much of interest in its own right, but has also an `external’ interest (for example, these groups, or twisted versions of them, appear in the theory of conformal blocks). This cohomology is naturally acted on by the mapping class group of the surface, but is as a representation of the latter still poorly understood.
In this talk we present joint work with Andreas Stavrou about the structure of this representation.
Among other things, we show that these cohomology groups come with a natural filtration of length n that is compatible with the Johnson filtration of the mapping class group. This implies that that the kernel of this representation contains the n-th member of the Johnson filtration. The latter is due to Bianchi-Miller-Wilson in the case of a single puncture, but we also find that this inclusion is in general strict, even in the single punctured case, thus refuting a conjecture they had made.
Anand Patel:
TITLE: On some special divisors in Hurwitz space
ABSTRACT: Motivated by the old problem of exhibiting low slope effective divisors on the moduli space of curves, my objective in this talk is to explore some intriguing questions about the codimension 1 geometry of Hurwitz spaces parametrizing branched covers of elliptic curves. Familiarity with the topic will not be assumed.
Andy Putman
TITLE: The second rational cohomology of the Torelli group
ABSTRACT: I will discuss some recent work with Dan Minahan in which we compute the second rational cohomology group of the Torelli group.
Tomer Schlank
TITLE: Random links and arithmetic statistics
ABSTRACT: Étalé homotopy is a theory that enables the study of algebraic and arithmetic concepts through geometric perspectives. Barry Mazur observed that, from this viewpoint, square-free integers are analogous to links in three-dimensional space. We will explore this analogy and propose a way to give it a statistical flavor. Informally, we assert that a random square-free integer of size approximately X resembles the closure of a random braid on roughly logX strands. We will make this statement more precise by introducing a number-theoretical analog for a family of numerical link invariants (called Kei’s) and use this analogy to propose a conjecture regarding their asymptotic behavior. We will also present a few cases where this conjecture has been proven. This is a joint work with Ariel Davis.
Alexandru Sucui
TITLE: Lower central series and Alexander invariants in group extensions
ABSTRACT: I will present a study of the lower central series, the Alexander invariants, and the cohomology jump loci of groups occurring as extensions with trivial monodromy in first homology with appropriate coefficients. I will illustrate these concepts with examples arising from the Bestvina-Brady groups associated to right-angled Artin groups and with the Milnor fibrations of complements of hyperplane arrangements.
Jesse Wolfson
TITLE: Hilbert’s 13th problem for braid groups
ABSTRACT: Resolvent problems all ask some version of the question, “How hard is it to solve some algebraic problem?” While this originated in the context of polynomials in one variable, following work of Buhler, Reichstein, Merkurjev, Farb and the speaker, the context was broadened to include finite groups, local systems, algebraic groups, birational invariants and more. In this framework, Hilbert’s 13th problem asks one to prove that RD(A_7)=3, and one could ask more generally about the resolvent degree RD(G) for any finite simple group G. Recent work of Reichstein proves that, in contrast to Hilbert’s expectations for finite groups, for *connected* algebraic groups G, RG(G)\le 5, and, assuming a conjecture of Tits, RD(G) is at most 1 for any connected algebraic group. In contrast, we describe ongoing joint work on the resolvent degree of finitely generated groups, and a proof-in-progress that RD(B_n)=n-2, for n\ge 5. Taken together, these results suggest the following rephrasing of Hilbert’s 13th: which is S_n more like, the connected algebraic group GL_n, or the braid group B_n? Some of the ideas in this project originated in joint work with Farb and Kisin. The current effort is joint work with Claudio Gómez-Gonzáles and Sidhanth Raman.
Chenglong Yu
TITLE: Commensurability among Deligne–Mostow Monodromy Groups
ABSTRACT: Deligne–Mostow constructed lattices in PU(1,n) as monodromy groups of certain hypergeometric functions. This provides examples of nonarithmetic lattics in PU(1,2) and PU(1,3). Thurston obtained the same groups via flat conic metrics on two-sphere. In this talk, we give the commensurability classification of Deligne–Mostow ball quotients and show that the 104 Deligne–Mostow lattices form 38 commensurability classes. Firstly, we find commensurability relations among Deligne–Mostow monodromy groups, which are not necessarily discrete. This recovers and generalizes previous work by Sauter and Deligne–Mostow in dimension two. In this part, we consider certain projective surfaces with two fibrations over the projective line, which induce two sets of Deligne–Mostow data. The correspondences among moduli spaces provide the geometric realizations of commensurability relations. Secondly, we obtain commensurability invariants from conformal classes of Hermitian forms and toroidal boundary divisors. This completes the commensurability classification of Deligne–Mostow lattices and also reproves Kappes–M\”oller and McMullen’s results on nonarithmetic Deligne–Mostow lattices. The talk is based on joint work with Zhiwei Zheng (YMSC, Tsinghua University).
Xiyan Zhong
TITLE: Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures
ABSTRACT: The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-ℓ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-ℓ mapping class group with coefficients in the Prym representation, and more generally in the r-tensor powers of the Prym representation. Our results also show that when r ≥ 2, this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus g.
