School of Mathematics

Integral points and curves on moduli of local systems

Junho Peter Whang
Princeton University
December 8, 2017
The classical affine cubic surface of Markoff has a well-known interpretation as a moduli space for local systems on the once-punctured torus. We show that the analogous moduli spaces for general topological surfaces form a rich family of log Calabi-Yau varieties, where a structure theorem for their integral points can be established using mapping class group descent. Related analysis also yields new results on the arithmetic of algebraic curves in these moduli spaces, including finiteness of imaginary quadratic integral points for non-special curves.

Spectral gaps without frustration

Marius Lemm
California Institute of Technology; Member, School of Mathematics
December 6, 2017
In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include physically relevant examples. We discuss "finite-size criteria", which allow to bound the spectral gap of the infinite system by the spectral gap of finite subsystems. We focus on the connection between spectral gaps and boundary conditions. Joint work with E. Mozgunov.

Motivic correlators and locally symmetric spaces IV

Alexander Goncharov
Yale University; Member, School of Mathematics and Natural Sciences
December 5, 2017

According to Langlands, pure motives are related to a certain class of automorphic representations.

Can one see mixed motives in the automorphic set-up? For examples, can one see periods of mixed motives in entirely automorphic terms? The goal of this and the next lecture is to supply some examples.

We define motivic correlators describing the structure of the motivic fundamental group $\pi_1^{\mathcal M}(X)$ of a curve. Their relevance to the questions raised above is explained by the following examples.

Automorphy for coherent cohomology of Shimura varieties

Jun Su
Princeton University
December 5, 2017
We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings’ B-G-G spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.

Algebraic combinatorics: applications to statistical mechanics and complexity theory

Greta Panova
University of Pennsylvania; von Neumann Fellow, School of Mathematics
December 4, 2017
We will give a brief overview of the classical topics, problems and results in Algebraic Combinatorics. Emerging from the representation theory of $S_n$ and $GL_n$, they took a life on their own via the theory of symmetric functions and Young Tableaux and found applications into new fields. In particular, these objects can describe integrable lattice models in statistical mechanics like dimer covers on the hexagonal grid, aka lozenge tilings.

Open Gromov-Witten theory of $(\mathbb{CP}^1,\mathbb{RP}^1)$ in all genera and Gromov-Witten Hurwitz correspondence

Amitai Zernik
Member, School of Mathematics
December 4, 2017

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to $(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

Locally symmetric spaces: $p$-adic aspects

Laurent Fargues
Institut de Mathématiques de Jussieu
November 30, 2017
$p$-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic $p$-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve.

Nonuniqueness of weak solutions to the Navier-Stokes equation

Tristan Buckmaster
Princeton University
November 29, 2017
For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy.