School of Mathematics

Motivic correlators and locally symmetric spaces III

Alexander Goncharov
Yale University; Member, School of Mathematics and Natural Sciences
October 31, 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.

Nonlinear descent on moduli of local systems

Junho Peter Whang
Princeton University
October 31, 2017
In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of nonlinear descent. Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics.

Fooling intersections of low-weight halfspaces

Rocco Servedio
Columbia University
October 30, 2017

A weight-$t$ halfspace is a Boolean function $f(x)=\mathrm{sign}(w_1 x_1 + \cdots + w_n x_n - \theta)$ where each $w_i$ is an integer in $\{-t,\dots,t\}.$  We give an explicit pseudorandom generator that $\delta$-fools any intersection of $k$ weight-$t$ halfspaces with seed length poly$(\log n, \log k,t,1/\delta)$. In particular, our result gives an explicit PRG that fools any intersection of any quasipoly$(n)$ number of halfspaces of any polylog$(n)$ weight to any $1/$polylog$(n)$ accuracy using seed length polylog$(n).$

High density phases of hard-core lattice particle systems

Ian Jauslin
Member, School of Mathematics
October 30, 2017
In this talk, I will discuss the behavior of hard-core lattice particle systems at high fugacities. I will first present a collection of models in which the high fugacity phase can be understood by expanding in powers of the inverse of the fugacity. I will then discuss a model in which this expansion diverges, but which can still be solved by expanding in other high fugacity variables. This model is an interacting dimer model, introduced by O.Heilmann and E.H.Lieb in 1979 as an example of a nematic liquid crystal.

Weinstein manifolds through skeletal topology

Laura Starkston
Stanford University
October 30, 2017
We will discuss how to study the symplectic geometry of $2n$-dimensional Weinstein manifolds via the topology of a core $n$-dimensional complex called the skeleton. We show that the Weinstein structure can be homotoped to admit a skeleton with a unique symplectic neighborhood. Then we further work to reduce the remaining singularities to a simple combinatorial list coinciding with Nadler's arboreal singularities.

Nematic liquid crystal phase in a system of interacting dimers

Ian Jauslin
Member, School of Mathematics
October 25, 2017
In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.

On the strength of comparison queries

Shay Moran
University of California, San Diego; Member, School of Mathematics
October 24, 2017

Joint work with Daniel Kane (UCSD) and Shachar Lovett (UCSD)

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry.

For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n \log^2 n)$ linear queries. This settles a problem studied by [Meyer auf der Heide ’84, Meiser ‘93, Erickson ‘95, Ailon and Chazelle ‘05, Gronlund and Pettie '14, Gold and Sharir ’15, Cardinal et al '15, Ezra and Sharir ’16] and others.

Motivic correlators and locally symmetric spaces II

Alexander Goncharov
Yale University; Member, School of Mathematics and Natural Sciences
October 24, 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.