A nearly optimal lower bound on the approximate degree of AC$^0$

Mark Bun
Princeton University
October 23, 2017

The approximate degree of a Boolean function $f$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. For any constant $\delta > 0$, we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi, and nearly matches the trivial upper bound of $n$ that holds for any function.

Geometry and arithmetic of sphere packings

Alex Kontorovich
Rutgers University
October 23, 2017
We introduce the notion of a "crystallographic sphere packing," which generalizes the classical Apollonian circle packing. Tools from arithmetic groups, hyperbolic geometry, and dynamics are used to show that, on one hand, there is an infinite zoo of such objects, while on the other, there are essentially finitely many of these, in all dimensions. No familiarity with any of these topics will be assumed.

Wrapped Fukaya categories and functors

Yuan Gao
Stonybrook University
October 23, 2017
Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated.

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.

Elliptic curves of rank two and generalised Kato classes

Francesc Castella
Princeton University
October 24, 2017
The generalised Kato classes of Darmon-Rotger arise as $p$-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over $Q$ of rank two. In this talk, we will report on a joint work in progress with Ming-Lun Hsieh concerning a special case of the conjectures of Darmon-Rotger.

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.