## Introductory Remarks by Alondra Nelson

Alondra Nelson

IAS School of Social Science

November 6, 2019

Emma Brunskill

Stanford University

November 7, 2019

Yuandong Tian

Facebook AI

November 7, 2019

Csaba Szepesvari

DeepMind & University of Alberta

November 7, 2019

Raffaella Margutti

Northwestern University

November 5, 2019

Baurzhan Mukhametzhanov

Member, School of Natural Sciences, IAS

November 1, 2019

We analyze modular invariance drawing inspiration from tauberian

theorems. Given a modular invariant partition function with a positive

spectral density, we derive lower and upper bounds on the number of

operators within a given energy interval. They are most revealing at high

energies. In this limit we rigorously derive the Cardy formula for the

microcanonical entropy together with optimal error estimates for various

widths of the averaging energy shell. Finally, we identify a new universal

Andrey Kupavskii

Member, School of Mathematics

November 5, 2019

Extremal set theory typically asks for the largest collection of sets satisfying certain constraints. In the first talk of these series, I'll cover some of the classical results and methods in extremal set theory. In particular, I'll cover the recent progress in the Erdos Matching Conjecture, which suggests the largest size of a family of k-subsets of an n-element set with no s pairwise disjoint sets.

Sara Tukachinsky

Member, School of Mathematics

November 4, 2019

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with various constraints on boundary and interior marked points. The presence of boundary poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions.

Anna Gilbert

University of Michigan; Member, School of Mathematics

November 4, 2019

In this work, we exploit the ill-posedness of linear inverse

problems to design algoithms to release differentially private data or

measurements of the physical system. We discuss the spectral

requirements on a matrix such that only a small amount of noise is

needed to achieve privacy and contrast this with the poor conditioning

of the system. We then instantiate our framework with several

diffusion operators and explore recovery via l1 constrained

minimisation. Our work indicates that it is possible to produce

Andrey Kupavskii

Member, School of Mathematics

October 29, 2019