## History of Science Lecture Series: Race by Numbers

Projit Bihari Mukharji

University of Pennsylvania

April 2, 2019

Li-Yang Tan

Stanford University

April 1, 2019

We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \mathrm{log}(n)$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program. Joint work with Ryan O'Donnell and Rocco Servedio.

Viswambhara Makam

Member, School of Mathematics

April 1, 2019

Invariant theory is a fundamental subject in mathematics, and is potentially applicable whenever there is symmetry at hand (group actions). In recent years, new problems and conjectures inspired by complexity have come to light. In this talk, I will describe some of these new problems, and discuss some positive and negative results regarding them.

Ben Filippenko

University of California, Berkeley

April 1, 2019

Liam McAllister

Cornell University

April 1, 2019

Xin Zhou

University of California, Santa Barbara; Member, School of Mathematics

March 27, 2019

I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one.

Matthew Stover

Temple University

March 27, 2019

Abstract: I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups, "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.

Patricia Clavin

March 27, 2019

Henry Wilton

Cambridge University

March 26, 2019

Abstract: There seems to be an analogy between the classes of fundamental groups of compact 3-manifolds and of one-relator groups. (Indeed, many 3-manifold groups are also one-relator groups.) For instance, Dehn’s Lemma for 3-manifolds (proved by Papakyriakopoulos) can be seen as analogous to Magnus’ Freiheitssatz for one-relator groups. But the analogy is still very incomplete, and since there are deep results on each side that have no analogue on the other, there is a strong incentive to flesh it out.

Shubhangi Saraf

Member, School of Mathematics

March 26, 2019

Are factors of sparse polynomials sparse? This is a really basic question and we are still quite far from understanding it in general.

In this talk, I will discuss a recent result showing that this is in some sense true for multivariate polynomials when the polynomial has each variable appearing only with bounded degree. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Caratheodory's Theorem.