## IAS Facility Video

## Continuous covers on symplectic manifolds

## The simplification of caustics

## Approximate counting and the Lovasz local lemma

We introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random.

## Efficient non-convex polynomial optimization and the sum-of-squares hierarchy

The sum-of-squares (SOS) hierarchy (due to Shor'85, Parrilo'00, and Lasserre'00) is a widely-studied meta-algorithm for (non-convex) polynomial optimization that has its roots in Hilbert's 17th problem about non-negative polynomials.

SOS plays an increasingly important role in theoretical computer science because it affords a new and unifying perspective on the field's most basic question:

What's the best possible polynomial-time algorithm for a given computational problem?

## Real Lagrangians in toric degenerations

manifolds with singularities in the joint work with Mark Gross, provide an ample source of

examples of Lagrangians that conjecturally are amenable to algebraic-geometric versions of

Floer theory. In the talk I will discuss joint work with Hülya Argüz on how the topology of the real

locus can be understood by means of the affine geometry and by Kato-Nakayama spaces

associated to log spaces.

## Lattices, Filtrations and some Applications

from a new prospective. Applications to geometry will be considered.

## Calabi-Yau geometry and quantum B-model

## Moduli spaces of elliptic curves in toric varieties

geometry and mirror symmetry. While they have several pleasing properties, they are often

quite singular, reducible, and non-equidimensional. When the source curves have genus 0, the

situation is markedly improved by adding logarithmic structure to the moduli problem. This

produces irreducible and non-singular moduli spaces of rational curves in toric varieties, whose