NOTE: Later start

Abstract: Minkowski's celebrated first theorem is one of the foundational results in the study of the geometry of numbers, and it has innumerable applications from basic number theory to convex geometry to cryptography. It tells us that a lattice (i.e., a linear transformation of Z^n) that is globally dense (i.e., has low determinant) must be locally dense (i.e., must have many short vectors). We will show a proof of a nearly tight converse to Minkowski's theorem, originally conjectured by Daniel Dadush, that is, that a lattice with many short points must have a sublattice with small determinant. This "reverse Minkowski theorem" has numerous applications in, e.g., complexity theory, additive combinatorics, cryptography, the study of Brownian motion on flat tori, algorithms for lattice problems, etc.

Based on joint work with Oded Regev.