Video Lectures

Erd\H{o}s unit distance problem asks the following: Let P

be a set of n distinct points in the plane, and let U(P)

denote the number of pairs of points in P that are at distance 1. How large can U(P) be? In 1946, Erd\H{o}s showed that for P=[n‾√]×[n‾√...

We show that for all functions t(n)≥n, every multitape Turing machine running in time t can be simulated in space only O(tlogt‾‾‾‾‾√). This is a substantial improvement over Hopcroft, Paul, and Valiant's simulation of time t in O(t/logt) space from...

Geometric and functional inequalities are fundamental in various mathematical areas, such as the calculus of variations, partial differential equations, and geometry. Classic examples encompass the isoperimetric inequality, Sobolev inequalities, and...

Geometric and functional inequalities are fundamental in various mathematical areas, such as the calculus of variations, partial differential equations, and geometry. Classic examples encompass the isoperimetric inequality, Sobolev inequalities, and...

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced...

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced...

The open Gromov-Witten (OGW) potential is a function defined  on the Maurer-Cartan space of a closed Lagrangian submanifold in a  symplectic manifold with values in the Novikov ring. From the values  of the OGW potential, one can extract so-called...

The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space Hn. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on...

Boosting is a fundamental and widely used method in machine learning, which determines that weak learnability of binary functions implies strong learnability. Traditionally, boosting theory has primarily focused on symmetric 0-1 loss functions...

Geometric and functional inequalities are fundamental in various mathematical areas, such as the calculus of variations, partial differential equations, and geometry. Classic examples encompass the isoperimetric inequality, Sobolev inequalities, and...