Video Lectures

A construction of Thurston assigns a hyperbolic 3-manifold to any polyhedron; a natural question is: which such are arithmetic? We report on ongoing work aiming to answer this question.

The ring of symmetric functions has a linear basis of Schur functions sλ indexed by partitions λ=(λ1≥λ2≥…≥0). Littlewood-Richardson coefficients cνλ,μ are the structure constants of such a basis.

A function is Schur nonnegative if it is a linear...

A remarkable result of Brändén and Huh tells us that volume polynomials of projective varieties are Lorentzian polynomials. The dual notion of covolume polynomials was introduced by Aluffi by considering the cohomology classes of subvarieties of a...

Motivated by the quantum unique ergodicity conjecture, we examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic n

-manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic...

In this talk, I will discuss some results concerning the geometry and topology of manifolds on which the first eigenvalue of the operator -γΔ + Ric is bounded below. Here, γ is a positive number, Δ is the Laplacian, and Ric denotes the pointwise...

(joint work with Shaoyun Bai) Using a new version of transversality condition (the FOP transversality) on orbifolds, one can construct Hamiltonian Floer theory over integers for all compact symplectic manifolds. In this talk I will first describe...

Two matrices are called equivalent if one can be transformed into the other by multiplying with invertible matrices on the left and right. Extending this idea to 3-tensors, it is natural to define two 3-tensors as isomorphic if they can be...

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...