Video Lectures

The first structures of particle dark matter form by gravitationally condensing out of the smooth mass distribution of the early universe. This formation mechanism leaves these "prompt cusps" with uniquely compact r^-1.5 density profiles and links...

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...

The inverse Galois problem asks for finite group G, whether G is a finite Galois extension of the rational numbers. Malle’s conjecture is a quantitative version of this problem, giving an asymptotic prediction of how many such extensions exist with...

I will start with a broad overview of the theory of initial conditions of the universe, what we know and what we hope to know in the near future.

Then I will describe some recent work. Motivated by precision calculations in primordial cosmology, a...

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...