Video Lectures

A conjecture of Komlós states that the discrepancy of any collection of unit vectors is O(1), i.e., for any matrix A with unit columns, there is a vector x with -1,1 entries such that |Ax|∞=O(1). The related Beck-Fiala conjecture states that any set...

It is often said that a geometer has only two basic tools to study a space: to slice it or to project it. Cartography provides a vivid example: contour lines are simply the intersections of a landscape with horizontal planes. In the nineteenth...

Recently, a number of formulas reminiscent of Weyl's law have been discovered in the context of symplectic geometry. Various three-manifold invariants, defined by building on ideas originating in Morse theory, have been at the heart of these...

The algebraic structure of various groups of homeomorphisms and diffeomorphisms was studied extensively in the 1960s and 1970s, when it was shown that these groups are (mostly) simple. A notable open case concerned the group of area-preserving...

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for...

After ten years of gravitational-wave observations with the LIGO–Virgo–KAGRA interferometers, over 200 signals from merging black hole binaries have been detected, bringing us closer to understanding how these systems form. Addressing this question...

Differential Galois groups are algebraic groups that describe symmetries of some systems of differential equations. The solutions considered can live in any differential field and thus a natural framework to consider such symmetries is the setting...

In this talk, I will completely describe the decomposition of an oscillator representation under the tensor embedding of a product of a symplectic and orthogonal group in the case of finite fields via a correspondence proposed by Roger Howe...

Harmonic analysis studies functions on the real line by expanding them as sums of frequencies (exponentials) and analyzing how each term contributes to the whole. In many applications—such as speech recognition—only the low frequencies matter.

Over...