Video Lectures

In this talk, I will discuss a proof of a quantitative version of the inverse theorem for Gowers uniformity norms 𝖴5 and 𝖴6 in 𝔽n2. The proof starts from an earlier partial result of Gowers and myself which reduces the inverse problem to a study of...

A theorem by Kazhdan and Ziegler says that any property of homogeneous polynomials---of a fixed degree but in an arbitrary number of variables---that is preserved under linear maps is either satisfied by all polynomials or else implies a uniform...

This is joint work with Haolin Shi (Yale). 3-webs are bipartite, trivalent, planar graphs. They were defined and studied by Kuperberg who showed that they correspond to invariant functions in tensor products of SL_3-representations. Webs and...

Ellenberg and Gijswijt drastically improved the best known upper asymptotic bound for the cardinality of a cap set in 2016. Tao introduced the notion of slice rank for tensors and showed that the Ellenberg-Gijswijt proof can be nicely formulated...

Let V be a complex vector space and consider symmetric d-linear forms on V, i.e., linear maps Symd(V)→>C. When V is finite dimensional and d>2, the structure of such forms is very complicated. Somewhat surprisingly, when V has countably infinite...

Several equivalent definitions of rank for matrices yield non-equivalent definitions of rank when generalized to higher order tensors. Understanding the interplay between these different definitions is related to important questions in additive...

The Alon-Jaeger-Tarsi conjecture states that for any finite field F of size at least 4  and any nonsingular  matrix M over F there exists a vector x such that neither x nor Mx has a 0 component. In this talk we discuss the proof of this result for...

Let f:0,1n to 0,1 be a boolean function. It can be uniquely represented as a multilinear polynomial. What is the structure of its monomials? This question turns out to be connected to some well-studied problems, such as the log-rank conjecture in...

A problem from theoretical computer science posed by Buhrman asks to show that a certain class of circuits (NC0[+]) is bad at decoding error correcting codes under random noise. (This would be in contrast with an analogous class of quantum circuits...

Suppose we are given a random rank-r order-3 tensor---that is, an n-by-n-by-n array of numbers that is the sum of r random rank-1 terms---and our goal is to recover the individual rank-1 terms. In principle this decomposition task is possible when r<<n3/2. Is this a fundamental barrier for efficient algorithms?

In recent years, the average-case complexity of various high-dimensional statistical tasks has been resolved in restricted-but-powerful models of computation such as statistical queries, sum-of-squares, or low-degree polynomials. However, the hardness of tensor decomposition has remained elusive, and I will explain what makes this problem difficult. We show the first formal hardness for average-case tensor decomposition: when r>>n3/2, the decomposition task is hard for algorithms that can be expressed as low-degree polynomials in the tensor entries.