# Video Lectures

### Quantitative Inverse Theorem for Gowers Uniformity Norms 𝖴5 and 𝖴6 in 𝔽n2

Luka Milicevic

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...

### Restriction-closed tensor properties

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...

### Dimers and 3-Webs

Richard Kenyon

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...

### G-stable Rank and the Cap Set Problem

Harm Derksen

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...

### Tensorial Forms in Infinite Dimensions

Andrew Snowden

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...

### Ranks of Tensors

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 via Group Ring Identities

Peter Pach

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...

### The Monomial Structure of Boolean Functions

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...

### Polynomial Maps With Noisy Input-Distributions

Jop Briet

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...

### Average-Case Computational Complexity of Tensor Decomposition

Alex Wein

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...