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
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
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...