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...
A subset of a group is said to be product free if it does not
contain the product of two elements in it. We consider how large
can a product free subset of the alternating group An be?
In the talk we will completely solve the problem by...
Approximate lattices in locally compact groups are approximate
subgroups that are discrete and have finite co-volume. They provide
natural examples of objects at the intersection of algebraic
groups, ergodic theory and additive combinatorics... with...
Work of Mark Shusterman and myself has proven an analogue of
Chowla's conjecture for polynomial rings over finite fields, which
controls k-points correlations of the Möbius function for k bounded
by a certain function of the finite field size...
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...
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...
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...
