Special Year 2022-23: Dynamics, Additive Number Theory and Algebraic Geometry

Abstract: I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemer\’edi’s theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form $(x,y), (x,y+z),...

Abstract: 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...

Abstract: 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...

Abstract: Let V be a complex vector space and consider symmetric d-linear forms on V, i.e., linear maps $Sym^d(V) \rightarrow > C$. When V is finite dimensional and $d>2$, the structure of such forms is very complicated. Somewhat surprisingly, when...

Abstract: 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...

Abstract: Let $f:{0,1}^n$ 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...

Abstract: 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...

Abstract: 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...

Abstract: Translational tiling is a covering of a space using translated copies of a building block, called a "tile", without any positive measure overlaps. What are the possible ways that a space can be tiled?
The most well known conjecture in this...

Abstract: We will discuss some results on high-degree varieties designed to expand the reach of the polynomial method and explain their applications in incidence geometry.