Abstract: Sharp restriction theory and the finite field extension problem have both received much attention in the last two decades, but so far they have not intersected. In this talk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts, new phenomena arise which are related to the underlying arithmetic and discrete structures. The talk is based on recent joint work with Cristian González-Riquelme.

Abstract: An integer distance set is a set in the Euclidean plane with the property that all pairwise distances between its points are integers. In this talk we will show that any integer distance set has all but very few of its points lying on a single line or circle. This helps us address some questions by Erdős. In particular, we deduce that integer distance sets in general position (no 3 points on a line, no 4 points on a circle) are very sparse, and we derive a near-optimal lower bound on the diameter of any non-collinear integer distance set of a given size. Our proof uses existing refinements of the Bombieri-Pila determinant method. This is joint work with Rachel Greenfeld and Sarah Peluse.

Abstract: The Brunn-Minkowski inequality is a fundamental geometric inequality, closely related to the isoperimetric inequality. It states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Here A+B={a+b: a \in A, b \in B}. Equality holds if and only if A and B are convex and homothetic sets (one is a dilation of the other) in R^d. The stability of the Brunn-Minkowski inequality is the principle that if we are close to equality, then A and B must be close to being convex and homothetic. We prove a sharp stability result for the Brunn-Minkowski inequality, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on this problem. This is joint work with Alessio Figalli and Peter van Hintum.

Abstract: Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau. 

Abstract: The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It’s been known since work of Ferenczi that linear word complexity highly constrains the dynamical behavior of a subshift.

In recent work with Darren Creutz, we show that if X is a transitive subshift with limsup p(n)/n < 3/2, then X is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, limsup p(n)/n = 3/2 can occur even for X measurably weak mixing. Our proofs rely on a substitutive/S-adic decomposition for such subshifts.

I’ll give some background/history on linear complexity, discuss our results, and will describe several ways in which 3/2 turns out to be a key threshold (for limsup p(n)/n) for several different types of dynamical behavior.

Abstract: In 1977, Furstenberg introduced an alternative proof for Szemerédi's theorem by transforming the combinatorial problem into a dynamical one, a method now known as the Furstenberg correspondence principle. Using this principle, along with modern tools, Kra, Moreira, Richter, and Robertson recently proved that, up to shifting by $t \in \{0,1\}$, any set $A \subset \mathbb{N}$ with positive upper Banach density contains a pattern of the form $B \oplus B = \{b_1 + b_2 : b_1, b_2 \in B, b_1 \neq b_2\}$ for some infinite set $B$.

In this presentation, we will explore a closely related  pattern, namely $B + B = \{b_1 + b_2 : b_1, b_2 \in B\}$ for some infinite set $B$, which, despite its similarity, exhibits very different behavior. In joint work with Ioannis Kousek, we introduce a slight modification to the Furstenberg correspondence. Combined with recent ideas from Kra, Moreira, Richter, and Robertson, this modification allows us to characterize, in terms of density, when a set $A \subset \mathbb{N}$ contains this new pattern.

Abstract: The Geometry Ramsey Conjecture is a question raised by Graham in 1994, which says that given any finite configuration X which lies on a sphere, for any finite coloring of the Euclidean space, there always exists a monochromatic congruent copy of tX for any large enough scalar t. One can also formulate a similar question for the finite field setting. While the study of the Geometry Ramsey Conjecture in literature focuses on the harmonic analysis approach, in this talk, we will explain how the higher order Fourier analysis method can be used to answer the Geometry Ramsey Conjecture in the finite field setting.

Abstract:  The Miyachi-Peral fixed-time Lp estimates with loss of derivatives are among the classical results for the Euclidean wave equation. It is well known that analogues of those estimates hold true for wave equations driven by more general elliptic operators than the standard Laplacian. Notably, a celebrated theorem by Seeger, Sogge and Stein on Fourier integral operators yields such estimates for the Laplace-Beltrami operator on any compact Riemannian manifold. In comparison, the wave equation driven by a (non-elliptic) sub-Laplacian on a sub-Riemannian manifold is much less understood, and sharp Lp estimates are known only in few particular cases. We report on recent progress based on joint work with Detlef Müller.

Abstract: Given an element in SL_n(Z/qZ) what is the smallest element of SL_n(Z) that projects to it? We show that a lift with entries bounded by O(q^2 log q) always exists, and that the exponent 2 is best possible. Time permitting we may discuss the analogous problem of finding integer matrices with prescribed determinant that approximates a given matrix with real entries. Joint work with Amitay Kamber.