Online ETA(η) Seminar

Abstract: Every subset of the positive integers with bounded gap length contains arbitrarily long arithmetic progressions.  This equivalent formulation of van der Waerden's theorem (1927) and its generalization to sets of positive density by Szemeredi (1975) are foundational theorems in partition and density Ramsey Theory.  In this talk, we will discuss an inhomogeneous version of van der Waerden's theorem that addresses where arithmetic progressions in syndetic sets are guaranteed to start.  We will discuss the theorem's proof -- featuring the recent "topological characteristic factor" machinery of by Glasner, Huang, Shao, Weiss, and Ye -- and some of its applications:  multiplicative structure in nilBohr sets and the disjointness between additive and multiplicative actions of the positive integers by homeomorphisms.  This talk is primarily based on work in https://arxiv.org/pdf/2207.00098.pdf.

Abstract: Boolean analysis has evolved into a multifaceted field of mathematics, blending techniques and intuition from analysis, probability and combinatorics. In this talk, we shall survey a line of recent developments in the field that has been motivated by problems in functional analysis and discrete geometry. Time permitting, selected applications in theoretical computer science will also be discussed.

Abstract: Let S be a subset of 1,…,N avoiding the nontrivial progressions x, x+y^2-1, x+2(y^2-1). We prove that |S|<N/loglog…log(N), where we have a fixed constant number of logarithms. This answers a question of Green, and is the first effective polynomial Szemerédi over the integers where the polynomials involved are not homogeneous of the same degree and the underlying pattern has linear complexity. Joint work with Sarah Peluse and Mehtaab Sawhney.

Abstract: We present a fundamentally new proof of the dimensionless L^p boundedness of the Bakry Riesz vector on manifolds with bounded geometry. The key ingredients of the proof are sparse domination and probabilistic representation of the Riesz vector. This type of proof has the significant advantage that it allows for a much stronger conclusion, giving us a new range of weighted estimates. The talk is based on joint work with K. Domelevo and S. Petermichl.

Abstract: The Brascamp—Lieb inequalities form a class of multilinear inequalities that includes a variety of well-known classical results, such as Hölder’s inequality, Young’s convolution inequality, and the Loomis—Whitney inequality, for example. Their theory is surprisingly multifaceted, involving ideas from semigroup interpolation, convex optimisation, and abstract algebra. In the first half of this talk, we will discuss some of the key aspects of this theory and some important variants on the Brascamp—Lieb framework; in the second half, we will talk specifically about how these inequalities arise in harmonic analysis, in particular about their use in fourier restriction theory and in recent results on the boundedness of the helical maximal function. If time permits, we will then talk about some more far-reaching connections with other areas of mathematics and the sciences. 

Abstract: Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance. Here, distance can be thought of as a simple two point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as three point configurations. In this talk I will go through some of the history of such point configuration questions, show how geometric averaging operators arise naturally and give some recent results.

Abstract:  I will present a survey about the current status of Sarnak's conjecture from the point of view of Veech's conjecture proved in a joint work with A. Kanigowski, M. Lemańczyk and T. de la Rue.

Abstract:  What is the maximum proportion of d-dimensional space that can be covered by disjoint, identical spheres? In this talk I will discuss a new lower bound for this problem, which is the first asymptotically growing improvement to Rogers' bound from 1947. Our proof is almost entirely combinatorial and reduces to a novel theorem about independent sets in graphs with bounded degrees and codegrees.

This is based on joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

Abstract: Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.

Abstract: We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with a Borel probability measure $\mu$ of Hausdorff dimension $d$, one needs at least $d$ measurements of a typical (prevalent) Lipschitz observable for reliable $\mu$-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of $\mu$ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. This allows us to establish a a variant of a lower bound in the prediction error conjecture of Schroer--Sauer--Ott--Yorke from 1998.

Based on joint works with Krzysztof Barański and Adam Śpiewak.

Abstract: Determining the limit in the weighted and subsequence ergodic theorem is the key to many applications, such as in recurrence, almost sure convergence, the Hardy-Littlewood-Vinogradov circle method---hence in number theory.  Instead of asking for the limit for specific cases, we try to determine what can be the limit in any situation: for any weighted or subsequence ergodic averages. 

Though the motivation comes from ergodic theory, the talk will be accessible to any graduate student who is familiar with the concept of Borel measures on the torus and their Fourier coefficients.