NUET Conference Schedule

Abstract: Answering a question of Erdős, Kra—Moreira—Richter—Roberston showed that every positive density subset of the integers contains a shift of a sumset configuration {b1 + b2 : b1 ≠ b2 ∈ B} for some infinite set B. I will address a polynomial variant of this problem: does every set of positive density in the integers contain a shift of a polynomial sumset configuration {p(b1) + b2 : b1 < b2 ∈ B} for some infinite set B, where p is a given intersective polynomial?

Abstract: Shannon introduced entropy for discrete random variables in his foundational work on information theory. Within about a decade Kolmogorov and Sinai had adapted it to the study of general measure-preserving systems in ergodic theory. Many variants and other applications have appeared since, connecting probability, combinatorics, dynamics and other areas.

I will survey a few recent developments in this story. I will focus largely on (i) Lewis Bowen’s “sofic entropy”, which helps us to study the dynamics of “large” groups such as free groups, and (ii) a cousin of sofic entropy in the world of unitary representations, which leads to new connections with operator algebras and random matrices.

Abstract: The purpose of this talk is to discuss some new polynomial ergodic theorems for commuting strongly mixing measure preserving transformations. (Joint work with Rigo Zelada).

Abstract: A measure-preserving G-system X is said to be an Abramov system of order at most k if the linear span of the polynomial phase functions of degree at most k on X is dense in L^2(X). Abramov systems feature prominently in the ergodic-theoretic approach of Bergelson–Tao–Ziegler to the inverse theorem for Gowers norms on vector spaces 𝔽_p^n. A central part of this approach is the analysis of the Host–Kra factors of G-systems with G equal to the (additive group of the) infinite-dimensional vector space 𝔽_p^ω. An appealing conjecture, which was proved by Bergelson–Tao–Ziegler in the high characteristic case but left open in low characteristic, was whether every ergodic 𝔽_p^ω-system of order k (i.e. isomorphic to its k-th Host–Kra factor) is Abramov of order at most k. More recently, Jamneshan, Shalom and Tao disproved this conjecture, and raised an interesting follow-up question: is every ergodic 𝔽_p^ω-system of order k a factor of some Abramov system of order at most k. I will discuss a positive answer to this question given in recent joint work with González-Sánchez and Szegedy. Our approach uses as a key ingredient a recent theory of certain dynamical systems which form a useful generalization of the classical nilsystems: the so-called nilspace systems. The talk will thus be an opportunity also to discuss some central aspects, applications and questions of this recent theory.

Abstract: In recent years, the joint ergodicity property, which states that averages of T_1^(a_1(n))f_1 ⋯ T_k^(a_k(n))f_k converge (in the L^2 norm) to the product ∏_(i=1)^k ∫ f_i dμ, has been extensively investigated in ergodic theory. The talk will focus on an analogous property in the topological dynamics setting, referred to as Δ-transitivity or joint transitivity. The problem is to determine conditions under which the sequence (T_1^(a_1(n))x, …, T_k^(a_k(n))x)_(n ∈ ℕ) is dense in X^k for a dense set of x ∈ X. In recent work with Andreas Koutsogiannis and Wenbo Sun, we established conditions under which the sequence (T_1^n, T_2^n, …, T_d^n) is jointly transitive. In this talk, we will review the main ideas of the proof and, if time allows, state possible future questions and directions.

Abstract: An important question in arithmetic Ramsey theory is whether Pythagorean triples are partition regular, i.e. whether every partition of the integers into finitely many pieces contains solutions of the equation x^2+y^2=z^2, where x,y,z belong to the same cell of the partition. I will present a recent result where we prove partition regularity of Pythagorean pairs, i.e. we can guarantee that there are two numbers x,y in the same cell of the partition such that x^2+y^2=z^2 for some integer z that can be in another cell (and a similar result with x,z instead of x,y). After some initial maneuvers inspired by ergodic theory, the proof consists in studying the asymptotic behavior of bounded multiplicative functions, and their products, along certain binary quadratic forms. This is a problem of independent interest that we solved using a combination of tools such as Gowers uniformity, to cover the random-like case, and concentration estimates, to cover the structured case. I will also present some recent developments concerning more general homogeneous quadratic equations in three variables. This is joint work with O. Klurman and J. Moreira.

Abstract: A tiling (or monotiling) is a covering of a space using copies of one building block, called a tile, without any overlaps. For instance, a planar tiling can be as boring as a regular square tilling or as interesting as the Escher lizard tiling or even the recently discovered aperiodic hat tiling. There seems to be a mysterious divide between “structured” tiling problems, in which the tilings are well behaved, to “wild” tiling problems, where almost anything can happen; e.g., translational tilings vs. isometric tilings, low dimension tilings vs. high dimension tilings, discrete tilings vs. continuous tilings. In the talk we will discuss this phenomenon, and present some recent progress towards solving the mystery of the structure of tilings.

Abstract: Furstenberg’s multiply recurrent theorem states that any dynamical system has multiply recurrent points. In this talk, we discuss multiple recurrence without commutativity. More precisely, let (X,T) and (X,S) be two minimal systems. It turns out that there is a residual subset X_0 of X such that for any x ∈ X_0 and any finite nonlinear integral polynomials p_1, …, p_d vanishing at 0, there is some subsequence {n_i} of ℕ with n_i → \infty satisfying

S^(n_i)x → x, T^(p_1(n_i))x → x, …, T^(p_d(n_i))x → x, i → \infty.

This based on joint works with Prof. Song Shao and Xiangdong Ye.

Abstract: This talk consists of two parts: the first presented by myself and the second by Or, both parts featuring joint work with Terence Tao.

In the first part, we review structural results concerning Host–Kra–Ziegler factors of 𝔽_p^ω-systems and the corresponding inverse Gowers theory of finite vector spaces due to Bergelson, Tao, and Ziegler. Specifically, we provide a counterexample to a conjecture by Bergelson, Tao, and Ziegler regarding the Abramov property of such factors. We conclude the first part with a discussion and speculation on the implications of the counterexample and the connection to the combinatorial theory.

In the second part, we delve into the structure theory of the more general setting of totally disconnected Host–Kra–Ziegler factors and the inverse Gowers theory of bounded torsion groups. Additionally, we examine the opposite case of connected Host–Kra–Ziegler factors and its relationship to the Green–Tao–Ziegler inverse theorem for cyclic groups. These results pave the way for a potential unified inverse Gowers theorem that incorporates both cyclic groups and bounded torsion groups.

Abstract: We discuss the orbits of the horocycle flow acting on X = SL(2, ℝ)/Γ when sampled at sparse subsets of the integers. We will focus on the case of polynomial (sub-quadratic) times and also at numbers which are products of at most two prime numbers. We will show that for every δ > 0 the orbit {h_(n^(2−δ)) x} is dense for every non-periodic point x ∈ SL(2, ℝ)/SL(2, ℤ) (joint work with M. Radziwill). We will also show that if Γ is arithmetic, then {h_(p·q) x}_(p,q−primes) equidistributes towards the only invariant measure on the regular orbit when sampled over ℤ (joint with G. Forni, M. Radziwill).

Abstract: We discuss norm-convergence results for multiple ergodic averages along sequences of polynomial growth evaluated at primes. We do so by comparing the latter averages with the corresponding ones along positive integers. Combining our results with Furstenberg’s correspondence principle, we derive several applications in combinatorics. Namely, we show that positive density subsets of positive integers contain arbitrarily long arithmetic progressions with common difference coming from the aforementioned sequences evaluated at primes (or shifted primes), obtaining far-reaching extensions to Szemerédi’s theorem. One of the main tools in the proof, that covers the polynomial sequences case, is the uniformity of the von Mangoldt function in intervals of the form [1,N] which follows from a result of Green and Tao, and, for sparse sequences, a recent result of Matomäki, Shao, Tao, and Teräväinen on the uniformity of the stated function in short intervals.

We will present results of individual and joint (with Dimitris Karageorgos, and Konstantinos Tsinas) work.

Abstract:
In this talk I will discuss the highlights of pointwise ergodic theory, beginning with the work of Birkhoff in the 1930s, then turning to the seminal work of Bourgain in the late 80s-early 90s, and concluding with more modern work, joint with Mirek and Tao.

Abstract: The utility of ergodic methods in additive combinatorics lies primarily in their robustness: no other approach has so far managed to detect as many patterns in dense sets of integers as the ergodic one. This however comes at a price: ergodic theory is incapable of providing quantitative information on the size of finite sets avoiding a given pattern. The last decade has witnessed considerable progress in narrowing the gap between what we know qualitatively and quantitatively thanks to an outpour of new bounds in the polynomial Szemerédi theorem of Bergelson and Leibman. In this talk, I will present two papers written jointly with Noah Kravitz and James Leng in which we contribute to this program of quantifying the polynomial Szemerédi theorem. Specifically, I will discuss our recent bounds for sets with no polynomial corners, one of the simplest 2-dimensional polynomial configurations. While the result is more combinatorial than ergodic, a substantial part of the toolbox that we develop has applications to ergodic theory.

Abstract: A subset of the positive integers is dynamically central syndetic if it contains the times of return of a point to a neighborhood of itself in a minimal dynamical system. This class of syndetic sets forms an important bridge between dynamics and combinatorics. We show that a set is dynamically central syndetic if and only if it is a member of a syndetic, idempotent filter. We elaborate on the consequences of this characterization for the dual family (sets of pointwise recurrence) and the localized dual family (dynamically central piecewise syndetic sets). For example, we provide several combinatorial characterizations of sets of pointwise recurrence, show that these sets do not have the Ramsey property, and they are sets of measurable multiple polynomial recurrence for commuting transformations. These results answer several questions asked by Host, Kra, and Maass. This talk is based on an ongoing joint work with Daniel Glasscock (University of Massachusetts Lowell).

Abstract: In this talk, we discuss the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal ℤ^d-system (X, T_1,. . . , T_d). We discuss the structural properties of systems that satisfy the so-called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a ℤ^d-system (X, T_1, . . . , T_d) that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal ℤ^d-systems that enjoy the unique closing parallelepiped property and provide explicit examples.

Abstract:
I’ll talk about recent work with Sean Prendiville and Fernando Shao in which we prove the first quantitative bounds in Bergelson—Leibman’s multidimensional polynomial Szemerédi theorem for the configuration (x,y), (x,y+d), (x+d^2,y), which we call “sqorners”. We also prove a “popular difference” version of this result, with effective bounds.

Abstract: This talk consists of two parts: the first presented by Asgar and the second by myself, both parts featuring joint work with Terence Tao.

In the first part, we review structural results concerning Host–Kra–Ziegler factors of 𝔽_p^ω-systems and the corresponding inverse Gowers theory of finite vector spaces due to Bergelson, Tao, and Ziegler. Specifically, we provide a counterexample to a conjecture by Bergelson, Tao, and Ziegler regarding the Abramov property of such factors. We conclude the first part with a discussion and speculation on the implications of the counterexample and the connection to the combinatorial theory.

In the second part, we delve into the structure theory of the more general setting of totally disconnected Host–Kra–Ziegler factors and the inverse Gowers theory of bounded torsion groups. Additionally, we examine the opposite case of connected Host–Kra–Ziegler factors and its relationship to the Green–Tao–Ziegler inverse theorem for cyclic groups. These results pave the way for a potential unified inverse Gowers theorem that incorporates both cyclic groups and bounded torsion groups.

Abstract: A map defined on the unit interval is called piecewise monotone if there exists a finite or countably infinite partition of pairwise disjoint subintervals such that the restriction of the map to each subinterval is monotone and continuous. In this talk, we explore the phenomenon of joint ergodicity of piecewise monotone maps. Specifically, under some natural conditions, piecewise monotone maps possessing distinct entropy values exhibit joint ergodicity. This means that the multiple ergodic averages regarding these maps converge to the product of their respective space averages.

This is a joint work with Vitaly Bergelson

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 scaler 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: We discuss the pointwise convergence of multiple ergodic averages weighted by two arithmetic weights: the primes and the Möbius function. For the Möbius weight, we show that the averages converge pointwise for any polynomial iterates. For the prime case, in ongoing joint work with Krause, Mousavi and Tao, we obtain results for two polynomial iterates. For the proofs, recent quantitative bounds for the Gowers norms of these weights, due to Leng, play a key role.

Abstract: The saturation theorems play crucial roles to deduce certain problems to the same ones on nilsystems. In this talk I will explain how to prove a saturation theorem for the product of finitely many minimal systems. Then I will give the applications of the theorem to problems related to independence. Namely, for a minimal system (X,T) and a factor map from X to its maximal d-step pronilfactor X_d, we explore the complexity of T on fibers in terms of independence. This is a joint work with Qiu Jiahao, Xu Hui and Yu Jiaqi.