Organizers: Adi Glücksam and Sasha Sodin
Time and Place: Weekly on Thursdays 12:00-14:00 at Ross 70.
Meant for faculty as well as advanced students (younger students may contact the organizers above), the seminar aims to expose its participants to various new branches of analysis. The uniqueness of the seminar is that is aspires to combine proofs in every lecture to deepen our understanding of the topic and the main tools used.
From kinetic PDEs to incompressible fluids Immanuel Ben-Porath Abstract: The quasi-neutral limit is a hydrodynamical limit offering a derivation of a PDE of incompressible fluid from a kinetic PDE . In the specific context of the 2D Vlasov-Poisson equation, the quasi-neutral limit leads to the 2D incompressible Euler equation. The purpose of this talk would be to concentrate on the modulated energy method, as developed by Brenier, in order to rigorously justify this limit. Well-posedness of the underlying equations will be discussed as well. The talk will be mostly expositional in nature, with some indication towards contemporary research themes.
Persistence and entropic repulsion for stationary Gaussian processes Naomi Feldheim Abstract: A real stationary Gaussian process (SGP) is a shift-invariant distribution over continuous functions f on R^d, whose finite marginals are multi-normal. Such a function is characterized by its spectral measure, that is, a probability measure on R^d whose Fourier transform yields the covariance kernel r(t) = cov(f(0),f(t)). Persistence of a stochastic process is the event of remaining above a fixed level on a large ball of radius T (a "hard wall" event). For a SGP, we ask two basic questions: What is the asymptotic behavior of the persistence probability, as T grows? Conditioned on the persistence event, what is the typical shape of the process (if there is one)? These questions, posed by physicists and applied mathematicians decades ago, have been successfully addressed only in the last few years, by exploiting tools from real, complex and harmonic analysis. After a survey of some past results, we shall focus on the regime for which entropic repulsion occurs. This is the phenomenon that, conditioned on a hard wall, the process is "pushed away" from the wall by a macroscopic amount, and fluctuates around some deterministic shape. We show that this phenomenon occurs universally for the class of SGPs with spectral blow-up at the origin. Based on joint work with Ohad Feldheim and Stephen Muirhead.
Thermodynamic formalism out of equilibrium Snir Ben Ovadia Abstract: We introduce the study of an adapted thermodynamic formalism for dynamical systems whose potential depends randomly on a Gibbs process. We introduce the notion of an associated ``pressure out of equilibrium”, and present a formula via a variational principle (the proof we will present will be analytic rather than dynamical, in the spirit of the seminar). If time allows, we will present an application involving a version of Azuma’s inequality for Gibbs processes (which was also the original motivation for this program).
On Almost Polynomially Convex Sets and Applications Adi Glücksam Abstract: In this talk I will describe what an almost polynomially convex set is, show how any union of disjoint cubes in R^d is almost polynomially convex and (in the end) show how to use this to construct a stationary random entire function with dense orbit almost surely. This talk is based on joint work with B. Weiss. While it is part (b) of last week, most of the talk is independent.
Stationary Random Entire Functions in High Dimensions Adi Glücksam Abstract: A translation $T_w$ is the action of translating an entire function by $w$, defined by $(T_w f)(z) = f(z+w)$. In 1996, B. Weiss demonstrated an abundance of translation-invariant probability measures on the space of entire functions. In this talk, we extend this result to holomorphic functions on $\mathbb{C}^d$ for any $d \ge 1$, answering a question posed by T-C. Dinh and N. Sibony in 2018. This talk is based on joint work with B. Weiss.
Critical sets of harmonic functions Eugenia Malinnikova Abstract: Naber and Valtorta showed that if a harmonic function in the unit ball in R^d has frequency bounded by N, then the (d−2)-dimensional Hausdorff measure of its critical set in the half ball is bounded by exp(CN^2). It is conjectured that the optimal bound should instead be polynomial in N. This is known in dimension two, but the higher-dimensional case remains surprisingly subtle. In this talk, we will review the Naber–Valtorta approach and show that in dimension three the estimate can be improved to a bound that is much closer to polynomial. The talk is based on ongoing joint work with Ben Foster and Josef Greilhuber.
Non-round Blashke-Santalo inequalities and applications to strengthened isoperimetry under symmetry assumption Galyna Livshyts Abstract: We will discuss the exciting subject of the Blashke-Santalo inequality, for sets as well as for functions, connections of this inequality to other topics, and a novel family of ``non-round’’ Blashke-Santalo inequalities. We will explore applications to a strengthened Brascamp-Lieb inequality for even functions, thereby extending this phenomenon beyond the Gaussian setting, and, more generally, beyond the rotation-invariant setting. Based on joint works with Colesanti, Kolesnikov and Rotem.
Dynamical and Dimensional Properties of Schrödinger Operators Under Finite-Rank Perturbations Netanel Levi Abstract: In this lecture, we will present several dynamical and fractal-dimensional ways of characterizing the spectral measures of Schrödinger operators, such as Rajchman behavior and Hausdorff/packing dimensions, and discuss the extent to which these properties are stable under rank-one perturbations. We begin with the concrete setting of half-line Schrödinger operators, where a theorem of Gordon shows that generic rank-one perturbations eliminate pure point spectrum, ruling out the most extreme dynamical and dimensional behavior. I will then describe constructions demonstrating that properties only slightly weaker than pure point spectrum can, in fact, be entirely stable: for certain sparse half-line models, both packing-dimension-zero and non-Rajchman behavior persist for every rank-one perturbation. In the second part, we examine how spectral dimensions behave when passing from the whole line to the half-line. I will present an operator whose spectral measure on the line has Hausdorff dimension one, whereas every half-line restriction - under any boundary condition - has dimension zero, even though the two settings differ only by a finite-rank perturbation.
A brief introduction to Young measures Cy Maor Abstract: One of the basic questions in the calculus of variations is the existence of minimizers for integral functionals. However, in many cases minimizers do not exist: the minimizing sequence must oscillate faster and faster on small amplitudes, but these oscillations vanish in the limit (think of $\frac{\sin(nx)}n\rigtarrow 0$). Around 1940, L.C. Young developed a finer notion of convergence of functions to deal with this problem; 50 years later, these "Young measures" reappeared as a tool to study new problems, e.g., if $\phi :\Omega\subset\mathbb R^n\rightarrow\mathbb R^n$ satisfies $dist(\nabla\phi , SO(n))\rightarrow 0$ in $L^p$, does fi converge to a single element in $SO(n)$? In this introductory talk I'll introduce Young measures and their original motivation, answer the above question (spoiler: yes!), and, if time permits, discuss some new developments.
Dry Ten Martini Problem for Sturmian Hamiltonians Siegfried Beckus Abstract: Are all possible spectral gaps, predicted by the Gap labelling theorem, open for a given Schr ̈odinger operator? This is the so-called ”Dry Ten Martini problem (Dry TMP)” motivated by the ”Ten Martini Problem (TMP)”. The name TMP was coined by Barry Simon after Mark Kac offered in 1981 ten Martinis to anyone who solves it. Originally, the TMP was proposed for the Almost Mathieu operator conjecturing Cantor spectrum for all couplings and all irrational frequencies. The TMP for the Almost Mathieu operator was solved by Artur Avila and Svetlana Jitomirskaya in 2005. In this talk, we discuss the Dry TMP for so-called Sturmian dynamical systems. These systems define a one-dimensional Schr ̈odinger operator where the potential is characterized in terms of two parameters: a frequency paramter and strength of the coupling constant. Like for the Almost-Mathieu operator one asks if all predicted spectral gaps are open for all irrational frequencies and all couplings. For large couplings, the Dry TMP for Sturmian systems was solved by Raymond in 1997. In 2016, the Inventiones paper by David Damanik, Anton Gorodetski and William Yessen provided a solution if the frequency is the golden mean for all non-zero couplings. In a current project with Ram Band and Raphael Loewy we solve the Dry TMP for all irrational frequencies and all couplings by a detailed control of suitable periodic approximations. In the talk we present the problem and the route to its resolution.
A lonely weak tile Itay Londner Abstract: Let A be a bounded measurable subset of R^d. Fuglede conjectured that A tiles R^d by translations if and only if it is a spectral set. Though the conjecture is false in full generality, it has been confirmed for convex sets. A key ingredient of the proof of the latter result is the notion of weak tiling. It has been proved by Lev and Matolcsi that every spectral set tiles its complement weakly with a suitable Borel measure. In my talk I will answer positively a question raised by Kolountzakis, Lev and Matolcsi whether there exists a set in R^d tiling its complement weakly but is neither spectral nor a tile. Based on joint work with Gergo Kiss, Mate Matolsci and Gabor Somlai
The transcendental Bézout problem and coarse zero count Lev Buhovsky Abstract: By the Bézout theorem, for a generic collection of complex polynomials p_1, ..., p_n in C[z_1, ..., z_n], the number of their common zeroes is bounded above by the product of the degrees. The transcendental Bézout problem asks whether an analogous statement holds for analytic functions of several complex variables. The counterexample of Cornalba and Shiffman shows that in higher dimensions the answer is negative if one counts the common zeroes in the "usual way". In my talk I will introduce a coarse count of the common zeroes, and will explain how it leads to a positive answer to the transcendental Bézout problem. Based on a work in progress joint with Iosif Polterovich, Leonid Polterovich, Egor Shelukhin and Vukašin Stojisavljević.
A gap labelling theorem for Schrödinger operators on graphs Gilad Sofer Abstract: Given a Schrödinger operator on Rn or Zn, we are often interested in its integrated density of states, which roughly measures the number of states per unit volume in the system below a given energy. Special attention is often given to the values the integrated density of states attains at spectral gaps, known as gap labels, which are of physical importance. For instance, in the integer quantum Hall effect, the gap labels correspond to the quantized values of the Hall conductance. However, predicting these gap labels often requires the use of complicated machinery, such as K-theory, making the proofs quite challenging. In this talk, we present a more accessible approach to developing gap labelling theorems, based on a method developed by Johnson and Moser. This is done using an object known as the Schwartzman group, and involves computing the rotation of the Prüfer angle for the associated generalized eigenfunctions. Mainly, we present a gap labelling theorem for Schrödinger operators on discrete and metric graphs, providing a simple way to predict the possible gap labels of graphs with the geometric structure of one-dimensional aperiodic tilings. Time permitting, we will also discuss the implications of this to the 'Dry Ten Martini Problem' for Schrödinger operators on graphs with a Sturmian structure. Based on joint work with Ram Band.
Stochastic fractals: conformal dimension. Ilia Binder Abstract: The conformal dimension of a set is defined as the infimum of the Hausdorff dimensions of all its quasisymmetric images. In this talk, I will explore the conformal dimensions of a variety of both deterministic and stochastic fractal sets. These include classical examples such as Bedford–McMullen carpets, as well as more complex random constructions like self-affine fractal percolation clusters. A highlight of the talk will be a proof that the graph of Brownian motion is minimal—its conformal dimension equals its Hausdorff dimension of . The talk is based on joint work with Hrant Hakobyan (Kansas State University) and Wenbo Li (Tsinghua University).