# Faculty Activities

*Abstract:*

This talk will be a review of the rudiments of the theory of reproducing kernel Hilbert spaces, intended as preparation for a series of talks in the OA/OT learning seminar about reproducing kernel Hilbert spaces in noncommutative analysis.

This talk is suitable for graduate students, no background is needed besides the basics of functional and complex analysis.

*Announcement:*

ALL TALKS WILL BE HELD AT AMADO 232

**Speakers and schedule:**

09:30 -10:00 Coffee and refreshments at the 8th floor lounge

10:00-11:00 Ron Peled (Tel Aviv University)

11:00-11:30 Coffee break

11:30-12:30 Nir Lev (Bar Ilan University)

12:30-14:00 Lunch break

14:00-15:00 Amitay Kamber (Hebrew University)

15:00-15:30 Coffee/Tea

15:30-16:30 Anatoly Vershik (Steklov Institute, St. Petersburg).

**TITLES AND ABSTRACTS:**

**1. Ron Peled:**

**Title**: Rigidity of proper colorings of Z^d (and other graph homomorphisms)

**Abstract**: A proper q-coloring of Z^d is an assignment of one of q values to each vertex of Z^d such that adjacent vertices are assigned different values. Such colorings arise naturally in combinatorics, ergodic theory (as a subshift of finite type) and statistical physics (as the ground states of the antiferromagnetic q-state Potts model). How does a "uniformly picked" proper q-coloring of Z^d look like? To make sense of this question, one may sample uniformly a proper q-coloring of a large finite domain in Z^d and seek possible patterns in the resulting coloring. Alternatively one may seek to classify the translation-invariant probability measures on proper q-colorings having maximal entropy. Work of Dobrushin (1968) implies that when q is large compared with d, q-colorings support a unique measure of maximal entropy. We focus on the opposite regime, when d is large compared with q, and prove that long-range order emerges in typical colorings, leading to multiplicity of measures of maximal entropy. Concretely, the extremal measures of maximal entropy are in correspondence with partitions (A,B) of the q colors into two equal-sized sets (near-equal if q is odd), and in the measure corresponding to (A,B) most of the vertices of the even (odd) sublattice of Z^d are assigned values from A (from B). The results address questions going back to Berker-Kadanoff (1980), Kotecký (1985) and Salas-Sokal (1997). The methods extend to the study of other graph homomorphism models on Z^d satisfying a certain symmetry condition and to their "low temperature" versions. Joint work with Yinon Spinka.

**2. Nir Lev:**

**Title**: On tiling the real line by translates of a function

**Abstract**: If f is a function on the real line, then a system of translates of f is said to be a << tiling >> if it constitutes a partition of unity. Which functions can tile the line by translations, and what can be said about the structure of the tiling? I will give some background on the problem and present our results obtained in joint work with Mihail Kolountzakis.

**3. Amitay Kamber:**

**Title**: Optimal Lifting in SL3

**Abstract**: Sarnak proved in one of his letters that as q goes to infinity, for every $\epsilon>0$ almost every matrix in $SL_2(F_q)$ can be lifted to a matrix in $SL_2(Z)$, where every coordinate is bounded by q^(3/2+\epsilon). The exponent 3/2 is optimal, since the number of matrices in $SL_2(Z)$ with coordinates bounded by $T$ is asymptotic to $T^2$. We prove a similar theorem, with optimal exponent, in the context of the action of $SL_3(Z)$ on the projective 2-dimensional space over $F_q$. Our work is based on lattice point counting arguments as in the work of Sarnak and Xue, and on property T for SL3. We will also explain the relation of this theorem to the work of Ghosh, Gorodnik and Nevo, and to some general conjectures of Sarnak whose aim is to "approximate" the Generalized Ramanujan Conjecture to deduce Diophantine results. Based on joint works with Konstantin Golubev and Hagai Lavner.

**4. Anatoly Vershik:**

**Title** : Graded graphs and central measures

**Abstract** : Graded graphs (Bratteli diagrams) as dynamical language : Markov compact of paths, central measures, connections to asymptotic representation theory of locally finite groups, characters as central measures, limit-shape type theorems and generalized law of large numbers.

*Abstract:*

Graded graphs (Bratteli diagrams) as dynamical language; Markov compact of paths, central measures. connection to asymptotic representation theory of locally finite groups, characters as a central measures. limit shape type theorems and generalized law of large numbers.

*Abstract:*

The spectrum of the Laplacian on graphs which have certain symmetry properties can be studied via a decomposition of the operator as a direct sum of one-dimensional operators which are simpler to analyze. In the case of metric graphs, such a decomposition was described by M. Solomyak and K. Naimark when the graphs are radial trees. In the discrete case, there is a result by J. Breuer and M. Keller treating more general graphs. We present a decomposition in the metric case which is derived from the discrete one. By doing so, we extend the family of (metric) graphs dealt with to also include certain symmetric graphs that are not trees. In addition, our analysis describes an explicit relation between the discrete and continuous cases. This is joint work with Jonathan Breuer.

*Abstract:*

The parabolic-elliptic Patlak-Keller-Segel system for $n$-populations describes the collective motion of cells interacting via a self-produced chemical agent (called chemoattractant). In two space dimensions and for a single population ($n=1$) it is known that the $L^1$-norm of the initial datum is a salient parameter which separates the dichotomy between global in time existence and finite time blow up (or, chemotactic collapse). In other words, there exists an (explicitly known) critical value $\beta_c$ such that a solution exists globally if and only if the initial $L^1$ norm is smaller than or equal to $\beta_c$. In this talk I will discuss the global existence of solutions for $n$-populations. Exploring the gradient flow structure of the system in Wasserstein space, we show that whenever the $L^1$ norm of the initial datum satisfies an appropriate notion of 'sub-criticality', the system admits a global solution.

This is joint work with Gershon Wolansky (arXiv: 1902.10736)

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We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. The failure is due to presence of what we call "exotic eigenvalues": eigenvalues whose eigenfunctions increasingly localize on the edges that are shrinking to a point.

We establish a sufficient condition for convergence which encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges. In some important special cases this condition is also shown to be necessary. Moreover, when the condition fails, it provides quantitative information on exotic eigenvalues.

Before formulating the main results we will review the setting of Schrödinger operators on metric graphs and the characterization of possible self-adjoint conditions, followed by numerous examples where the limiting operator is not obvious or where the convergence fails outright. The talk is based on a joint work with Yuri Latushkin and Selim Sukhtaiev, arXiv:1806.00561 and on work in progress with Yves Colin de Verdiere.

*Abstract:*

Graphene is an allotrope of carbon consisting of a single layer of carbon atoms that are densely packed in a honeycomb crystal lattice. Suppose that one take a graphene sheet with holes and start to change the magnetic flux through the holes. When the changes of the magnetic fluxes become integer, the energetic spectrum as a whole should return to its initial state. However, the individual eigenenergies are not necessarily periodic; they can cross the boundary between hole and electron states. The number of such crossings (counted with sign) is called the spectral flow. The situation of non-zero spectral flow is important for physicists and is called the Aharonov-Bohm effect.

Single-layer graphene is described by the Dirac operator acting on a two-or four-component spinor. There is also a bilayer form of graphene, which is described by self-adjoint differential operators of non-Dirac type. In the talk I will show how the spectral flow can be computed, using topological methods, for both single-layer and bilayer graphene (and, more generally, for a one-parameter family of arbitrary first order self-adjoint elliptic operators over a compact surface with classical boundary conditions). The talk is based on my papers arXiv:1108.0806 and arXiv:1703.06105.

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During the last 20 years there has been a considerable literature on a collection of related mathematical topics: higher degree versions of Poncelet’s Theorem, certain measures associated to some finite Blaschke products and the numerical range of finite dimensional completely non-unitary contractions with defect index 1. I will explain that without realizing it, the authors of these works were discussing Orthogonal Polynomials on the Unit Circle (OPUC). This will allow us to use OPUC methods to provide illuminating proofs of some of their results and in turn to allow the insights from this literature to tell us something about OPUC. This is joint work with Andrei Martínez-Finkelshtein and Brian Simanek. Background will be provided on the topics discussed.

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T.B.A.

*Abstract:*

Estimating a manifold from (possibly noisy) samples appears to be a difficult problem. Indeed, even after decades of research, there are no (computationally tractable) manifold learning methods that actually "learn" the manifold. Most of the methods try, instead, to embed the manifold into a low-dimensional Euclidean space. This process inevitably introduces distortions and cannot guarantee a robust estimate of the manifold.

In this talk, we will discuss a new method to estimate a manifold in the ambient space, which is efficient even in the case of an ambient space of high dimension. The method gives a robust estimate to the manifold and its tangent, without introducing distortions. Moreover, we will show statistical convergence guarantees.

It is on a work in progress, joint with Barak Sober.