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Geometric Functional Analysis and Probability Seminar

ThursdayJun 29, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Amir Dembo Title:The criticality of a randomly-driven front.Abstract:opens in new windowin html    pdfopens in new window
Consider independent continuous-time random walks on the integers to the right of a front R(t). Starting at R(0)=0, whenever a particle attempts to jump into the front, the latter instantaneously advances k steps to the right, absorbing all particles along its path. Sly (2016) resolves the question of Kesten and Sidoravicius (2008), by showing that for k=1 the front R(t) advances linearly once the particle density exceeds 1, but little is known about the large t asymptotic of R(t) at critical density 1. In a joint work with L-C Tsai, for the variant model with k taken as the minimal random integer such that exactly k particles are absorbed by the move of R(t), we obtain both scaling exponent and the random scaling limit for the front at the critical density 1. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the initial local fluctuations (with the scaling limit oscillating between instantaneous super and sub-critical phases).
ThursdayJun 15, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Piotr Nayar Title:Gaussian mixtures with applications to entropy inequalities and convex geometryAbstract:opens in new windowin html    pdfopens in new window
We say that a symmetric random variable X is a Gaussian mixture if X has the same distribution as YG, where G is a standard Gaussian random variable, and Y is a positive random variable independent of G. In the first part of the talk we use this simple notion to study the Shannon entropy of sums of independent random variables. In the second part we investigate, using Gaussian mixtures, certain topics related to the geometry of B_p^n balls, including optimal Khinchine-type inequalities and Schur-type comparison for volumes of section and projections of these sets. In the third part we discuss extensions of Gaussian correlation inequality to the case of p-stable laws and uniform measure on the Euclidean sphere. Based on a joint work with Alexandros Eskenazis and Tomasz Tkocz.
ThursdayJun 08, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Nishant ChandgotiaTitle:Irrational rotations, random affine transformations and the central limit theoremAbstract:opens in new windowin html    pdfopens in new window
It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.
ThursdayMar 09, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Sasha ShamovTitle:Conditional determinantal processes are determinantalAbstract:opens in new windowin html    pdfopens in new window

A determinantal point process governed by a locally trace class Hermitian contraction kernel on a measure space $E$ remains determinantal when conditioned on its configuration on an arbitrary measurable subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$.

Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if $K$ is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process.

ThursdayFeb 09, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Alexander FishTitle:The values of quadratic forms on difference sets, measure rigidity and equidistributionAbstract:opens in new windowin html    pdfopens in new window

Given a quadratic form Q in d variables over the integers, e.g. Q(x,y,z) = xy - z^2, and a set of positive density E in Z^d, we investigate what kind of structure can be found in the set Q(E-E). 
We will see that if d >= 3, and Q is indefinite, then the measure rigidity, due to Bourgain-Furman-Lindenstrauss-Mozes or Benoist-Quint, of the action of the group of the symmetries of Q implies that there exists k >=1 such that  k^2*Q(Z^d) is a subset of Q(E-E). 
We will give an alternative proof of the theorem for the case Q(x,y,z) = xy - z^2 that uses more classical equidistribution results of Vinogradov, and Weyl, as well as a more recent result by Frantzikinakis-Kra. The latter proof extends the theorem to other polynomials having a much smaller group of symmetries. Based on joint works with M. Bjorklund (Chalmers), and K. Bulinski (Sydney). 

ThursdayJan 19, 201711:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jay Rosen Title:Tightness for the Cover Time of S^2 Abstract:opens in new windowin html    pdfopens in new window

Let M be a smooth, compact, connected two-dimensional, Riemannian manifold without boundary, and let  C_epsilon be  the amount of time needed for the Brownian motion to come within (Riemannian) distance epsilon of all points in M. The first order asymptotics of C_epsilon as epsilon goes to 0 are known. We show that for the two dimensional sphere 

\sqrt{C_epsilon}-2\sqrt{2}\( \log \epsilon^{-1}- \frac{1}{4}\log\log \epsilon^{-1}\) is tight.

Joint work with David Belius and  Ofer Zeitouni.

ThursdayJan 12, 201711:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Ran Tessler and Assaf Naor Title:Double lecture !Abstract:opens in new windowin html    pdfopens in new window

First Speaker: Ran Tessler (ETH)
Time: 11:00
Title: A sharp threshold for Hamiltonian spheres in a random 2-complex.
Abstract: We define the notion of Hamiltonian sphere - a 2-complex homeomorphic to a sphere which uses all vertices. We prove an explicit sharp threshold for the appearance of Hamiltonian spheres in the Linial-Meshulam model for random 2-complexes. The proof combines combinatorial, probabilistic and geometric arguments. Based on a joint work with Zur Luria.

Second Speaker: Assaf Naor (Princeton)
Time: 12:00
Title: A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms
Abstract: In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its "vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on W^{1,1}), metric geometry (sharp nonembeddability into L_1) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.

ThursdayJan 05, 201711:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Amir DemboTitle:Walking within growing domains: recurrence versus transience Abstract:opens in new windowin html    pdfopens in new window
When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary. This talk is based on joint works with Ruojun Huang, Ben Morris, Yuval Peres, Vladas Sidoravicius and Tianyi Zheng.
ThursdayDec 29, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Alon Nishry Title:Gaussian complex zeros on the hole event: the emergence of a forbidden regionAbstract:opens in new windowin html    pdfopens in new window

Consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane.
I will show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. A remarkable feature of this limiting measure is the existence of a large 'forbidden region' between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. This answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result known to hold in the random matrix setting, where such a gap does not appear.
The talk is based on a joint work with S. Ghosh.

ThursdayDec 15, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Snir Ben OvadiaTitle:Symbolic dynamics for non uniformly hyperbolic diffeomorphisms of compact smooth manifolds Abstract:opens in new windowin html    pdfopens in new window

Given a dynamical system, a partition of the space induces a mapping to the space of sequences of the partition elements (a point is mapped to the partition elements containing its orbit terms). Such a duality is called Symbolic Dynamics, Markov partitions are an important tool, as the symbolic dynamics they induce enfold many of the important dynamical properties of the original system, and they allow an easier studying of them.
We show that general non uniformly hyperbolic C^{1+epsilon} diffeomorphism on compact manifolds of any dimension admit countable Markov partitions. Previously this was only known in dimension 2.

ThursdayNov 17, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Anirban Basak Title:Invertibility of sparse random matricesAbstract:opens in new windowin html    pdfopens in new window
We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = \Omega(\frac{\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. We also find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdos-Reyni graph whenever its edge connectivity probability is above the critical threshold $\Omega(\frac{\log n}{n})$. This is joint work with Mark Rudelson.
ThursdayNov 03, 201611:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:David Ellis Title:Some applications of the $p$-biased measure to Erd\H{o}s-Ko-Rado type problemsAbstract:opens in new windowin html    pdfopens in new window

If $X$ is a finite set, the $p$-biased measure on the power-set of $X$ is defined as follows: choose a subset $S$ of $X$ at random by including each element of $X$ independently with probability $p$. If $\mathcal{F}$ is a family of subsets of $X$, one can consider the {\em $p$-biased measure} of $\mathcal{F}$, denoted by $\mu_p(\mathcal{F})$, as a function of $p$; if $\mathcal{F}$ is closed under taking supersets, then this function is an increasing function of $p$. Seminal results of Friedgut and Friedgut-Kalai give criteria for this function to have a 'sharp threshold'. A careful analysis of the behaviour of this function also yields some rather strong results in extremal combinatorics which do not explicitly mention the $p$-biased measure - in particular, in the field of {\em Erd\H{o}s-Ko-Rado type problems}, which concern the sizes of families of objects in which any two (or three...) of the objects 'agree' or 'intersect' in some way. We will discuss some of these, including a recent proof of an old conjecture of Frankl that a symmetric three-wise intersecting family of subsets of $\{1,2,\ldots,n\}$ has size $o(2^n)$, and some 'stability' results characterizing the structure of 'large' $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots,n\}$. Based on joint work with (subsets of) Nathan Keller, Noam Lifshitz and Bhargav Narayanan.

WednesdaySep 14, 201614:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Assaf NaorTitle:The Lipschitz extension problem for finite dimensional normed spacesAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL DAY
ThursdayJun 23, 201612:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Shamgar Gurevich Title:Small Representations of Finite Classical GroupsAbstract:opens in new windowin html    pdfopens in new window
Many properties of a finite group G can be approached using formulas involving sums over its characters. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the "small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some conjectures which are currently out of reach.
ThursdayJun 23, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan HermonTitle:L_2 Mixing and hypercontractivity via maximal inequalities and hitting-timesAbstract:opens in new windowin html    pdfopens in new window

There are numerous essentially equivalent characterizations of mixing in $L_1$ of a finite Markov chain. Some of these characterizations involve natural probabilistic concepts such as couplings, stopping times and hitting times. In contrast, while there are several analytic and geometric tools for bounding the $L_2$ mixing time, none of them are tight and they do not have a probabilistic interpretation.

We provide tight probabilistic characterizations in terms of hitting times distributions for mixing in $L_2$ (for normal chains) and (under reversibility) in relative entropy. This is done by assigning appropriate penalty (depending on the size of the set) to the case that the chain did not escape from a certain set.

We also prove a new extremal characterization of the log-sobolev constant in terms of a weighted version of the spectral gap (where the weight depends on the size of the support of the function).

ThursdayJun 16, 201612:00
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eliran SubagTitle:Critical points and the Gibbs measure of pure spherical spin glassesAbstract:opens in new windowin html    pdfopens in new window
Recently, several results concerning the critical points of the energy landscape of pure $p$-spin spherical spin glasses have been obtained by means of moment computations and a proof of a certain invariance property. I will describe those and explain how they can be boosted by an investigation of the behavior around the critical points to obtain a geometric description for the Gibbs measure at low enough temperature. The talk is based on joint work with Ofer Zeitouni.
ThursdayJun 16, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Eviatar ProcacciaTitle:Can one hear the shape of a random walk?Abstract:opens in new windowin html    pdfopens in new window
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path's boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. Joint work with Marek Biskup.
ThursdayMay 05, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Ilya GoldsheidTitle:Recurrent Random Walks on a Strip: conditions for the CLTAbstract:opens in new windowin html    pdfopens in new window Double feature room 155
This is joint work with Dima Dolgopyat. We prove that a recurrent random walk (RW) in i.i.d. random environment (RE) on a strip which does not obey the Sinai law exhibits the Central Limit asymptotic behaviour. Moreover, there exists a collection of proper subvarieties in the space of transition probabilities such that: (a) If the RE is stationary and ergodic and the transition probabilities are concentrated on one of sub-varieties from our collection then the CLT holds; (b) If the environment is i.i.d then the above condition is also necessary for the CLT to hold. In particular, the CLT holds for the quasiperiodic environments with Diophantine frequencies in the recurrent case and complement this result by proving that in the transient case the CLT holds for all strictly ergodic environments.
ThursdayMay 05, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Tal OrenshteinTitle:One-dependent walks in hypergeometric-Dirichlet environmentsAbstract:opens in new windowin html    pdfopens in new windowDouble feature room 155
Dirichlet environments are one of the few examples in Random Walk in Random Environment in which some non-trivial random walk properties are fully and explicitly characterized in terms of the parameters. A key feature of the model is the so-called 'time reversal property', saying that inverting the time is resulting in the same class of models, with an explicit change of parameters. In this talk, which is based on a joint work in process with Christophe Sabot, I'll present a generalization of random walks in Dirichlet environments using hypergeometric functions having that nice feature, and discuss the question of existence of an invariant probability measure for the process on the environments from the point of view of the walker which is absolutely continuous with respect to the initial measure.
ThursdayApr 21, 201611:00
Geometric Functional Analysis and Probability Seminar
Speaker:Atilla YilmazTitle:Large deviations for random walk in space-time random environment: averaged vs. quenchedAbstract:opens in new windowin html    pdfopens in new windowRoom 155
I will present recent joint work with F. Rassoul-Agha (Utah) and T. Seppalainen (Madison) where we consider random walk on a hypercubic lattice of arbitrary dimension in a space-time random environment that is assumed to be temporally independent and spatially translation invariant. The large deviation principle (LDP) for the empirical velocity of the averaged walk (i.e., level-1) is simply Cramer’s theorem. We take the point of view of the particle and establish the process-level (i.e., level-3) averaged LDP for the environment Markov chain. The rate function $I_{3,a}$ is a specific relative entropy which reproduces Cramer’s rate function via the so-called contraction principle. We identify the unique minimizer of this contraction at any velocity and analyse its structure. When the environment is spatially ergodic, the level-3 quenched LDP follows from our previous work which gives a variational formula for the rate function $I_{3,q}$ involving a Donsker-Varadhan-type relative entropy $H_q$. We derive a decomposition formula for $I_{3,a}$ that expresses it as a sum of contributions from the walk (via $H_q$) and the environment. We use this formula to characterize the equality of the level-1 averaged and quenched rate functions, and conclude with several related results and open problems.
ThursdayMar 31, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir Yehudayoff Title:Geometric stability using information theoryAbstract:opens in new windowin html    pdfopens in new windowmoved to room 155

Projection inequalities bound the volume of a body in terms of a product of the volumes of lower-dimensional projections of the body. Two well-known examples are the Loomis-Whitney inequality and the more general Uniform Cover inequality. We will see how to use information theory to prove stability versions of these inequalities, showing that when they are close to being tight, the body in question is close to being a box (which is the unique case of equality). We will also see how to obtain a stability result for the edge-isoperimetric inequality in the infinite d-dimensional lattice. Namely, that a subset of Z^d with small edge-boundary must be close in symmetric difference to a d-dimensional cube.

Based on work with David Ellis, Ehud Friedgut and Guy Kindler.

ThursdayMar 17, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Asaf FerberTitle:Iterated Log Law for various graph parametersAbstract:opens in new windowin html    pdfopens in new window

We show that a version of the classical Iterated Log Law of Khinchin, and independently of Kolmogorov from the 1920's, holds for various parameters in the binomial random graph model and in a random 0/1 Bernoulli matrix. In particular, for a constant p, we show that such a law holds for the number of copies of a fixed graph H in G(n,p), we show a similar statement for the number of Hamilton cycles in a random k-uniform hypergraph, provided that k\geq 4. In the graph case (that is, k=2), since the number of Hamilton cycles in G(n,p), denoted by X_n, does not converge to a normal distribution but rather tends to a log-normal distribution (as has been first proved by Janson), we show that a version of the Iterated Log Law holds for \log X_n. We also obtain similar result for the permanent of a 0/1 bernouli random matrix.

No prior knowledge is required.

Joint with Daniel Motealegre and Van Vu.

ThursdayMar 10, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Mark RudelsonTitle:No-gaps delocalization for general random matricesAbstract:opens in new windowin html    pdfopens in new window

Heuristically, delocalization for a random matrix means that its normalized eigenvectors look like the vectors uniformly distributed over the unit sphere. This can be made precise in a number of different ways. We show that with high probability, any sufficiently large set of coordinates of an eigenvector carries a non-negligible portion of its Euclidean norm. Our results pertain to a large class of random matrices including matrices with independent entries, symmetric, skew-symmetric matrices, as well as more general ensembles.

Joint work with Roman Vershynin.

ThursdayMar 03, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Alexei V. PenskoiTitle:Recent advances in geometric optimization of eigenvalues of the Laplace-Beltrami operator on closed surfacesAbstract:opens in new windowin html    pdfopens in new window
Since a metric defines the Laplace-Beltrami operator on a closed surface, the eigenvalues of the Laplace-Beltrami operator are functionals on the space of Riemannian metrics on the surface. A metric is called maximal for i-th eigenvalue if the i-th eigenvalue attends its maximum on this metric. It turns out that the question about finding maximal metrics is very deep and related to analysis, topology, algebraic and differential geometry. In this talk several recent advances in this question will be exposed.
ThursdayFeb 18, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Evgeny StrahovTitle:Determinantal processes related to products of random matricesAbstract:opens in new windowin html    pdfopens in new window
I will talk about determinantal processes formed by eigenvalues and singular values of products of complex Gaussian matrices. Such determinantal processes can be understood as natural generalizations of the classical Ginibre and Laguerre ensembles of Random Matrix Theory, and the correlation kernels of these processes can be expressed in terms of special functions/double contour integrals. This enables to investigate determinantal processes for products of random matrices in different asymptotic regimes, and to compute different probabilistic quantities of interest. In particular, I will present the asymptotics for the hole probabilities, i.e. for probabilities of the events that there are no particles in a disc of radius r with its center at 0, as r goes to infinity. In addition, I will explain how the gap probabilities for squared singular values of products of random complex matrices can be described in terms of completely integrable Hamiltonian differential equations, and how to interpret these Hamiltonian differential equations as the monodromy preserving deformation equations of the Jimbo, Miwa, Mori, Ueno and Sato theory. Finally, I will discuss certain time-dependent determinantal processes related to products of random matrices.
ThursdayFeb 04, 201611:15
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gaultier Lambert Title:Fluctuations of linear statistics of determinantal processesAbstract:opens in new windowin html    pdfopens in new window
Determinantal point processes arise in the description of eigenvalues of unitary invariant Hermitian random matrices, as well as in many statistical mechanics models such as random tilings, non-intersecting paths, etc. I will explain a cumulant method developed by A. Soshnikov to analyze the asymptotics distributions of linear statistics of determinantal processes and certain combinatorial identities associated with the sine process. I will present some applications to orthogonal ensembles and, if time permits, to certain biorthogonal ensembles and discuss some models which exhibit a transition from Poisson to GUE.
ThursdayJan 28, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Nathan KellerTitle:Stability Versions of Erdös-Ko-Rado Type Theorems via Isoperimetry Abstract:opens in new windowin html    pdfopens in new window

Erdös-Ko-Rado (EKR) type theorems yield upper bounds on the size of set families under various intersection requirements on the elements. Stability versions of such theorems assert that if the size of a family is close to the maximum possible then the family itself must be close (in appropriate sense) to a maximum family. In this talk we present an approach to stability versions of EKR-type theorems through isoperimetric inequalities for subsets of the hypercube. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on t-intersecting families (for k < n/(t+1)), and to show that, somewhat surprisingly, both theorems hold when the "intersection" requirement is replaced by a much weaker requirement. Furthermore, we obtain stability versions of several recent EKR-type results, including Frankl's proof of the Erdös matching conjecture for n>(2s+1)k-s.

Joint work with David Ellis and Noam Lifshitz.

ThursdayJan 07, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Shamgar GurevitchTitle:Low Dimensional Representations of Finite Classical GroupsAbstract:opens in new windowin html    pdfopens in new window

Many questions about properties of a finite group such as random walks, spectrum of Cayley graphs, distribution of word maps etc., can be approached by using “generalized Fourier sum” formulas involving characters of the group. Numerical data show that characters of low dimensional representations of the group contribute the largest terms to these sums. However, relatively little seems to be known about these small representations so a systematic knowledge of them could lead to proofs of some of the properties. The talk will demonstrates, through concrete examples, and numerical simulations, a new method to construct and analyze those small representations, and hence hopefully to solve some of the aforementioned questions.

The talk is intended for non-experts.

This is part from a joint project with Roger Howe (Yale).

WednesdayJan 06, 201611:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Eyal Lubetzky Title:Effect of initial conditions on mixing for spin systemsAbstract:opens in new windowin html    pdfopens in new windownote unusual day

Recently, the "information percolation" framework was introduced as a way to obtain sharp estimates on mixing for spin systems at high temperatures, and in particular, to establish cutoff for the Ising model in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic.

Joint work with Allan Sly.

ThursdayDec 31, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Zemer KosloffTitle:Symmetric Birkhoff sums in infinite ergodic theoryAbstract:opens in new windowin html    pdfopens in new window

By a Theorem of Aaronson, normalized Birkhoff sums of positive integrable functions in infinite, ergodic systems either tend to 0 almost surely or there is a subsequence along which every further subsequence tends to infinity. This is not true for normalized symmetric Birkhoff sums where the summation is along a symmetric time interval as there are examples of infinite, ergodic systems for which the absolutely normalized symmetric Birkhoff sums of positive integrable functions may be almost surely bounded away from zero and infinity. In this talk I will start by explaining a variety of transformations (of different nature) satisfying this phenomena, discuss the case main result that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge point-wise and there even exists a universal divergence statement. Time permits I will show some examples of actions of other groups which converge and some recent (yet unwritten) results on actions by commuting skew products which are related to self intersection local times.

The contents of this talk are a combination of 3 papers, one of which is a joint work with Benjamin Weiss and Jon Aaronson and another one is work in progress with Jon Aaronson.

ThursdayDec 17, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir DemboTitle:Extremal Cuts of Sparse Random GraphsAbstract:opens in new windowin html    pdfopens in new window

The Max-Cut problem seeks to determine the maximal cut size in a given graph. With no polynomial-time efficient approximation for Max-Cut (unless P=NP), its asymptotic for a typical large sparse graph is of considerable interest. We prove that for uniformly random d-regular graph of N vertices, and for the uniformly chosen Erdos-Renyi graph of M=Nd/2 edges, the leading correction to M/2 (the typical cut size), is P∗sqrt(NM/2). Here P∗ is the ground state energy of the Sherrington-Kirkpatrick model, expressed analytically via Parisi's formula.

This talk is based on a joint work with Subhabrata Sen and Andrea Montanari.

ThursdayDec 10, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ohad FeldheimTitle:Double Roots of Random PolynomialsAbstract:opens in new windowin html    pdfopens in new window

We consider random polynomials of degree n whose coefficients are i.i.d. distributed over a finite set of integers, with probability at most 1/2 to take any particular value. We show that the probability that such a polynomial of degree n has a double root is dominated by the probability that 0,1 or -1 are double roots up to an error of o(n−2). Our result generalizes a similar result of Peled, Sen and Zeitouni for Littlewood polynomials.

Joint work with Ron Peled and Arnab Sen.

ThursdayDec 03, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ron RosenthalTitle:Eigenvalue confinement and spectral gap for random simplicial complexesAbstract:opens in new windowin html    pdfopens in new window
We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on $n$ vertices, where each $d$-cell is added independently with probability $p$ to the complete $(d-1)$-skeleton. From the point of view of random matrix theory, the adjacency matrix is a sparse, self adjoint random matrix with dependent entries. Under the assumption $np(1-p)>> log^4 n$, we prove that the spectral gap between the $\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$ eigenvalues is $np-2\sqrt{dnp(1-p)}(1+o(1))$ with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. Based on a joint work with Antti Knowles.
ThursdayNov 26, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yaar SolomonTitle:The Danzer problem and a solution to a problem of Gowers Abstract:opens in new windowin html    pdfopens in new window
Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].
ThursdayNov 12, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Christopher JoynerTitle:Random Walk approach to spectral statistics in random Bernoulli matricesAbstract:opens in new windowin html    pdfopens in new window
Random Bernoulli matrices (in which the matrix elements are chosen independently from plus or minus 1 with equal probability) are intimately connected to the adjacency matrices of random graphs and share many spectral properties. In the limit of large matrix dimension the distribution of eigenvalues from such matrices resembles that from matrices in which the elements are chosen randomly from a Gaussian distribution - the question is why? We take a dynamical approach to this problem, which is achieved by initiating a discrete random walk process over the space of matrices. Previously we have used this idea to analyse the corresponding eigenvalue motion but I will discuss some recent developments which involve the adaptation of Stein's method to this context.
ThursdayAug 06, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 208
Speaker:Balázs RáthTitle:Voter model percolationAbstract:opens in new windowin html    pdfopens in new windowplease note unusual room

The voter model on $\Z^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d \geq 3$, the set of (extremal) stationary distributions is a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a strongly correlated field of 0's and 1's on $\Z^d$ in which the density of 1's is $\alpha$.

We consider such a configuration as a site percolation model on $\Z^d$. We prove that if $d \geq 5$, the probability of existence of an infinite percolation cluster of 1's exhibits a phase transition in $\alpha$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d \geq 3$.

These results partially settle a conjecture of Bricmont, Lebowitz and Maes (1987).
Joint work with Daniel Valesin (University of Groningen)

ThursdayJul 30, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Eviatar ProcacciaTitle:Stationary Eden model on groupsAbstract:opens in new windowin html    pdfopens in new window

We consider two stationary versions of the Eden model, on the upper half planar lattice, resulting in an infinite forest covering the half plane. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. Using the mass transport principle, we generalize the result to Eden model in graphs of the form $G \times Z$, where G is a Cayley graph. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Häggström in 2007.

ThursdayJul 16, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 1
Speaker:Alexander FishTitle:Ergodic theorems for amenable groupsAbstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL ROOM

We will talk on the validity of the mean ergodic theorem along left Følner sequences in a countable amenable group G. Although the weak ergodic theorem always holds along any left Følner sequence in G, we will provide examples where the mean ergodic theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g. if G is virtually nilpotent, then we will prove that the mean ergodic theorem does indeed hold along any left Følner sequence. Based on the joint work with M. Björklund (Chalmers).

ThursdayJul 09, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Dan FlorentinTitle:Stability and Rate of Convergence of the Steiner SymmetrizationAbstract:opens in new windowin html    pdfopens in new window
We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of Steiner symmetrization.
ThursdayJul 02, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Dan FlorentinTitle:Stability and Rate of Convergence of the Steiner SymmetrizationAbstract:opens in new windowin html    pdfopens in new window
We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of Steiner symmetrization.
ThursdayJun 25, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yinon SpinkaTitle:Long-range order in random 3-colorings of Z^dAbstract:opens in new windowin html    pdfopens in new window

Consider a random coloring of a bounded domain in Zd with the probability of each coloring F proportional to exp(−β∗N(F)), where β>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecký conjecture is that in such a model, for d≥3 and high enough β, a sampled coloring will typically exhibit long-range order, placing the same color at most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case β=∞.

The main ingredient in our proof is a new structure theorem for 3-colorings which characterizes the ways in which different "phases" may interact, putting special emphasis on the role of edges connecting vertices of the same color. We also discuss several related conjectures. No background in statistical physics will be assumed and all terms will be explained thoroughly.

Joint work with Ohad Feldheim.

ThursdayJun 18, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Amir DemboTitle:The Atlas model, in and out of equilibriumAbstract:opens in new windowin html    pdfopens in new window

Consider a one-dimensional semi-infinite system of Brownian particles, starting at Poisson (L) point process on the positive half-line, with the left-most (Atlas) particle endowed a unit drift to the right. We show that for the equilibrium density (L=2), the asymptotic Gaussian space-time particle fluctuations are governed by the stochastic heat equation with Neumann boundary condition at zero. As a by product we resolve a conjecture of Pal and Pitman (2008) about the asympotic (random) fBM trajectory of the Atlas particle.

In a complementary work, we derive and explicitly solve the Stefan (free-boundary) equations for the limiting particle-profile when starting at out of equilibrium density (L other than 2). We thus determine the corresponding (non-random) asymptotic trajectory of the Atlas particle.

This talk is based on joint works with Li-Cheng Tsai, Manuel Cabezas, Andrey Sarantsev and Vladas Sidoravicius.

ThursdayJun 11, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Anton MalyshevTitle:Metric distortion between random finite subsets of the intervalAbstract:opens in new windowin html    pdfopens in new window

Consider a random finite metric space X given by sampling n points in the unit interval uniformly, and a deterministic finite metric space U given by placing n points in the unit interval at uniform distance. With high probability, X will contain some pairs of points at distance roughly 1/n^2, so any bijection from X to U must distort distances by a factor of roughly n. However, with high probability, two of these random spaces, X_1 and X_2, have a bijection which distorts distances by a factor of only about n^2/3. The exponent of 2/3 is optimal.

ThursdayJun 04, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Mark RudelsonTitle:Approximation complexity of convex bodiesAbstract:opens in new windowin html    pdfopens in new window
Consider the approximation of an n-dimensional convex body by a projection of a section of an N-dimensional simplex, and call the minimal N for which such approximation exists the approximation complexity of the body. The reason for such strange definition lies in computer science. A projection of a section of a simplex is the feasible set of a linear programming problem, and so it can be efficiently generated. We will discuss how large the approximation complexity of different classes of convex bodies can be.
ThursdayMay 28, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gerard Ben ArousTitle:The ant in the labyrinth: recent progressAbstract:opens in new windowin html    pdfopens in new window
ThursdayMay 21, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Lenya RyzhikTitle:The weakly random Schroedinger equation: a consumer reportAbstract:opens in new windowin html    pdfopens in new window
Consider a Schroedinger equation with a weakly random time-independent potential. When the correlation function of the potential is, roughly speaking, of the Schwartz class, it has been shown by Spohn (1977), and Erdos and Yau (2001) that the kinetic limit holds -- the expectation of the phase space energy density of the solution converges weakly (after integration against a test function, not in the probabilistic sense) to the solution of a kinetic equation. We "extend" this result to potentials whose correlation functions satisfy (in some sense) "sharp" conditions, and also prove a parallel homogenization result for slowly varying initial conditions. I will explain the quotation marks above and make some speculations on the genuinely sharp conditions on the random potential that separate various regimes. This talk is a joint work with T. Chen and T. Komorowski
WednesdayMay 20, 201511:00
Geometric Functional Analysis and Probability SeminarRoom 108
Speaker:Kate JuschenkoTitle:Amenability of subgroups of interval exchange transformation groupAbstract:opens in new windowin html    pdfopens in new window
ThursdayMay 07, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Ehud FriedgutTitle:An information theoretic proof of a hypercontractive inequalityAbstract:opens in new windowin html    pdfopens in new window

In the famous KKL (Kahn-Kalai-Linial) paper of 1988 the authors "imported" to combinatorics and theoretical computer science a hypercontractive inequality known as Beckner's ineqaulity (proven first, independently, by Gross and Bonami). This inequality has since become an extremely useful and influential tool, used in tens of papers, in a wide variety of settings. In many cases there are no proofs known that do not use the inequality.

In this talk I'll try to illuminate the information theoretic nature of both the inequality and its dual, touch upon a proof of the dual version from about a decade ago, (joint with V. Rodl), and a fresh (and unrelated) information theoretic proof of the primal version.

No prior knowledge will be assumed regarding discrete Fourier analysis, Entropy, and hypercontractivity.

ThursdayApr 16, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Piotr MilosTitle:Extremal individuals in branching systemsAbstract:opens in new windowin html    pdfopens in new window

Branching processes have been subject of intense and fascinating studies for a long time. In my talk I will present two problems in order to highlight their rich structure and various technical approaches in the intersection of probability and analysis.
Firstly, I will present results concerning a branching random walk with the time-inhomogeneous branching law. We consider a system of particles, which at the end of each time unit produce offspring randomly and independently. The branching law, determining the number and locations of the offspring is the same for all particles in a given generation. Until recently, a common assumption was that the branching law does not change over time. In a pioneering work, Fang and Zeitouni (2010) considered a process with two macroscopic time intervals with different branching laws. In my talk I will present the results when the branching law varies at mesoscopic and microscopic scales. In arguably the most interesting case, when the branching law is sampled randomly for every step, I will present a quenched result with detailed asymptotics of the maximal particle. Interestingly, the disorder has a slowing-down effect manifesting itself on the log level.
Secondly, I will turn to the classical branching Brownian motion. Let us assume that particles move according to a Brownian motion with drift μ and split with intensity 1. It is well-know that for μ≥2√ the system escapes to infinity, thus the overall minimum is well-defined. In order to understand it better, we modify the process such that the particles are absorbed at position 0. I will present the results concerning the law of the number of absorbed particles N. In particular I will concentrate on P(N=0) and the maximal exponential moment of N. This reveals new deep connections with the FKPP equation. Finally, I will also consider −2√<μ<2√ and Nxt the number of particles absorbed until the time t when the system starts from x. In this case I will show the convergence to the traveling wave solution of the FKPP equation for an appropriate choice of x,t−>∞.
The results were obtained jointly with B. Mallein and with J. Berestycki, E. Brunet and S. Harris respectively.

ThursdayMar 26, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Tom HutchcroftTitle:Hyperbolic and Parabolic Random MapsAbstract:opens in new windowin html    pdfopens in new window
We establish a sharp division of infinite random planar graphs into two types, hyperbolic and parabolic, showing that many probabilistic and geometric properties of such a graph are determined by the graph's average curvature, a local quantity which is often easy to compute. Work in progress with Omer Angel, Asaf Nachmias and Gourab Ray.
ThursdayMar 26, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Tom HutchcroftTitle:Hyperbolic and Parabolic Random MapsAbstract:opens in new windowin html    pdfopens in new window
We establish a sharp division of infinite random planar graphs into two types, hyperbolic and parabolic, showing that many probabilistic and geometric properties of such a graph are determined by the graph's average curvature, a local quantity which is often easy to compute. Work in progress with Omer Angel, Asaf Nachmias and Gourab Ray.
ThursdayMar 12, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Toby JohnsonTitle:The frog model on treesAbstract:opens in new windowin html    pdfopens in new window

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model.

I'll talk about a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge.

ThursdayFeb 12, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Chaim Even ZoharTitle:Invariants of Random Knots and LinksAbstract:opens in new windowin html    pdfopens in new window
We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-three knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. All terms above will be defined and explained. Joint work with Joel Hass, Nati Linial, and Tahl Nowik.
ThursdayFeb 05, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Yuval PeledTitle:On the phase transition in random simplicial complexesAbstract:opens in new windowin html    pdfopens in new window
It is well-known that the model of random graphs undergoes a dramatic change around p=1/n. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant connected component. Several years ago, Linial and Meshulam have introduced the X_d(n,p) model, a probability space of n-vertex d-dimensional simplicial complexes, where X_1(n,p) coincides with G(n,p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from X_d(n,p), and show that it is strictly greater than the threshold of d-collapsibility. In addition, we compute the real Betti numbers, i.e. the dimension of the homology groups, of X_d(n,p)for p=c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d=1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d > 1 the emergence of the giant shadow is a first order phase transition. The talk will contain the necessary toplogical backgorund on simplicial complexes, and will focus on the main idea of the proof: the local weak limit of random simplicial complexes and its role in the analysis of phase transitions. Joint work with Nati Linial.
ThursdayJan 29, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Gady KozmaTitle:Random walk in random environment: the operator theory approachAbstract:opens in new windowin html    pdfopens in new window

Examine random walk in a stationary, ergodic, random environment which is bistochastic i.e. the sum of probabilities to enter any fixed vertex is 1. Consider the drift as a function on the probability space on the environments, and assume it belongs to domain of definition of where D is the symmetrized generator of the walk (this is the famous  The Actual Formula condition). We show that under these conditions the walk satisfies a central limit theorem. The proof uses the "relaxed sector condition" which shows an unexpected connection to the spectral theory of unbounded operators.

All terms will be explained in the talk. This is joint work with Balint Toth.

ThursdayJan 22, 201511:05
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Thomas LebleTitle:Large deviations for the empirical field of Coulomb and Riesz systemsAbstract:opens in new windowin html    pdfopens in new window
We study a system of $N$ particles with Coulomb/Riesz pairwise interactions under a confining potential. After rescaling we deal with a microscopic quantity, the associated empirical point process, for which we give a large deviation principle whose rate function is the sum of a relative entropy and of the "renormalized energy" defined by Sandier-Serfaty. We also present applications to point processes emerging from random matrix theory. This is joint work with S. Serfaty.
ThursdayDec 18, 201410:30
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Alexander FishTitle:Plunnecke inequalities in countable abelian groups - general caseAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL TIME
Plunnecke inequalities for sumsets of finite sets in abelian groups are extended to measure -preserving systems (mps). For a set A in a group, and a set B of positive measure in mps, we estimate the measure of the union of translations along the set A of B. To prove the new inequalities we extend the graph-theoretic method recently developed by Petridis to "measure graphs". As an application, through Furstenberg's correspondence principle, we obtain the new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Based on joint works with M. Bjorklund, Chalmers, and with Kamil Bulinski, Sydne
ThursdayAug 28, 201411:00
Geometric Functional Analysis and Probability SeminarRoom 261
Speaker:Laurie FieldTitle:Two-sided radial SLE and length-biased chordal SLEAbstract:opens in new windowin html    pdfopens in new windowFilter test
Models in statistical physics often give measures on self-avoiding paths. We can restrict such a measure to the paths that pass through a marked point, obtaining a "pinned measure". The aggregate of the pinned measures over all possible marked points is just the original measure biased by the path's length. Does the analogous result hold for SLE curves, which appear in the scaling limits of many such models at criticality? We show that it does: the aggregate of two-sided radial SLE is length-biased chordal SLE, where the path's length is measured in the natural parametrisation.