מכון איינשטיין למתמטיקה Einstein Institute of Mathematics

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Annual lecture series in memory of Prof. Aryeh Dvoretzky

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In 2009, the Einstein Institute of Mathematics at the Hebrew University of Jerusalem established an annual lecture series in memory of Professor Aryeh Dvoretzky (1916-2008).

Each lecturer will deliver two lectures. It is suggested that one lecture be accessible to the university scientific community at large (colloquium-style) and the additional lecture may be more specialized and of interest to professional mathematicians and specialists.

   2017    Prof. Alex Eskin    Chicago

1. Lecture 1. Thursday, May 18th, 14:30    Department Colloquium    Lecture Hall 2
   Polygonal billiards and dynamics on moduli spaces      Abstract
   Billiards in polygons can exhibit some bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry (and in particular Hodge theory), Teichmüller theory and ergodic theory on homogeneous spaces.
   I will attempt to give a gentle introduction to the subject.
   A large part of this talk will be accessible to undergraduates.
   
   This lecture is aimed at a general audience.

2. Lecture 2. Tuesday, May 23rd, 14:00    Ross 70
   On stationary measure rigidity and orbit closures for actions of non-abelian groups      Abstract
   I will describe joint work in progress with Aaron Brown, Federico Rodriguez-Hertz and Simion Filip. Our aim is to find some analogue, in the context of smooth dynamics, of Ratner's theorems on unipotent flows. This would be a (partial) generalization of the results of Benoist-Quint and my work with Elon Lindenstrauss in the homogeneous setting, the results of Brown and Rodriguez-Hertz in dimension 2, and the my results with Maryam Mirzakhani in the setting of Teichmüller dynamics.

Past lectures

• 2016    On the centenary of Prof. Dvoretzky's birth
  Prof. Lai-Sang Young    Courant Institute of Mathematical Sciences
  Ergodic theory of differentiable dynamical systems
  1. Lecture 1. Thursday, April 7th, 14:30    Department Colloquium    Lecture Hall 2
     Measuring dynamical complexity      Abstract
     I will discuss, for differentiable dynamical systems, three ways to capture dynamical complexity:

     1. hyperbolicity, which measures the sensitivity of dependence on initial conditions.
        
     2. entropy, which measures the predictability of future dynamical events in the sense of information theory.
        
     3. the speed of correlation decay or equivalently the rate at which memory is lost.

     I will review these ideas in nontechnical terms, present theorems showing how they are related, and give a very brief (and somewhat personal) survey of the progress made in the last decades. For illustration, I will show how these results apply to a concrete example: shear- induced chaos in periodically kicked oscillators, a phenomenon closely related to that observed by van der Pol nearly 100 years ago.
     
     This lecture is aimed at a general audience.

  2. Lecture 2. Sunday, April 10th, 16:00    Lecture Hall 2
     Proving the positivity of Lyapunov exponents      Abstract
     A signature of chaotic behavior in dynamical systems is sensitive dependence on initial conditions, and Lyapunov exponents measure the rates at which nearby orbits diverge. One might expect that geometric expansion or stretching in a map would lead to positive Lyapunov exponents. This, however, is very difficult to prove - except for maps with invariant cones (or a priori separation of expanding and contracting directions). An example that has come to symbolize the challenge is the standard map: there is a parameter that corresponds to the amount of stretching, and for no value of this parameter, no matter how large, has anyone been able to prove the positivity of Lyapunov exponents. In this talk, I would like to explain the reasons behind this challenge, and to show that a tiny amount of noise can make the problem much more tractable.

• 2015      Prof. Terence Tao    UCLA
  1. Lecture 1, Wednesday March 18th at 14:00    Watch on
     Local universality for random matrices and zeroes of random polynomials         Abstract
     The eigenvalues of random matrices, as well as the zeroes of random polynomials, often obey remarkable universality properties: the asymptotic distribution of these point processes (after suitable normalisation) is often independent of the distribution of the component random variables comprising the matrices or polynomials. In particular, the asymptotics of discrete random matrix or polynomial models often agree with those of their continuous counterparts.
     There are now several techniques to rigorously establish this universality.
     Here we focus on one such technique, based on the classical Lindeberg exchange strategy, first used to give a non-Fourier-analytic proof of the central limit theorem, and survey some of the results obtained by this method.
  2. Lecture 2, Thursday March 19th at 14:30 (at the colloquium)    Watch on
     Small and large gaps between the primes         Abstract
     There are many questions about the gaps between consecutive prime numbers which are not completely solved, even after decades of effort. For instance, the twin prime conjecture, which asserts that the gap between primes can equal 2 infinitely often, remains open.
     However, there has been recent progress in understanding both very small and very large gaps between primes. Last year, in a breakthrough work of Yitang Zhang, it was shown that there were infinitely many gaps between primes of bounded size; Zhang's original bound here was 70 million, but it has since been cut down to 246 thanks to the efforts of James Maynard and an online collaborative Polymath project.
     In the opposite direction, recent work of Kevin Ford, Ben Green, Sergei Konyagin, and myself have shown the existence of prime gaps significantly larger than their average spacing, improving upon earlier work of Rankin, Pintz, and others. The two proofs are somewhat different in nature; the first argument relies on sieve theory methods, whereas the second argument uses probabilistic arguments combined with the results of Ben Green and myself in arithmetic progressions in the primes.
     We present the main ideas of both of these results in this lecture.

• 2013      Prof. H.-T. Yau    Harvard
  1. Lecture 1, Tuesday, May 21th, 2013, at 14:00
     Random matrices, Dyson Brownian motion and De Georgi-Nash-Moser theory of parabolic regularity         Abstract
     Eugene Wigner’s revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior depending only on the symmetry type of the model.
     One class of such models are invariant matrix ensembles for which the eigenvalue distribution distributions are given by Coulomb-gases with potential $V$ and inverse temperature beta.
     In this lecture we will sketch the proof of the universality of Coulomb gases both in the bulk and at the edges of the spectrum.
     We will link the universality problem to the decay of correlation functions of Coulomb gases and show that such a decay follows from a Holder regularity of a discrete parabolic equation with random coefficients.
     The parabolic regularity will be established partly using the recent argument of Caffarelli-Chan-Vasseur, which is a De Giorgi-Nash-Moser type method.
     The singularities in random coefficients pose major challenges; We will use optimal level repulsion estimates in random matrices to control them.
  2. Lecture 2, Thursday, May 23rd, 2013, at 14:30 (at the colloquium)    Universality of random matrices         Abstract
     Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior. These universal statistics represent a new paradigm of point processes that are characteristically different from the Poisson statistics of independent points.
     A prominent example of Wigner's thesis is the Wigner-Dyson-Gaudin-Mehta conjecture asserting that the spectral statistics of random matrices with independent entries depend only on the symmetry classes but are independent of the distributions of matrix elements.
     In this lecture, we will outline the recent solution to this conjecture and demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality.
     Related topics such as delocalization of eigenvectors for random matrices and Erdős-Renyi graphs will also be discussed.

• 2012      Prof. S.R. Srinivasa Varadhan    (NYU)  :  Large deviations    Abstracts
  1. Lecture 1, Thursday, May 17th, 2012, at 16:00 (at the colloquium)    A general view on large deviations
  2. Lecture 2, Monday, May 21st, 2012 at 14:00    Large deviations for nonconventional sums

• 2011      Prof. Ofer Zeitouni    (Weizmann Institute of Science and the University of Minnesota).      March 31st and April 5th, 2011.      Abstracts

  Lecture 1, March 31st:      Branching random walks, Gaussian free fields and thick points: second moments

  The distribution of the location of the rightmost particle of branching Brownian motions in dimension 1 is described by the Kolmogorov-Petrovskii-Piscounov equation. Bramson's work (1982) showed that probabilistic methods, and specifically a variant of the second moment method, are extremely effective in analyzing the solution. Recently, a surprising link between branching Brownian motions and the behavior of the maximum of certain Gaussian fields has been established, shedding light on the fluctuations of the maximum. I will describe this common thread in a discrete setup, starting with one of Dvoretzky's favorite objects: multiple points for Brownian motion.

  Lecture 2, April 5th:      Recursions, Branching Random Walks, and tightness

  I will describe the role that recursions play in the analysis of maxima of branching random walks and of Gaussian random fields; other applications will be pointed out.

• 2010      Prof. Y. Sinai    (Princeton).      April 22nd and 25th, 2010.      Abstracts

  1st lecture, April 22nd:      Navier-Stokes System and Renormalization Group Method

  Renormalization group method is used in dynamics for constructing solutions with special properties. We shall discuss the general problem of constructing solutions with singularities in finite time in equations of fluid dynamics using this method. So far it was successful for dealing with complex-valued solutions which do not satisfy energy inequality.

  2nd lecture, April 25th:      Decay of Fourier Modes in Solutions of the 2-dimensional Navier-Stokes System

  Fourier modes decay exponentially in the case of periodic boundary conditions. In more general cases the behaviour depends on the boundary conditions. The most difficult case is the unit square with Dirichlet boundary conditions. I shall explain the difficulties and some methods how to overcome them.

Prof. Dvoretzky was one of the founding fathers of the institute. He received his Ph.D. in Mathematics from the Hebrew University in 1941, under the supervision of Michael Fekete. Dvoretzky was appointed Professor of Mathematics at the Hebrew University in 1951, and was the first alumnus of the Hebrew University to become a full professor in it. He was a leading analyst and probabilist and had contributed greatly to the development of Israeli mathematical research, including functional analysis, mathematical statistics, probability, game theory and Higher-dimensional algebraic geometry. He was deeply involved in university administration, and among his contributions were Dean of the Faculty of Sciences (1955) and Vice President of the Hebrew University (1959). He was the founder and first director of the Institute for Advanced Studies of the Hebrew University. He also served as chief scientist for the Ministry of Defense. For his many achievements, he received the Israel Prize for Exact Sciences in 1973. Later he became president of the Israeli Academy of Sciences (1974), of which he was one of the founding members, as well as serving as president of The Weizmann Institute of Sciences (1985-1988). External links Aryeh Dvoretzky (1916-2008)  /  Israel Journal of Mathematics 167 (2008). Obituary / Joseph Yahav, Hebrew University of Jerusalem. IMS Bulletin, Volume 37, Issue 7, p.13. Haaretz May 18th, 2008 Aryeh Dvoretzky, 1916 - 2008  /  The MacTutor History of Mathematics archive Aryeh Dvoretzky  /  Oberwolfach photo collection Aryeh Dvoretzky  /  The Mathematics Genealogy Project