The principal research interests of the Department lie in the three general areas of Analysis (understood in the broadest sense, including applications), Geometry and Algebra. ## Mathematics

Sergei Yakovenko, Head

The Gershon Kekst Professorial Chair

Topics covered in Analysis include operator and matrix theory, function theory on the plane, graphs and Riemann surfaces, spectral theory, several aspects of probability, and some applications of statistics, linear and nonlinear ordinary and partial differential equations, harmonic analysis, dynamical systems, ergodic theory, control theory in its various manifestations, optimization, game theory and mathematical economics, approximation and complexity of functions, numerical analysis, singularity theory, and robotics.

The areas of Geometry studied at the Department are the structure of finite and infinite dimensional spaces, geometric aspects of random walks and percolation, real analytic geometry and o-minimal structures, topology of singular holomorphic foliations.

The direction of Algebra includes some aspects of algebraic geometry, non-Archimedian analytic spaces, representation theory, quantum groups, combinatorics, number theory, automorphic forms, ring theory, and enveloping algebras. Although the approach taken is primarily that of pure mathematics, some of the research leans toward possible applications. Listed below is a sample of some of the specific topics that the department's members have pursued lately or are involved in now.

Algebraic Geometry: Work has be continued on so called algebraic and analytic geometry over the field of one element. Roughly speaking, algebraic geometry over the field of one element studies algebraic varieties defined by binomial equations in a way that does not regard the addition operation. This way reveals additional structures of combinatorial nature on binomial algebraic varieties, which are not seen in the presence of the addition operation. Analytic geometry over the field of one element is related to the corresponding algebraic geometry in a similar way as in the classical situation, and studies objects that encode a skeletal structure of algebraic varieties and non-Archimedean analytic spaces.

Analytic theory of ordinary differential equations: A comlete solution of the Infinitesimal Hilbert 16th Problem was achieved: the number of isolated zeros of complete Abelian integrals is explicitly bounded by the double exponential of the degree. A similar bound for the oscillarion of solutions of Fuchsian systems with real spectra of all residues was achieved; in this case the bound is non-uniform but growing polynomially near the frontier of Fuchsian class.

A general finiteness result for zeros of pseudo-Abelian integrals was achieved for perturbation of non-Hamiltonian (Darbouxian) integrable systems.

Deep relations between the Hilbert's problem (as well as another closely connected one - Poincare's Center-Focus problem) and several fields in Classical and modern Analysis and Algebra have been found. Among them Generalized Moments, Several Complex variables, Composition Algebra and D-modules. These promising relations are now investigated.

Analytic and real algebraic geometry: Polynomials associated with geometric objects (smooth manifolds and convex sets) are studied. The volume of a tubular neighborhood of a manufold, as a function of the radius of the manifold, is one example of such a polynomial. Special functions appear naturally in this study. Interesting relations are discovered. In particular, distribution of zeros of such polinomials is investigated for some manifolds.

The Maxwell problem (the question about the number of equilibrium points for the electrostatic field created by three charges) was advanced and an upper bound 12 was achieved (the conjectured number is 4 and previous results were in the range of billions).

A new demonstration of the Gromov theorem was achieved.

Automorphic forms: On the one hand, joint work is finishing up on lower bounds for automorphic L-functions at the edge of the critical strip; on the other, we are seeing how these results fit into the general scheme of bounds for these L-functions.Recently, Langlands's theory of automorphic forms has been looked at from the point of view of p-adic analysis. Although some results on Hecke's L-functions have been successful, much remains to be done. We have been trying to understand what the theory of Langlands-Shahidi might tell us from this point of view. So far, we have worked out the p-adic analysis of the inverse of Riemann's zeta-function.

Banach spaces: The geometry of finite and infinite dimensional normed spaces and maps between them is investigated. A topic of particular interest is classification problems in the class of Banach spaces under Lipschitz and uniform homeomorphisms, and under Lipschitz and uniform quotient maps. Another main subject is tight embedding results, particularly for finite dimensional subspaces and subsets ofL.^{p}

Combinatorics: The study of various permutation statistics on the symmetric groups and on related groups continues. New refinements and extensions of MacMahon's classical equidistribution theorem are found, relating that sub-area of Enumerative Combinatorics to the sub-area of Shape-Avoiding Permutations.

Differential and integral operators: The well-known asymptotic formulas for the Heat Kernel on the Heisenberg Group (for small time) are not uniform near the conjugate line. In particular the power laws are different for generic points and conjugate points. We establish an integral formula whose Laplace asymptotics clearly show how the asymptotic law is deformed as one approaches the conjugate line. The sub-Riemannian nature of the geometry dictates application of the Hamiltonian formalism (on the cotangent bundle) rather than the more conventional approach involving connections on the tangent bundle. Explicit formulas for wave kernels of degenerate hyperbolic and elliptic operators are obtained, using Laplace transformation and sophisticated inversion formulas for products of confluent hypergeometric functions. The resulting formulas involve hypergeometric functions of rational functions of the arguments.

Game theory and mathematical economics: Costs of time and negotiations were incorporated into a dynamic system leading to the Nash bargaining solution for cooperative games. A strategic model of financial markets, in which a central bank determines interest rates and creates money, is developed. Nash equilibria for the price-taking agents are studied. As the number of individual agents increases, the price making strategic behavior becomes indistinguishable from that of price takers, a-la Walrasian models.

Dynamical systems: Coupled slow and fast ordinary differential equations were examined via singular perturbations analysis. Attention was paid to the structure of switching between manifolds of equilibria of the fast motion. Invariant measures of multi-valued dynamics were characterized. Ergodic theory on non-compact spaces is investigated.

Modeling biological system: How long does it take in average for a random particle to escape from a sphere with a small hole? this question is at the basis of many problems related to chemical reactions in microstructures. In particular, it is possible to redefine the forward binding rate constant in confined geometry. Applications go from dendritic spine and synapses, to the analysis of noise in photoreceptors.

Operator theory and Matrix Function theory: Finite dimensional reproducing kernel Krein spaces were used to obtain necessary and sufficient conditions for the existence of solutions to a number of biangential interpolation problems in the extended Schur class of meromorphic matrix valued functions with a finite number of poles in the domain of interest. Linear fractional descriptions of these solutions were also obtained when the conditions for existence were met.

A monograph devoted to the theory of J-contractive and J-inner matrix valued functions and a number of applications of this theory was completed and published by Cambridge University Press.

Optimization and control: The control of coupled slow and fast motions was examined. The model is of singular perturbations with, possibly, measure-valued variables representing the limit of the fast variables. Design of switching modes between manifolds of equilibria or invariant measures of the fast dynamics were examined. The possibility to ignite impulses of the slow dynamics was demonstrated. The limit occupational measures of controlled dynamics were examined utilizing general convexification techniques.

Partial Differential Equations and global analysis: Influence of small noise on dynamical systems on Riemannian compact manifolds can be studied using the asymptotic of the probability density function. As the noise goes to zero, the ground state solution gets concentrated on the subsets of the recurrent set of the dynamical system, where the topological pressure (formulated as a variational problem) is achieved.

Probability and geometry: Several subjects relating probability and geometry of sets in finite dimensional space or in discrete structures are investigated. These include problems pertaining to Statistical Physics; in particular, percolation, random walks on diverse geometrical structures, motion in random media, the study of convex sets in high dimensional Euclidean space, as well as the study of random matrices. Also studied are various aspects of stochastic analysis and filtering theory.

Representation theory and related topics: This concerns the representation theory of algebraic groups, enveloping algebras and quantum groups -- specifically at present, the construction of adapted pairs leading to affine slices for coadjoint orbits, the path model of crystals for Borcherds algebras, embedding of reflection groups in Weyl groups and constructing invariants by reverse transgression.

Another direction is the representation theory of classical Lie superalgebras and related vertex algebras. Specifically, a criterion of simplicity of vacuum module was conjectured and it was proven for simple Lie algebras and some Lie superalgebras. This was applied to study of W-algebras. Vacuum Shapovalov-Kac determinant for Virasoro and Neveu-Schwarz algebras was computed.

For both associative and Lie algebras with polynomial identities, the study of their codimension growth is continued, via the applications of the representation theory of the Symmetric groups. The Vershik-Kerov representation theory of the infinite symmetric group, together with Probability and with the Theory of Symmetric Functions, are applied to the study of combinatorial identities.

Spectral theory of differential operators: The Dirichlet Laplacian in a class of narrow planar domains is considered. The asymptotic behaviour of its spectrum is studied when the width of the domain tends to zero. It is shown that this behaviour is determined, up to the second term of asymptotics, by the germ of the function defining the shape of domain, at its maximal point. Applications to the spectrum of thin waveguides are given.

http://www.wisdom.weizmann.ac.il/

Research Staff, Visitors and Students## Professors

Zvi Artstein, Ph.D., The Hebrew University of Jerusalem, Jerusalem, Israel (on extension of service)

The Hettie H. Heineman Professorial Chair of Mathematics

Itai Benjamini, Ph.D., The Hebrew University of Jerusalem, Jerusalem, Israel

The Renee and Jay Weiss Professorial Chair

Vladimir Berkovich, Ph.D., Moscow State University, Moscow, Russian Federation

The Matthew B. Rosenhaus Professorial Chair

Stephen Gelbart, Ph.D., Princeton University, Princeton, United States

The Nicki and J. Ira Harris Professorial Chair

Anthony Joseph, Ph.D., University of Oxford, United Kingdom (on extension of service)

The Donald Frey Professorial Chair

Yakar Kannai, Ph.D., The Hebrew University of Jerusalem, Jerusalem, Israel (on extension of service)

The Erica and Ludwig Jesselson Professorial Chair of Theoretical Mathematics

Victor Katsnelson, Ph.D., Kharkov University, Kharkov, Ukraine (on extension of service)

The Ruth and Sylvia Shogam Professorial Chair

Gideon Schechtman, Ph.D., The Hebrew University of Jerusalem, Jerusalem, Israel

The William Petschek Professorial Chair of Mathematics

Sergei Yakovenko, Ph.D., Institute of Control Science, Moscow, Russian Federation

The Gershon Kekst Professorial Chair

Yosef Yomdin, Ph.D., Novosibirsk State University, Russian Federation

The Moshe Porath Professorial Professorial Chair in Mathematics

Ofer Zeitouni, Ph.D., Technion - Israel Institute of Technology, Haifa, Israel

The Herman P. Taubman Professorial Chair of Mathematics

## Professors Emeriti

Harry Dym, Ph.D., Massachusetts Institute of Technology, Cambridge, United States

Amitai Regev, Ph.D., The Hebrew University of Jerusalem, Jerusalem, Israel

Michael Solomyak, Ph.D., University of Leningrad, Russian Federation

## Associate Professors

Maria Gorelik, Ph.D., Weizmann Institute of Science, Rehovot, Israel

Gady Kozma, Ph.D., Tel Aviv University, Tel-Aviv, Israel

Omri Sarig, Ph.D., Tel Aviv University, Tel-Aviv, Israel

The Theodore R. and Edlyn Racoosin Professorial Chair

## Senior Scientist

Dmitry Novikov, Ph.D., Weizmann Institute of Science, Rehovot, Israel

## Consultants

Gil Alon, The Open University, Raanana, Israel

Yevgenia Apartsin, Bar-Ilan University

Yosef Bernstein, Tel Aviv University, Tel-Aviv, Israel

Lucas Fresse, The Hebrew University of Jerusalem, Jerusalem, Israel

Crystal Hoyt, Bar-Ilan University, Ramat-Gan, Israel

Elena Litsyn, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Anna Melnikov, University of Haifa, Haifa, Israel

Shahar Nevo, Bar-Ilan University, Ramat-Gan, Israel

Andrei Reznikov, Bar-Ilan University, Ramat-Gan, Israel

Nina Roytvarf

Marshall Slemrod, University of Wisconsin, Madison, WI, USA

Victor Zalgaller

## Visiting Scientists

Avraham Aizenbud, MIT, Mass. Inst. of Tech., U.S.A.

Damir Arov, S. Ukrainian University , Odessa, Ukraine

Michael Bjoerklun, ETH Zurich, Switzerland

Marcin Bobienski, Warsaw University, Poland

Fred Brauer, U. of British Columbia, Canada

Alexander Brudny, University of Calgary, Canada

Yury Burago, Steklov Institute, St Petersburg, RAS, Russia

Laurent Clozel, Universite de Paris -Sud

Amadeu Delshams, Universitat di Catalunia, Barcelona, Spain

Vladimir Derkach, Donetsk National University, Ukraine

Alex Eremenko, Purdue University , W. Lafayette, IN, U.S.A.

Florence Fauquant-Millet, University Jean-Monnet, France

Alexander Fish, University of Wisconsin-Madison, USA

Andrei Gabrielov, Purdue University , W. Lafayette, IN, U.S.A.

Antonio Giambruno, Universita di Palermo, Italy

Vladimir Golubyatnikov, Russian Acad. of Sci., Novosibirsk, Russia

Jacob Greenstein, University of California at Riverside, U.S.A.

Shamgar Gurevitch, U. of Wisconsin, Madison, WI, U.S.A.

William B. Johnson, Texas A&M University , U.S.A.

Anatoly Katok, University of Pennsylvania, U.S.A.

Svetlana Katok, University of Pennsylvania, U.S.A.

Lubomir Gavrilov, Universite Paul Sabatier, Toulouse, France

Boris Khesin, University of Toronto, Canada

Anton Khoroshkin, Independent U. of Moscow, Russia

Nathan Enoch Lewis, Vacciguard, Misgav, Israel

Alexander Loskutov, Moscow State University , Russia

Gennady Lyubeznik, University of Minnesota, Minneapolis, U.S.A.

Leonid Makar-Limanov, Wayne State University , U.S.A.

Pavao Mardesic, Universite de Bourgogne, Dijon, France

Pierre Milman, University of Toronto, Canada

Nikolai Mnev, Steklov Institute, St Petersburg, RAS, Russia

Sebastian Mueller, Technische Unniversitaet Graz, Germany

Asaf Nachmias, MIT, Mass. Inst. of Tech., U.S.A.

Alexander Nazarov, State University , St. Petersburg, Russia

Marina Prokhorova, Russian Acad. of Sci., Ekaterinburg, Russia

Petr Pushkar, Universite Libre de Bruxelles, Belgium

Joseph Romanovsky, Steklov Institute, St Petersburg, RAS, Russia

Vladimir Roubtsov, Universite d'Angers, Angers, France

Grigori Rozenblioum, Chalmers Inst. of Technology, Goteborg, Sweden

Peter Sarnak, Princeton U., Princeton, U.S.A.

Arnab Sen, University of Berkeley, U.S.A.

Bruno Shapira, Universite' Paris-Sud 11

Boris Shapiro, Stockholm University, Sweden

Senya Shlosman, CNRS, France

Adi Shraibman, Tel Aviv University

Vladas Sidoravicius, IMPA, Brazil

Marshall Slemrod, University of Wisconsin, Madison, WI, U.S.A.

Allan Sly, Microsoft Research, USA

Augusto Teixeira, ETHZ, Switzerland

Nina Uraltseva, Steklov Institute, St Petersburg, RAS, Russia

Anatoly Vershik, Russian Acad. of Sci., St. Petersburg, Russia

Jonathan Widom, Northwestern University, Evanston, IL, U.S.A.

Doron Zeilberger, Rutgers University, USA

## Postdoctoral Fellows

Gideon Amir, Weizmann Institute of Science, Israel

Lucas Fresse, Ph.D., Universit Claude Bernard Lyon 1

Ouziel Hadad, Hebrew University of Jerusalem, Israel

Crystal Hoyt, Bar-Ilan University, Ramat-Gan, Israel

Nathan Keller, Hebrew University of Jerusalem, Israel

Veniamin Kisunko, University Of Toronto

Dalia Krieger, Ph.D., University Of Waterloo

Polyxeni Lamprou, Weizmann Institute of Science, Israel

Nir Lev, Tel-Aviv University, Israel

Alexander Rahm, Institut Fourier, Grenoble

Noam Solomon, Ben-Gurion University, Israel

Tamir Tuller, Tel-Aviv University, Israel

Johan Harald Tykesson, Chalmers University Of Technology

David Windisch, ETH, Chalmers Institute, Zurich, Sweden

## Research Students

Avraham Aizenbud Dmitry Batenkov Gal Binyamini Ido Bright Dominik Reinhard Freche Yair Hartman Dvir Haviv Alon Ivtsan Tal Orenshtein Eviatar Procaccia Shifra Reif Niv Moshe Sarig Eric Shellef Omer Tamuz Ekaterina Zavyalova ## Administrator

Raanan Michael