Mathematics

Amitai Regev, Head
The Herman P. Taubman Professor of Mathematics

The principal research interests of the Department lie in the two general areas of mathematical analysis and its applications, and of algebra, mainly representation theory, algebraic geometry, and number theory. Topics covered in analysis include the structure of finite and infinite dimensional spaces, 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, control theory in its various manifestations, optimization, game theory and mathematical economics, approximation and complexity of functions, numerical analysis, singularity theory, and robotics. The algebraic direction includes some aspects of algebraic geometry, 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: Study has been started on integration on p-adic analytic spaces. The purpose is to extend the class of analytic functions on a smooth p-adic analytic space to a broader class of locally analytic functions so that the corresponding de Rham complex is an exact resolution of the constant sheaf. Some progress has been made on integration of one-forms.

Automorphic forms: Here the representation theory of these forms was studied from two separate points of view. First, it was (finally) shown that for the quasi-split unitary group in three variables, every tempered L-packet of cuspidal automorphic representations contains a globally generic representation. Second, work was finished on analyzing the entirety and boundedness in vertical strips of the automorphic L-functions that appear in constant terms of Eisenstein series a la Langlands.

Banach spaces: The geometry of infinite dimensional normed spaces and maps between them is investigated, particularly the classification of Banach spaces under Lipschitz and uniform homeomorphisms, and under Lipschitz and uniform quotient maps.

Differential and integral operators: An explicit functional calculus for various degenerate operators related to the Heisenberg group was studied. The cosine of the square root of the sum of squares of self adjoint operators was computed in terms of the cosines of those operators, both in the commutative and in the non-commutative case. Formulas for higher order heat invariants of the Laplacian were suggested.

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.

Discrete spectrum below the bottom of the essential spectrum was studied for elliptic periodic waveguide-type operators, perturbed by a decaying potential. It was shown that the classical Weyl eigenvalue asymptotic formula for this sort of problems is, in general, violated, and its substitute was found.

Dynamical systems: The singular limit of slow and fast interacting components was modeled as coupled vector and measure-valued dynamics.

Hilbert 16th problem: Considerable progress was achieved towards solving the so called tangential form of Hilbert sixteenth problem, concerning an upper bound for the number of isolated complex zeros of complete Abelian integrals. Following the pattern established earlier, a significant step was made towards the solution of the problem for arbitrary bivariate Hamiltonians. A redundant system of the Picard-Fuchs equations for Abelian integrals was derived and investigated.

Deep relations between Hilbert's problem (as well as another closely connected one - Poincare's Centre-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.

Operator theory and Matrix Function theory: The theory of the joint system realization of rational matrix function is developed. Applications to the Fuchsian differential systems are carried out. Interpolation problems for matrix valued functions in the Schur class are studied. In particular, a comprehensive study of the case in which interpolation constraints are imposed on the boundary of the region of interest was undertaken. The investigation of the inverse monodromy problem for canonical integral and differential system continues. In particular a parameterization of the set of all solutions to this problem was given under the assumption that the monodromy matrix is strongly regular. A number of classes of uniqueness were also delineated.

Optimization and control: Several aspects of stability, control and optimizations are under investigation. The control of coupled slow and fast motions is examined. The model is of singular perturbations with measure-valued variables forming the relaxation of the fast variables. Stabilization by hybrid control is developed, where discrete digital interventions combine with continuous analog feedback. The role of information in dynamic optimization is examined, and is applied to decision processes.

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, and the study of convex sets in high dimensional Euclidean space.

Representation theory and related topics: This concerns the representation theory of algebraic groups, enveloping algebras and quantum groups -- specifically, at present, the contruction and determination of the KPRV determinants in the semisimple, affine, quantum and super frameworks, the determination of semi-invariants for parabolic subalgebras and the topology of primitive ideals in the semisimple case.

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.

Research Staff, Visitors and Students

Professors

Zvi Artstein, Ph.D., The Hebrew University of Jerusalem

The Hettie H. Heineman Professor of Mathematics

Vladimir Berkovich, Ph.D., University of Moscow

The Matthew B. Rosenhous Professor (from March 2000)

Aryeh Dvoretzky, Ph.D., The Hebrew University of Jerusalem

Institute Professor

Harry Dym, Ph.D., Massachusetts Institute of Technology

The Renee and Jay Weiss Professor

Stephen Gelbart, Ph.D., Princeton University

The Nicki and J. Ira Harris Professor

Anthony Joseph, Ph.D., University of Oxford

The Donald Frey Professor

Yakar Kannai, Ph.D., The Hebrew University of Jerusalem

The Erica and Ludwig Jesselson Professor of Theoretical Mathematics

Victor Katsnelson, Ph.D., Kharkov University, Kharkov

The Ruth and Sylvia Shogam Professor

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

The Herman P. Taubman Professor of Mathematics

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

The William Petschek Professor of Mathematics

Oded Schramm, Ph.D., Princeton University, Princeton, NJ

The Sam and Ayala Zacks Professor

Yosef Yomdin, Ph.D., Novosibirsk State University

The Moshe Porath Professor of Mathematics

Professors Emeriti

Michail Solomyak, Ph.D., University of Leningrad

Associate Professors

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

Incumbent of the Louis and Ida Rich Career Development Chair

Sergey Yakovenko, Ph.D., Moscow State University, Moscow

Senior Scientist

Victor Vinnikov, Ph.D., Ben-Gurion University of the Negev, Beersheva (left January 2000)

Incumbent of the Graham and Rhona Beck Career Development Chair (until January 2000)

Junior Staff Scientist

Nina Roytvarf, Ph.D., The Weizmann Institute of Science

Consultants

Boris Freydin, Ben-Gurion University of the Negev, Beer-Sheva
Vladimir Hinich, University of Haifa, Haifa
Anna Melnikov, Center for Technological Education, Holon
Andre Reznikov, Tel Aviv University
Yosef Stein, Max-Planck-Institut fur Mathematik, Bonn, Germany (left September 2000)
Victor Zalgaller

Visiting Scientists

El Alaoui Abdelhafid, University of Paris & M. M. Curie, Paris, France
Damir Z. Arov, South Ukranian Pedagogical University, Odessa, Ukraine
Vladimir Dubovoy, Kharkov State University, Kharkov, Ukraine
Vladimir Gurari, Institute for Chemical Physics, Moscow, Russia
Yurii Lyubarskii, Norwegian University of Science & Technology, Trondheim, Norway
Pierre Milman, University of Toronto, Toronto, Ontario, Canada

Postdoctoral Fellows

Michael Entov, Ph.D., Stanford University, Stanford, CA.
Subhajit Ghosechowdry, Ph.D., Purdue University, Lafayette, IN
Maria Gorelik, Ph.D., Weizmann Institute of Science
Cyril Grunspan, Ph.D., Ecole Polytechnique, Palaiseau, France
Ofer Hadas, Ph.D., Tel Aviv University, Ramat Aviv (until August 2000)
Anne Henke, Ph.D., University of Oxford, Oxford, England
David Holcman, Ph.D., Pierre et Marie Curie University, Paris, France

Michael Kaltenbaeck, Ph.D., Institute fuer Analysis und Technische Math., Wien, Austria

(until August 2000)

Emmanuel Lanzmann, Ph.D., Universite Paris 6, Paris, France (until September 2000)

Michael Margaliot, Ph.D., Tel Aviv University, Ramat Aviv (until September 2000)
Paul Mezo, Ph.D., University of Toronto, Toronto, Ontario, Canada
Shahar Nevo, Ph.D., Bar Ilan University, Ramat Gan
Fedor Pakovitch, Ph.D., L'Universite Grenoble 1, Cedex, France
Ekaterina Shulman, Ph.D., Moscow State Institute of Electronics and Mathematics Technical University, Moscow, Russia

(until May 2000)

Roman Vershynin, Ph.D., University of Missouri, Columbia, Missouri
Suiping Zhou, Ph.D., Beijing University of Aeronautics and Astronautics, Beijing, P.R. China

Research Students

Yevgenia Apartsin	Yuri Bazlov
Mikhail Blinov	Ilan Degani
Jacob Greenstein	Alexei Grigoriev
Nadya Gurevich	Olga Maleva
Katherine Naimark	Elena Perelman
Iosif Polterovich	Constantin Cristian Popa
Tamar Seeman-Emerson	Alexander Shapiro
Dragana Todoric	Dan Volok
Artem Zvavitch

Administrator

Raanan Michael