Workgroups

Mark Elin, Fiana Jacobzon, Marina Levenshtein, David Shoikhet, Filippo Bracci ( Universitá di Roma “Tor Vergata”, Italy), Manuel D.Contreras (Universidad de Sevilla, Spain), Santiago Díaz-Madrigal (Universidad de Sevilla, Spain), and Simeon Reich (Technion, Israel).

 

The development of complex dynamical systems has been the subject of research since the beginning of the 20th century. One of the first applicable models for complex dynamical systems arose from studies of stochastic branching processes in the growth of families and populations. Interest in these models has further increased because of their connections to chemical and nuclear chain reactions, the theory of cosmic radiation, and many other biological and physical problems. The examination of these problems is based on one-parameter semigroups of holomorphic self-mappings of the unit disk of a complex plane, which is our sphere of interest. We study the asymptotic behavior of discrete and continuous time semigroups (in one-dimensional and multi-dimensional settings), rates of convergence of semigroups to their attractive fixed points, and boundary rigidity problems for semigroups and their generators.

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Keywords: Semigroup, infinitesimal generator, iteration theory, asymptotic behavior, rigidity

Aviv Gibali, Valery Y. Glizer, Alexander Goldvard, Vladimir Turetsky, Gideon Avigad (Department of Mechanical Engineering), Emilia Fridman (Tel-Aviv University), Leonid Fridman (National Autonomous University of Mexico, Mexico) and Galina A. Kurina (Voronezh State University, Russia), Yair Censor (University of Haifa), Karl-Heinz Küfer (Fraunhofer Institute for Industrial Mathematics (ITWM), Germany) and Philipp Süss (Fraunhofer ITWM, Germany).

 

Control theory examines ways to manipulate input to a dynamic system in order to obtain desired behavior and output. Differential games theory focuses on the optimal strategies of several agents subject to their differing and often opposing goals. Optimization theory studies methods for choosing an optimal element from a given admissible set.

Prof. Glizer ׳s research focuses on the following: control problems and differential games with singularly perturbed dynamics; cheap control problems; singular control problems; robust control problems; differential games with perfect and imperfect information; differential games with hybrid dynamics; singular differential games; multi-objective differential games; singularly perturbed ODE, PDE, functional-differential equations, difference equations; non-linear stochastic differential and difference equations; nonlinear theory of generalized functions and its applications.

Dr. Goldvard focuses on the solution of multi-criteria (multi-objective) optimization problems via evolutionary algorithms.

Prof. Turetsky is engaged in studying the following: pursuit-evasion games with perfect and imperfect information; robust control; generalized linear-quadratic games; optimal control; cheap control problems; differential games with hybrid dynamics; invariant sets for feedback strategies; inverse problems of signal restoration and differentiation.

Dr. Gibali ׳s research area is Nonlinear Analysis and Optimization Theory. In particular developing and modifying iterative projection methods for solving variational inequalities, feasibility and fixed-points problems with applications to real-world problems such as intensity-modulated radiation therapy (IMRT) treatment planning.

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Keywords: System analysis, control design, non-cooperative and antagonistic games, multi-objective optimization

Mark Elin, Fiana Jacobzon, Marina Levenshtein, David Shoikhet, Dov Aharonov (Technion, Israel), Lev Aizenberg (Bar-Ilan University, Israel), Vladimir Bolotnikov (College of William and Mary, USA), Dmitry Khavinson (University of South Florida, USA) Nikolai Tarkhanov (Universität Potsdam, Germany) and Lawrence Zalcman (Bar-Ilan University, Israel).

 

Geometric function theory focuses on the geometric properties of univalent mappings. This subject has been examined with changing emphasis for over a hundred years. Well-known results in this field include the Riemann mapping theorem, hyperbolic geometry, the Schwarz Lemma, the Julia-Wolff-Caratheodory Theorem and others.

Our research focuses on biholomorphic mappings on a unit ball (in one-dimensional and multi-dimensional complex spaces). We study the geometric structures of these mappings, including starlike and spiral-like mappings with respect to an interior point or a boundary point, convex functions in one direction and so on. Geometric characteristics of images involve distortion and covering theorems and boundary behavior of different classes of mappings, as well as interpolation problems.

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Keywords: Starlike, spirallike functions, distortion theorems, boundary behavior, angular derivative

Mark Berman, Gabriel Ben-Simon, Dietmar Salamon (ETH Zürich, Switzerland), Marc Burger (ETH Zürich, Switzerland),Tobias Hartnick (Technion), Frol Zapolsky (University of Haifa), Egor Shelukhin (Tel Aviv University), Leonid Polterovich (University of Tel Aviv), Michael Schein (Bar-Ilan University), Uri Onn (Ben Gurion University), Benjamin Klopsch (University of Düsseldorf, Germany) and Christopher Voll (University of Bielefeld, Germany).

 

Groups encapsulate symmetry and are fundamental objects in both pure and applied mathematics. The Theory of Groups constitutes a fundamental element of Pure Mathematics – appearing in Algebra, Number Theory, Combinatorics, Algebraic Topology, Algebraic Geometry, Galois Theory and Class Field Theory – as well as in many other areas of science such as Physics, Chemistry, Cryptography and Computer Science.

Dr. Ben-Simon’s research focusses on geometric group theory, finite and infinite dimensional Lie groups, the study of Newton’s equations via Hamiltonian dynamics and the application of geometric group theory to symplectic manifolds via quasi-morphisms.

Dr. Berman’s research focusses on zeta functions of finitely generated nilpotent groups; these are analytic functions that encode the numbers of finite index subgroups in a given group. In particular, techniques are employed from the theory of algebraic groups, Lie algebras, representation theory, p-adic integration and combinatorics to determine special features of such zeta functions.

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Keywords: geometric group theory, Lie groups, symplectic manifolds, finitely generated nilpotent groups, algebraic groups,representation theory, combinatorics, zeta functions of groups, subgroup growth, p-adic integration

Buma Abramovitz, Miryam Berezina, Ludmila Shvartzman, Fiana Yacobzon, Abraham Berman (Technion, Israel) and Boris Koichu (Technion, Israel).

 

The main purpose of this research is to develop methods for teaching mathematics at the undergraduate level in order to improve students’ understanding.

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Keywords: Mathematical education, understanding, undergraduate level

Mark Elin, Victor A. Khatskevich, David Shoikhet, Daniel Alpay (Ben-Gurion University of the Negev, Israel), Tomas Azizov (Voronezh State University, Voronezh, Russia), Lawrence Harris (University of Kentucky, USA), Jurii Kozicki (University of Maria Curie-Sklodowska, Poland), Tadeusz Kuczumov (University of Maria Curie-Sklodowska, Poland),Simeon Reich (Technion, Israel), Victor Shulman (Vologda State Technical University, Russia) and Jaroslav Zemanek (Polish Academy of Sciences, Warsaw, Poland).

 

This research focuses on fixed point theory and operator methods and their applications to holomorphic semigroups, dichotomy in spaces of operators, Abel and Schroeder type functional equations and others. We aim to give some parametric descriptions of bistrict plus operators and some groups of automorphisms of the unit operator ball. We are also interested in the structure of images of operator linear fractional relations.

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Keywords: Hilbert space, orthogonal systems of vectors, linear operator, spectrum of operator