Seminars, 2018

  • November 6, 2018
    The group ring and its units
    Prof. Leo Margolis, Vrije Universiteit Brussel

  • August 8, 2018
    L-functions and the Failure of the Local-Global Principle for G_2
    Dr. Avner Segal, University of British Columbia, Canada

    A common question in mathematics is "Can global questions be answered by local means?", this is usually referred to as the "local-global principle". A famous example is the Hasse-Minkowski theorem on quadratic forms. On the other and, it is also known that the Hasse-Minkowski theorem cannot be extended to cubic forms. In this talk, I will present the local-global principle for automorphic represntations, describe its success in the cuspidal spectrum of the group GL_n and its failure in the cuspidal spectrum of the exceptional group of type G_2.
  • June 5, 2018
    Reflexivity in non-commutative Hardy Algebras'
    Dr. Leonid Helmer, ORT Braude College

    Let H(E) be a non-commutative Hardy algebra associated with a W-correspondence E. These algebras were introduced in 2004 by Muhly and Solel, and generalize the classical Hardy algebra of the unit disc H(D). As a special case one obtains also the algebra Fn of Popescu, which is H(Cn) in our setting. In this paper we view the algebra H(E) as acting on a Hilbert space via an induced representation. We write it ρπ(H(E)) and we study the reflexivity of ρπ(H(E)). This question was studied by Arias and Popescu in the context of the algebra Fn , and by other authors in several other special cases. As it will be clear from our presentation, the extension to the case of a general W-correspondence E over a general W-algebra M requires new techniques and approach.
    We obtain some partial results in the general case and we turn to the case of a correspondence over a factor.
    Under some additional assumptions on the representation : M B(H) we show that ρπ(H(E)) is reflexive. Then we apply these results to analytic crossed products ρπ(H(αM)) and obtain their reflexivity for any automorphism α Aut(M) whenever M is a factor.
  • May 22, 2018
    Nonlinear Schrödinger equations, Lotka-Volterra models, and control of soliton collisions in broadband optical waveguide systems
    Dr. Avner Peleg

    Transmission rates in broadband optical waveguide systems are significantly enhanced by launching many pulse sequences through the same waveguide. Since pulses from different sequences propagate with different group velocities, intersequence pulse collisions are very frequent, and can lead to severe transmission degradation. On the other hand, the energy exchange in pulse collisions can be beneficially used for realizing fast control of the transmission.
    In the current work we develop a general approach for exploiting the energy exchange in intersequence collisions for transmission stabilization and switching, using solitons as the optical pulses. The approach is based on showing that collision-induced amplitude dynamics of N sequences of solitons of the nonlinear Schrödinger (NLS) equation in the presence of dissipative perturbations can be described by N-dimensional Lotka-Volterra (LV) models. To derive the LV models, we first analyze the effects of a single two-soliton collision, using a perturbative expansion in the eigenmodes of the linear operator describing small perturbations about the NLS soliton. We then use stability and bifurcation analysis for the equilibrium points of the LV models to develop ways for achieving robust transmission stabilization and switching that work well for a variety of optical waveguides. Further enhancement of transmission stability is enabled by suppression of resonant emission of radiation in nonlinear waveguide couplers with frequency dependent linear gain-loss. Furthermore, we show that supercritical Hopf bifurcations of the equilibrium points of the LV models can be used to induce large stable oscillations of soliton amplitudes along ultra-long distances. The latter finding is an important step towards realizing spatio-temporal chaos with multiple sequences of colliding solitons in nonlinear optical waveguides.
  • May 15, 2018
    Pseudo-tropical curves
    Dr. Sergei Lanzat, Technion

    We consider a generalization of tropical curves, removing requirements ofrationality of slopes and integrality and discuss the resulting theory andits interrelations with other areas. Balancing conditions are interpretedas criticality of a certain action functional. A generalized Bezout theoreminvolves Minkowsky sum and mixed area. A problem of counting curves passing through an appropriate collection of points turns out to be related to the Plucker relations of Grassmanians. We also discuss new recursive relations for this count.
    This is the joint work with Michael Polyak.
  • May 8, 2018
    Quasi-normal surfaces and approximations of least area surfaces in 3-maniflods
    Dr. Eliezer Appleboim, Technion

    This work presents an approximation method by simple pieces of least area surfaces in 3-manifolds.
    In [JR] a piecewise linear (PL) version of minimal and least area surfaces is presented. This PL version is based on normal surfaces. A normal surface in a 3-manifold is a surface that intersects a given triangulation of the manifold in a special simple manner. It was shown in [JR] that PL-minimal and least area surfaces share many of the properties of classical differential geometric least area surfaces. However, the extent at which this analogy holds if far from being fully understood. For instance, the following question is very natural in this context, yet it is still open:
    Question: Can every minimal surface in the smooth sense, be presented as a limit surface of a sequence of PL-minimal surfaces, appropriately constructed?
    Addressing the above question is the main theme of this work. Specifically, a method of approximating a least area surface by simple pieces will be given. It will be shown that there exist obstructions that prevent the approximating simple pieces from being the PL- minimal surfaces as defined in [JR]. A modified version of PL-minimal surfaces will be defined, and it will be shown that this family of surfaces yields a good approximation of least area surfaces, i.e., a least area surface is a limit surface, in an appropriate sense, of a sequence of simple approximating surfaces, and the area of the given least area surface is also the limit of the areas of the approximating surfaces.
    This study is of major interest of its own sake, since it gives a new topological insight and understanding of known geometric results about the global behavior of least area surfaces. Potential applications of this study in the areas of Image Processing and Computer Graphics will also be discussed. Examples are the problem of removing noise from an image, and finding the shortest path on a triangulated surface. Some preliminary experimental results will be shown.
    Reference:  [JR] W. Jaco and H. Rubinstein, PL-Minimal Surfaces in 3-Manifolds, J. Differntial Geometry, 27, 1988.
  • May 2, 2018
    Spaces of real places of ordered fields
    Dr. Michael Machura, ORT Braude College

    We consider a famous open problem: Does every compact Hausdorff space is  a space of real places of some ordered field? 
    Until now not many examples of spaces of real places are known. What about so simple spaces as interval, sphere, torus? Are all spaces of real places metrizable?
    How algebraic and transtendental extention affects of space of real places? 
    We shall provide partiall answers to these questions. The fields of rationals functions of one and more variable shall be analized.
  • April 24, 2018
    Umbral calculus and identities for orthogonal polynomials
    Dr. Orli Herscovici, Haifa University

    The roots of Umbral calculus go back to the XVII century. It attracted an attention of many scientists in the second half of the XVIII century, but a rigorous study of the Umbral calculus belongs to 70-80 years of the XIX century. Since then the Umbral calculus is a powerful technique for study of orthogonal polynomials of the Sheffer type among them are the Bernoulli, Euler, Hermite, Laguerre, and many other polynomials.
    By using these techniques, one can obtain new identities for Sheffer orthogonal polynomials and establish connections between different kind of polynomials.