Seminars, 2019

Lectures:
  • July 10, 2019
    Improving Hardy's inequality 
    Prof. Matthias Keller, Potsdam University

    Abstract
    About 100 years ago Hardy proved his famous inequality. Since then such inequalities have been proven in various contexts. In this talk we not only address the question of the sharp constant but rather the question of an optimal Hardy weight. This is joint work with Yehuda Pinchover and Felix Pogorzelski.
  • June 25, 2019
    Cluster Algebras on SL_n 
    Dr. Idan Eisner

    Abstract
    Cluster algebras are commutative rings with a distinguished set of generators that are grouped into overlapping finite sets of the same cardinality. Among many other examples, cluster algebras appear in coordinate rings of various algebraic varieties.
    Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For some of these classes a compatible cluster structure can be constructed. We will discuss a family of such cases on G=SL_n.
    The talk will assume no prior knowledge of cluster algebras. Definitions and examples will be given.
  • June 18, 2019
    Application of reflections to a class of uniform non-Laplacian growth 
    Prof. Tatiana Savin, Ohio University, USA

    Abstract
    We start the talk with a review of statements of Hele-Shaw problems as examples of Laplacian growth.
    Then, we discuss the generalizations of the Schwarz symmetry principle to the case of solutions to elliptic equations subject to non-homogeneous Dirichlet and Neumann conditions given on a real-analytic curve. Next, we show how the latter is applied to the Laplacian growth problems. Finally, we apply the reflections to solve some non-Laplacian growth problems.
  • May 7, 2019
    On the Lorenz flow and the modular surface 
    Dr. Tali Pinsky, Technion

    Abstract
    The talk describe two famous flows in dimension three: The geodesic flow on the modular surface and the chaotic Lorenz equations on R^3.
    I will explain how each of these is defined, and show some elusive connections between them. This will be an introductory talk.
  • March 26, 2019
    Stability of some super-resolution problems 
    Dr. Dmitry Batenkov, MIT, USA

    Abstract
    The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization.  I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth.  I will also discuss optimal recovery algorithms, and some open questions.
  • March 13, 2019
    Short-time existence of geometric evolution of 2nd and 4th order 
    Dr. Jakob Ruben
  • February 5, 2019
    Globally Solving the Trust Region Subproblem Using Simple First-Order Methods
    Dr. Yakov Vaisbourd, Tel-Aviv University

    Abstract
    We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds to one of two possible scenarios that occur in this problem, commonly referred to in the literature as the presence or absence of the ``hard case". We consider a family of first-order methods, which includes the projected and conditional gradient methods. We show that any method belonging to this family produces a sequence which is guaranteed to converge to a stationary point of the trust region subproblem. Based on this result and the established sufficient optimality conditions, we show that convergence to an optimal solution can be also guaranteed as long as the method is properly initialized. In particular, if the method is initialized with the zeros vector and reinitialized with a randomly generated feasible point, then the best of the two obtained vectors is an optimal solution of the problem in probability 1.
  • January 22, 2019
    Classification Problems of Infinite Type (in Differential Geometry)
    Dr. Ori Yudilevich, Catholic University of Leuven, Belgium

    Abstract
    The notions of a Lie groupoid and Lie algebroid (generalizations of the notions of a Lie group and Lie algebra) have proven to be powerful tools in differential geometry. One notable application, which takes its roots in the classical work of Élie Cartan, is their use in classifying geometric structures. For example, it was recently shown by Fernandes and Struchiner that the classification of so-called Kähler-Bochner manifolds, a result due to Robert Bryant (2001), can be understood as the integration of a certain Lie algebroid (that encodes the classification problem) to a Lie groupoid (that encodes the solutions). This method, however, is limited to a special class of classification problems known as problems of "finite type". Roughly speaking, this means that the space of solutions is finite dimensional. To address problems of "infinite type", one needs slightly more general tools.
    In this talk (based on joint work with Rui Loja Fernandes), I will present the notions of a Bryant groupoid and Bryant algebroid that can be used to tackle classification problems that are both "finite" and "infinite". These notions, which stem from the work of Robert Bryant, exhibit interesting behavior that brings together ideas from the theory of Lie groupoids and Lie algebroids, on the one hand, and from the theory of Partial Differential Equations, on the other. This talk is aimed at a broad mathematical audience and will not assume any previous knowledge of differential geometry.