New advances on the existence and regularity of Brakke’s mean curvature flows

The next seminar:  New advances on the existence and regularity of Brakke’s mean curvature flows

Lecturer: Salvatore Stuvard  
Date & Time:  28.11.2023 , 12:00 AM
Place: Department of Mathematics,University of Milan 
Abstract: In this talk, I aim at giving an overview of what we know, and also of what we don’t know, about the existence and the regularity properties of Brakke flows. These are measure-theoretic, weak solutions to the mean curvature flow capable of describing the evolution of surfaces through singularities and topology changes, with applications ranging from Materials Science to Biology. If time permits, I will also try to explain how the analysis of the parabolic flow opens a new avenue of research into the regularity of its stationary, elliptic counterpart: minimal surfaces. Joint work with Yoshihiro Tonegawa (Tokyo Tech).

Exploring the space of graphs with a fixed discrete curvature

The next seminar:  

Lecturer:  Michelle Roost
Date & Time: 07.11.2023 
Place: Maks Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Abstract: In practical network analysis, we are often interested how the structure of a network influences its
function. Structural information is not encoded by the nodes, but by the relations between them.
These relations are expressed in edges that possess properties like direction and weight. In this
work, we want to explore the space of graphs with a given Ricci curvature sequence which is an
edge-based classifier. Having its roots in differential geometry, several discrete Ricci curvatures
have been shown to capture geometrical properties of graphs and have been successfully applied
to complex networks.
In this talk, we want to explore the space of graphs with a given discrete Ricci curvature sequence.
For this, we will first analyse a graph ensemble created by a Markov Chain Monte Carlo type
algorithm approximating a given curvature sequence. Second, we will obtain theoretical results
on the collection of all graphs with a fixed curvature and degree sequence for the simple notion
of Forman–Ricci curvature and see that these graphs are connected by a set of ‘moves’.

Combinatorial wall-crossing via the Mullineux involution

  • The next seminar: Combinatorial wall-crossing via the Mullineux involution

    Lecturer: Title: Galyna Dobrovolska
    Date & Time: 31.10.2023
    Place: Ariel University
    Abstract: The rational Cherednik algebra H_c is an algebra of interest in modern representation theory, which is a degeneration of the double affine Hecke algebra, introduced by Cherednik to prove Macdonald’s conjectures about properties of Macdonald polynomials. For values of c lying in chambers separated by walls, representations of H_c are labeled by partitions. Combinatorial wall-crossing is a bijection from the set of irreducible representations of H_c to the set of irreducible representations of H_c’, where c and c’ lie in adjacent chambers separated by a wall. Combinatorial wall-crossing across one wall was proven by Losev to be equal in large positive characteristic to an extension of the Mullineux involution from modular representation theory of the symmetric group. We will exhibit interesting patterns in combinatorial wall-crossing, both proven and observed in computer experiments, and use them to prove and refine parts of a conjecture of Bezrukavnikov.

Seminars, 2019

  • 2019
    • May 7, 2019
      On the Lorenz flow and the modular surface
      Dr. Tali Pinsky, Technion
      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
      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
      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
      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.
    • January 15, 2019
      An embedded Cartesian scheme for the Navier-Stokes equations
      Prof. Dalia Fishelov, Afeka Tel Aviv Academic College of Engineering
      In this talk the two-dimensional Navier-Stokes system in stream function formulation is considered.
      We describe a fourth-order compact scheme for regular domains in 2D. We then proceed to irregular domains.
      First, the irregular domain is embedded in a Cartesian grid. Then, an interpolating polynomial is built for regular elements inside the domain as well as for irregular elements near the boundary.
      A compact high-order scheme is then constructed for the Navier-Stokes equations by applying the differential operators involved in the Navier-Stokes equations to the interpolating polynomial.
      Numerical results will be presented for various irregular domains. A particular attention is devoted to flows in elliptical domains.

      In the case of the ellipse, we also demonstrate the ability of the scheme for computations of the eigenvalues and the eigenfunctions of the biharmonic problem on the ellipse