Journée de l'équipe Math-Physique

La journée d'équipe se tiendra en hybride: en salle rené Baire et sur TEAMS.

Programme:

Titres et Résumés:

• Nicolas Babinet: Supersymmetric matrix models, ABJM theory and quantum curves

  Matrix models have a long and fruitful story both in physics and pure mathematics. In physics Wigner has introduced random matrix formalism to describe energy levels of heavy nuclei for instance, while in mathematics there are strong evidences of a deep connection between eigenvalues of random matrices and distribution of prime numbers, thus leading to some conjectures about Riemann’s zeta function.
  Many properties have been proven for the associated partition function and the corresponding free energy, even if some are still conjectures. One which still leads to rich developments is the so-called quantum curve. In this framework the partition function behaves as a wave function and the quantum curve acts on it as an operator, i.e. promoting classical coordinates to operators.

  Starting from this framework and after presenting the basic random matrix model, I want first to focus on the supersymmetric case that I’m interested in. I will present some properties that we have found and structures that still need to be elucidated. A very similar model that I will then present is the ABJM theory, first introduced in string theory, whose partition function can be studied with matrix model. An interesting method was developed to explore this theory yet some problems are still unsolved. I will finally review conjectures that I’m interesting in, more specifically those in relation with quantum curves.

• Eddy Brandon De León: Computational approach to the Schottky problem
  
  The characterization of jacobians of algebraic curves among all principally polarized Abelian varieties (PPAV) is known as the Schottky problem. A PPAV can be considered as the quotient ℂᵍ/λ, where λ=ℤᵍ+Ωᵍ and Ω is a symmetric complex matrix with a positive-definite imaginary part (Riemann matrix). We aim to develop a computational tool such that given a PPAV A we can tell for a given precision whether it is the jacobian of an algebraic curve or not. For this purpose, we make use of the Welter’s conjecture, which states that A=Jac(X), for some algebraic curve X, if and only if there exists a trisecant of its kummer variety, which is the image of the embedding Kum : A/{±1}↪ℙ²ˆᵍ⁻¹. This becomes an optimization problem, since we consider an auxiliary non-negative function f : A×A×A→ℝ whose global minimum is zero if and only if A=Jac(X).
• Sion ChanLang: To Be Announced
• Edgar Gasperin-Garcia: The Role of the Energy Scalar Product in the QNM Spectral Instability Problem
  
  The impact of the scalar product in the Quasinormal mode (QNM) spectral instability problem is discussed. It is illustrated in a simple example how the use of different norms can lead to different looking pseudospectra for the same operator. With this motivation, we will study QNMs of a scalar field on a spherically symmetric spacetime background. The relation between the physical energy and the effective energy (used in a recent result of Jaramillo, Panosso-Macedo and Al Sheikh) is obtained and the boundary terms are identified. The energy inner product is exploited to obtain a weak formulation of the associated eigenvalue problem which opens the possibility of using finite elements methods to numerically explore the spectrum. Other applications such as using Keldysh theorem to obtain asymptotic resonant expansions exploiting the scalar product are also discussed.
• Francisco Hernandez-Iglesias: Dubrovin equation and canonical coordinates for the 2D-Toda Frobenius manifold.
  
  Integrable systems of N evolutionary PDEs that admit a bi-Hamiltonian formulation, i.e., that are endowed with a pair of compatible Poisson brackets, have an underlying geometric structure known as a Frobenius manifold. This N-dimensional Frobenius manifold has two compatible flat metrics on its tangent space, which define the Poisson brackets of the system. Extending this theory to the 2D-Toda hierarchy, a 2+1 integrable system of bi-Hamiltonian type, we define an infinite-dimensional Frobenius manifold and study its main elements: the product and the metric on the tangent spaces, the unit and Euler vector fields. Then we introduce the deformed flat connection and derive from this definition the  Dubrovin equation. The main difference with respect to the finite-dimensional case is that the cotangent spaces are no longer isomorphic to the tangent spaces, meaning the Dubrovin equation will take a weak form. Finally, we will analyze the (weak) solutions of the Dubrovin equation around infinity and link them to the canonical coordinates.
• Ajinkya Kulkarni:Algebraic and topological invariants of fusion categories
  
  I will talk about invariants of fusion categories coming from finite groups. A topological invariant called the B-tensor (along with the T-matrix) is shown to suffice to distinguish twisted Drinfeld doubles of square-free groups of odd order. We also define a character theory for objects in group-theoretical fusion categories (GTCs) and use it to compute fusion rings of GTCs of dimensions less than 21. We find that there are 34 non-pointed fusion rings coming from GTCs of dimension less than 21, of which 26 are singly generated and only 3 are non-hyperrings. In the process, we obtain weak lower bounds for non-equivalent categorifications of each of these non-pointed fusion rings. Further, we find several examples of Morita equivalent GTCs which have the same fusion ring but distinct Frobenius-Schur indicators (including one case where both the GTCs are pointed), thereby refuting a conjecture of H. Tucker.
• Oscar Meneses Rojas: Caustics and fronts in General Relativity
  
  The concept of caustic naturally arises when systems of light rays are studied. They are defined as the envelope of a family of light rays. In gravitational lensing for example, when light rays issuing from a surface tends to re converge after passing close to a strong gravitational field, light rays will start to intersect each other and the surfaces equidistants to the initial surface generically will have singularities. The propagation of a front can be described by the study of a single hypersurface in the space time, which generically has singularities, is a null hypersurface and is called Big Front. In a general way caustics and fronts belong to the symplectic and contact worlds respectively and there is a caustic associated to the propagation of the front.
  I will introduce what a front is and its generic singularities to finally conclude what is the caustic associated to the propagating front. This constitutes a preliminary step in developing a setting for understanding the stability properties and patterns of generic caustics in the birth of event horizons in general relativity.
• Nikola Stoilov: Numerical studies of the Zakharov-Kuznetsov equations
  
  In this work we look at the behaviour of the Zakharov Kuznetsov (ZK) equations, using ad- vanced numerical tools. As a nonlinear dispersive PDE, initially used to model magnetized plasma, ZK has solutions that develop a singularity in finite time from smooth initial data. We demonstrate its behaviour and will look at phenomena including blow-up, soliton resolution and soliton interaction and discuss how the non-integrability transpires in these cases. We propose several conjectures for the long term behaviour.
  Based on joint works with Christian Klein and Svetlana Roudenko.