In the framework of the functional analysis, this talk aims to illustrate the research we have developed during the last years: from the study of geometric and algebraic properties of triple structures to a one-to-one correspondence with TKK Lie algebras. We shall highlight the last novelties achieved in the Lie setting.
Spin glasses are models of statistical mechanics in which a large number of elementary units interact with each other in a disordered manner. In the simplest case, there are direct interactions between any two units in the system, and I will start by reviewing some of the key mathematical results in this context. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. I will then discuss some of the technical challenges that arise in this case, as well as recent progress.
The classical contraction principle is one of those basic results in Analysis with many fundamental applications. In this talk, we will examine a variational interpretation of it which turns out to be more flexible. In particular, it can be used to deal with situations where existence and uniqueness of solutions is known or expected. Though many classical situations can be treated, due to time restrictions we will focus on two representative examples: that of initial-value Cauchy problems for autonomous ODE systems, and the case of non-linear, non-variational monotone PDE equations in divergence form. It remains to be seen if this perspective could help in new situations.
The tone of the talk will be elementary. No specialized background is required.
The talk describes an approach to the investigation of normalized solutions for nonlinear Schrödinger equations based on the analysis of the masses of ground states of the corresponding action functional. We begin with a brief overview of some recent results on the relation between action and energy ground states. Then, we provide a complete characterization of the masses of action ground states, obtained via a Darboux-type property for the derivative of the action ground state level, and we exploit this result to tackle normalized solutions with a twofold perspective. First, we prove existence of normalized nodal solutions for every mass in the $L^2$-subcritical regime, and for a whole interval of masses in the $L^2$-critical and supercritical cases. Then, we show when least energy normalized solutions/least energy normalized nodal solutions are action ground states/nodal action ground states.
In quantum physics, and more specifically in quantum optics, several notions of ``classical’’ and hence ``nonclassical’’ state are in use. They rely on the positivity of quasi-probability distributions, specifically the Glauber-Sudarshan or Wigner functions of the state. Characterizing the classical states is in general a difficult task, involving interesting questions of functional and harmonic analyss. In this talk, we will, after reviewing the subject, report on some recent progress.
In this talk we explore the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincaré reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.
In the 1960's, W. Leavitt studied a class of universal algebras which do not have a well-defined rank, i.e., algebras $L$ for which $L^m\cong L^n$ as $L$-modules with $m$<$n$, later known as the Leavitt algebra $L(m,n)$. In two simultaneuous but independent studies by G. Abrams and G. Pino, and P. Ara et. al., an algebra arising from a directed graph $E$ and a field $K$ has been introduced called the Leavitt path algebra $L_K(E)$. This algebra turned out to be the generalization of $L(1,n)$. In fact, $L(1,n)\cong L_K(R_n)$ where $R_n$ is the graph having one vertex and $n$ loops.
In 2013, R. Hazrat formulated the Graded Classification Conjecture for Leavitt path algebras which claims that the so-called talented monoid is a graded Morita invariant for the class of Leavitt path algebras. In this talk, we will see Leavitt Path algebra modules as a special case of quiver representation with relations and how it will take a role in proving the conjecture.
More concretely, we will provide evidences and a confirmation in the finite-dimensional case of the conjecture.
This is a compilation of joint works with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela.
When M is a Fano variety and D is an anticanonical divisor in M, mirror symmetry suggests that the quantum cohomology of M should be a deformation of the symplectic cohomology of M \ D. We prove that this holds under even weaker hypotheses on D (although not in general), and explain the consequences for mirror symmetry. We also explain how our methods give rise to interesting symplectic rigidity results for subsets of M. Along the way we hope to give a brief introduction to Varolgunes’ relative symplectic cohomology, which is the key technical tool used to prove our symplectic rigidity results, but which is of far broader significance in symplectic topology and mirror symmetry as it makes the computation of quantum cohomology “local”. This is joint work with Strom Borman, Mohamed El Alami, and Umut Varolgunes.
