Oberseminar Mathematische Physik
Jointly with Johannes Alt, Margherita Disertori, Luca Fresta, Illia Karabash and Eveliina Peltola, I am organizing a mathematical physics seminar which takes place a few times per semester. The seminar usually takes place on Mondays from 2.15  3.15 pm in seminar room 0.006 (Endenicher Allee 60).
Talks Spring 2024:
 March 11, 2024: Théo Pinet (IMJPRG, Université ParisCité, UdeM)
Existence of inflations for representations of shifted quantum affine algebrasIt is well known that the only simple finitedimensional Lie algebra admitting a 2dimensional irreducible representation is sl(2). The restriction functors arising in classical Lie theory, from inclusions of Dynkin diagrams, are therefore not essentially surjective on finitedimensional simple modules. This talk aims to specify whether or not this surjectivity defect remains in the setting of FinkelbergTsymbaliuk’s shifted quantum affine algebras (SQAs for short).
SQAs are infinitedimensional algebras parametrized by a finitedimensional Lie algebra and a coweight of this Lie algebra. They are natural variations of the usual quantum loop algebras which are in turn algebras of critical importance in geometry, in quantum integrable systems and in the study of cluster algebras. In this talk, we will give a pedagogical introduction to the remarkable representation theory of SQAs and will state an existence theorem for some notable modules, that we call inflations. We will construct these modules as special preimages for natural restriction functors (associated to inclusions of Dynkin diagrams) and will discuss important applications of their existence to the study of monoidal categorifications of cluster algebras and to integrable systems.

April 15, 2024: Per Moosavi (Stockholm University)
Anisotropic quantum Hall dropletsI will discuss recent work on 2D droplets of noninteracting electrons in strong magnetic fields, confined by an anisotropic trapping potential. Using semiclassical methods, we obtain the oneparticle energy spectrum and wave functions in the lowest Landau level by deriving and solving a transport equation inspired by standard WKB theory. This shows that energy eigenstates are localized on equipotentials of the trap, generalizing the rotationalsymmetric situation for isotropic traps. From these microscopic firstprinciple considerations, we obtain explicit results for manybody observables for anisotropic quantum Hall droplets in the thermodynamic limit. In particular, we show that correlations along the droplet's edge are longranged, in agreement with lowenergy edge modes described by a free chiral conformal field theory in terms of the canonical angle variable of the trapping potential.

May 13, 2024: Alexis LangloisRémillard (University of Bonn)
The Dunkl total angular momentum algebra and its representationsThe Dirac operator and its dual symbol can be seen as the generators of a realisation the Lie superalgebra osp(12) inside the tensor product of the Weyl algebra and a Clifford algebra. The algebra of operators supercommuting with this realisation is called the total angular momentum algebra (TAMA). The polynomial nullsolutions of the Dirac equations form a family of irreducible representations of the TAMA than can be expressed via special functions. For a (complex) reflection group, we can deform the derivatives into the Dunkl operators. Then, inside the associated rational Cherednik algebra tensored by a Clifford algebra, the DunklDirac operator and its dual symbol also generate a realisation of the Lie superalgebra osp(12). The Dunkl TAMA is then the symmetry algebra of this realisation. We will present an overview of this algebra and explore its representations via the few known examples.

May 27, 2024: Runan He (University of Halle)
Analysis of PDE Models in Natural Sciences: MicroElectroMechanical Systems, Surface Plasmon Polaritons and Sintering
The first part of the talk introduces the study of some mathematical models for a MicroElectroMechanical System (MEMS) capacitor, consisting of a fixed plate and a flexible plate separated by a fluid. It investigates the wellposedness of solutions to the resulting quasilinear coupled systems, as well as the finitetime blowup (quenching) of solutions. The models considered include a parabolicdispersive system modelling the fluid flow under an elastic plate, a parabolichyperbolic system for a thin membrane, as well as an ellipticdispersive system for quasistatic fluid flow under an elastic plate. Shorttime existence, uniqueness and smoothness are obtained by combining wellposedness results for a single equation with an abstract semigroup approach for the system. Quenching is shown to occur, if the solution ceases to exist after a finite time.
The second part of the talk introduce linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric and proves the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. We also show that the real frequency exists in the nonlinear setting in the case of $PT$symmetric materials.
The third part of the talk introduces the Mullins equation modelling sintering process. We show that the existence of the selfsimilar solution to the Mullins equation for large groove angle and the nonexistence of selfsimilar solution for small groove angle.

June 3, 2024: Arnaud Triay (LMU Munich)
The excitation spectrum of a dilute Bose gas with an impurity
We study a dilute system of N interacting bosons coupled to an impurity particle via a pair potential in the GrossPitaevskii regime. We derive an expansion of the ground state energy up to order one in the boson number and show that the difference of excited eigenvalues to the ground state is given by the eigenvalues of the renormalized BogoliubovFröhlich Hamiltonian in the large N limit.