Workshop for young researchers in analysis and mathematical physics

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Monday, October 9, 20239:00 AM RegistrationRegistration9:00 AM - 9:15 AMRoom: B1019:15 AM WelcomeWelcome9:15 AM - 9:30 AMRoom: B1019:30 AM Isoperimetric inequalities and the critical mass in nuclear fission - Phan Thanh Nam (LMU)Isoperimetric inequalities and the critical mass in nuclear fission
- Phan Thanh Nam (LMU)

9:30 AM - 10:30 AMRoom: B101 I will discuss the connection from the classical isoperimetric inequality to the critical mass in nuclear fission reactions predicted by the liquid drop model. In particular, I will address several open questions and some recent results on the existence and nonexistence of minimizers. The talk is based on joint work with Rupert Frank.10:30 AM Coffee BreakCoffee Break10:30 AM - 11:00 AMRoom: B10111:00 AM Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases - Marwa Shahine (IMB)Fredholm Property of the Linearized Boltzmann Operator for a Mixture of Polyatomic Gases- Marwa Shahine (IMB)

11:00 AM - 12:00 PMRoom: B101 In this talk, we consider the Boltzmann equation that models a mixture of polyatomic gases assuming the internal energy to be continuous. Under some convenient assumptions on the collision cross-section, we prove that the linearized Boltzmann operator L is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator. We prove that the collision frequency is coercive and that the perturbation operator is Hilbert-Schmidt integral operator.12:00 PM LunchLunch12:00 PM - 1:45 PMRoom: B1011:45 PM Sub-Riemannian fluids - Adrien Tendani Soler (Université de Bordeaux)Sub-Riemannian fluids- Adrien Tendani Soler (Université de Bordeaux)

1:45 PM - 2:45 PMRoom: B101 In this presentation, I will introduce the Euler and Navier-Stokes systems with anisotropic velocity fields, where this anisotropy is defined by a geometric structure. Additionally, I will discuss the issues of well-posedness and regularity for the subRiemannian Navier-Stokes system. The analysis is based on the propagation of carefully selected families of vector fields through the energy and relies on hypoellipticity, geometry and analysis on stratified Lie groups.2:45 PM Coffee BreakCoffee Break2:45 PM - 3:15 PMRoom: B1013:15 PM Poster SessionPoster Session3:15 PM - 4:30 PMRoom: B1014:30 PM Welcome drinksWelcome drinks4:30 PM - 5:30 PMRoom: B101 -
Tuesday, October 10, 20239:00 AM Effective descriptions of interacting Bose gases - Lea Boßmann (LMU Munich)Effective descriptions of interacting Bose gases
- Lea Boßmann (LMU Munich)

9:00 AM - 10:00 AMRoom: B101 Since the first realization of a Bose-Einstein condensate in a laboratory in 1995, low-temperature Bose gases have been studied from many different perspectives. In this talk, I will give a basic introduction into the mathematical description of an interacting Bose gas and explain the concept of effective evolution equations.10:00 AM Coffee BreakCoffee Break10:00 AM - 10:30 AMRoom: B10110:30 AM Bose Gases — Results and Methods - Florian Haberberger (LMU)Bose Gases — Results and Methods- Florian Haberberger (LMU)

10:30 AM - 11:30 AMRoom: B101 I will begin by discussing the significant results concerning Bose Gases, tracing their origins back to Albert Einstein's groundbreaking work on ideal gases in 1924. Our exploration will also emphasize the theoretical underpinnings and methodologies involved, such as the introduction of the one-particle density matrix by Roger Penrose and Lars Onsager in 1956, along with the utilization of the Fock space formalism. Finally, I will provide an overview of ongoing research and the unresolved questions in this field.11:30 AM LunchLunch11:30 AM - 1:15 PMRoom: B1011:15 PM High frequency controllability cost of linear hyperbolic systems, with a long gaze at localized data. - Vincent Laheurte (Institut de Mathématiques de Bordeaux)High frequency controllability cost of linear hyperbolic systems, with a long gaze at localized data.- Vincent Laheurte (Institut de Mathématiques de Bordeaux)

1:15 PM - 2:15 PMRoom: B101 We revisit the classical issue of the controllability/observability cost of linear first order evolution systems, starting with ODEs, before turning to some linear first order evolution PDEs in several space dimensions, including hyperbolic systems and pseudo-differential systems obtained by linearization in fluid mechanics. In particular we investigate the cost of localized initial data, and in the dispersive case, of initial data which are semi-classically microlocalized.2:15 PM Coffee BreakCoffee Break2:15 PM - 2:45 PMRoom: B1012:45 PM A Poincaré-Steklov map for the MIT bag model - Mahdi ZreikA Poincaré-Steklov map for the MIT bag model- Mahdi Zreik

2:45 PM - 3:45 PMRoom: B101 In this talk, I will present some study of the Poincaré-Steklov (PS) operator associated with the MIT bag operator on a smooth domain $\Omega\subset \mathbb{R}^{3}$ with a compact boundary $\Sigma:=\partial\Omega$. This operator can be seen as the analogue of the Dirichlet-to-Neumann mapping, where the free Dirac operator $D_m := -i\alpha\cdot\nabla +m\beta$ plays the role of the Laplace operator, and the Dirichlet and the Neumann traces are replaced by orthogonal projections of the Dirichlet traces along the boundary $\Sigma.$ More precisely, this operator is associated with the following boundary value problem \begin{equation} (D_m-z)v =0, \quad \text{ in } \Omega, \qquad P_{\pm}t_{\Sigma}v = g\in H^{1/2}(\Sigma)^{4}, \end{equation} where $P_{\pm}$ are the orthogonal projections along the boundary $\Sigma$ and $t_{\Sigma}$ is the classical trace operator. In the first part of this talk, I will explain how the PS operator fits well into the framework of classical pseudodifferential operators and determine its principal symbol. In the second part, I will discuss the properties of the PS operator when the mass $m$ becomes large enough. Namely, I will show that it is a $1/m$-pseudodifferential operator and I will give its main properties, in particular its semiclassical principal symbol. Then, we apply these results to establish a Krein-type resolvent formula for the Dirac operator $D_M= D_m +M\beta 1_{\mathbb{R}^{3}\setminus\overline{\Omega}}$ in terms of the resolvent of the MIT bag operator when $M > 0$ is large enough. Finally, we show that the operator $D_M$ in the limit of the large coupling ($M\longrightarrow \infty$) converges to the MIT bag operator in terms of the norm-resolvent with a convergence rate of $\mathcal{O}(M^{-1}).$3:45 PM Discussion sessionDiscussion session3:45 PM - 5:15 PMRoom: B1016:30 PM Conference dinnerConference dinner6:30 PM - 8:00 PMRoom: B101 -
Wednesday, October 11, 20239:00 AM Negative spectrum of Schrödinger operators with oscillating potentials - Larry Read (LMU)Negative spectrum of Schrödinger operators with oscillating potentials
- Larry Read (LMU)

9:00 AM - 10:00 AMRoom: B101 For a Schrödinger operator $-\Delta-\alpha V$, the decay of the potential $V$ towards infinity determines the finiteness of its negative spectrum. In the particular case where $V$ is asymptotically homogeneous of degree -2, the size of the coupling constant $\alpha$ distinguishes between the generation of finitely or infinitely many negative eigenvalues. In this talk we will show that a similar property holds when the potential has slower decay but oscillates at infinity.10:00 AM Coffee BreakCoffee Break10:00 AM - 10:30 AMRoom: B10110:30 AM Szegő-type asymptotics for the half-filled lowest Landau level - Leonhard Wetzel (LMU)Szegő-type asymptotics for the half-filled lowest Landau level- Leonhard Wetzel (LMU)

10:30 AM - 11:30 AMRoom: B101 Consider the Landau Hamiltonian $H=-\frac{\partial^2}{\partial x_1^2} + \left( - i \frac{\partial}{\partial x_2} - Bx_1 \right)^2$ in Landau gauge acting on $L^2(\mathbb R^2)$. Let $1_{\{B\}}(H)$ be the spectral projection onto the (infinitely degenerate) eigenspace corresponding to the eigenvalue $\lambda = B$. Leschke, Sobolev and Spitzer established an asymptotic expansion of the form \[ \operatorname{tr}h(1_{[-l,l]^2} 1_{\{B \}}(H) 1_{[-l,l]^2}) = \alpha l^2 + \beta l + o(l) \] as $l\to \infty$, for fairly general functions $h$ with $h(0) = 0$. Our main result is a corresponding asymptotic expansion when replacing the projecton $1_{\{B\}}(H)$ by a subprojection $P$ onto a subspace incorporating only ``half'' the eigenfunctions (in a sense that will be made precise). We will see that in this case the subleading behavior of the asymptotic expansion features an additional logarithmic enhancement term $\log l$.11:30 AM closing statementsclosing statements11:30 AM - 12:00 PMRoom: B10112:00 PM LunchLunch12:00 PM - 1:00 PMRoom: B101