# Mathematical Physics and Harmonic Analysis Seminar

## Fall 2024

Date: | September 6, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Qiaochu Ma, Texas A&M University |

Title: | Mixed quantization and quantum chaos |

Abstract: | Quantum Ergodicity (QE) is a classical topic in quantum chaos, it states that on a compact Riemannian manifold whose geodesic flow is ergodic, the Laplacian has a density-one subsequence of eigenfunctions that tends to be equidistributed. In this talk, we present a uniform version of QE for a certain series of unitary flat vector bundles. The key technique involves combining semiclassical and geometric quantizations. The holonomy of flat bundles provides fascinating geometrical phenomena. |

Date: | September 20, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Xueyin Wang, Texas A&M University |

Title: | Spectral analysis of non-self-adjoint quasi-periodic Schr\"odinger operators |

Abstract: | One of the main challenges in studying non-self-adjoint operators is the lack of a spectral theorem. By applying Avila’s global theory, we derive new spectral and isospectral results for non-self-adjoint quasi-periodic Schrödinger operators. This presentation is based on joint work with Zhenfu Wang, Jiangong You, and Qi Zhou. |

Date: | September 27, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Íris Emilsdóttir, Rice University |

Title: | Gap Labelling for Subshifts |

Abstract: | Johnson-Schwartzman theorems are fundamental in gap labelling, providing sets that contain all possible spectral gap labels for dynamically defined operators. However, it remains unclear whether all elements in these sets correspond to actual open gaps and, if so, which sampling functions generate them. In this talk, we explore these questions for operators defined by subshifts over compact alphabets, focusing on the conditions under which spectral gaps open and their relationship to the underlying dynamics. Additionally, we will discuss the computation of the Schwartzman group for certain subshifts and its role in gap labelling. |

Date: | October 4, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Peter Madsen, Ludwig Maximilian University of Munich |

Title: | On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas |

Abstract: | Describing the long-time dynamical behaviour of fermionic systems is a long-standing open problem. When the coupling constant $\lambda$ of the interaction is small, the dynamics of the system up to kinetic time $t \sim \lambda^{-2}$ is conjectured to be effectively governed by the Boltzmann-Nordheim kinetic equation. Considering a system of fermions on a cubic lattice at thermal equilibrium, we prove that the two-point time correlation function of the many-body quantum dynamics can be computed effectively using the collisional frequency of the Boltzmann-Nordheim collision operator. |

Date: | October 18, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Christof Sparber, University of Illinois Chicago |

Title: | Ground state (in-)stability and long-time behavior in multi-dimensional Schrödinger equations |

Abstract: | We consider Schrödinger equations with competing nonlinearities in spatial dimensions up to three, for which global existence holds (i.e. for which no finite time blow-up occurs). A typical example is the case of the (focusing- defocusing) cubic-quintic nonlinear Schrödinger equation. We recall the notions of energy minimizing and of action minimizing ground states and show that, in general, they are nonequivalent. The question of long-time behavior of solutions, in particular the problem of ground-state (in-)stability will be discussed using analytical results and numerical simulations. This is joint work with R. Carles and C. Klein. |

Date: | October 25, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Yi-Sheng Lim, Texas A&M University |

Title: | An operator approach to high-contrast homogenization |

Abstract: | Homogenization refers to the approximation of PDEs with rapidly oscillating coefficients with a nice (constant coefficient) PDE. Physically, we can think of a composite that is obtained by finely mixing together a "soft" and a "stiff" material. Mathematically, we can study the PDE $-div( a(x/\epsilon) grad u^\epsilon ) + u^\epsilon = f$, where the coefficient matrix is $a(y)$ is 1-periodic, takes values $c_{soft} I$ at the "soft" regions, and $c_{stiff} I$ at the "stiff" regions. We want to take \epsilon to 0. This talk focuses on the "high contrast" case $c_{soft} = \epsilon^2$ and $c_{stiff} = 1$. In other words, there is a loss of uniform ellipticity in \epsilon, and this poses fundamental mathematical challenges. To tackle this setting, I will explain the main ideas behind the operator theoretic framework developed by Cherednichenko, Ershova, and Kiselev (2020). The key object is that of a "boundary triple" in the sense of Ryzhov (2009). We obtain an effective limiting description with an $O(\epsilon^2)$ error in the operator-norm. |

Date: | November 1, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Christoph Fischbacher, Baylor University |

Title: | Slow propagation velocities of discrete Schrödinger operators in large periodic potential |

Abstract: | I will present some recent joint work with Abdul-Rahman, Darras, and Stolz (https://arxiv.org/abs/2401.11508). While periodic Schrödinger operators have purely ac spectrum and exhibit ballistic transport, I will show that if the potential is large enough, it is possible to make the velocity of this transport arbitrarily small. I will discuss the special case of period 2, where things can be computed explicitly and then talk about the case of general period p. |

Date: | November 15, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Iulia Cristian, University of Bonn |

Title: | Coagulation equations describing rain and sedimentation |

Abstract: | Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in space of clusters. In this talk, we focus on a model that is inhomogeneous in space and contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass-conserving solutions for a class of coagulation rates for which in the space homogeneous case instantaneous loss of mass occurs. This is based on a joint work with B. Niethammer and J J. L. Velázquez. |

Date: | December 13, 2024 |

Time: | 1:50pm |

Location: | BLOC 302 |

Speaker: | Abdul-Lateef Haji-Ali, Heriot-Watt University |

Title: | TBD |

Abstract: | TBD |