### TBA by Andreas Deuchert

- Series
- Math Physics Seminar
- Time
- Friday, December 6, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Andreas Deuchert – Virginia Tech – adeuchert@vt.edu

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- Series
- Math Physics Seminar
- Time
- Friday, December 6, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Andreas Deuchert – Virginia Tech – adeuchert@vt.edu

- Series
- Math Physics Seminar
- Time
- Friday, November 15, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Ian Jauslin – Rutgers University – ian.jauslin@rutgers.edu

As is well known, many materials freeze at low temperatures. Microscopically,

this means that their molecules form a phase where there is long range order

in their positions. Despite their ubiquity, proving that these freezing

transitions occur in realistic microscopic models has been a significant

challenge, and it remains an open problem in continuum models at positive

temperatures. In this talk, I will focus on lattice particle models, in which

the positions of particles are discrete, and discuss a general criterion

under which crystallization can be proved to occur. The class of models that

the criterion applies to are those in which there is *no sliding*, that is,

particles are largely locked in place when the density is large. The tool

used in the proof is Pirogov-Sinai theory and cluster expansions. I will

present the criterion in its general formulation, and discuss some concrete

examples. This is joint work with Qidong He and Joel L. Lebowitz.

- Series
- Math Physics Seminar
- Time
- Friday, November 8, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Michael Hott – University of Minnesota Twin Cities – mhott@umn.edu

As highly tunable platforms with exotic rich phase diagrams, moiré materials have captured the hearts and minds of physicists. Moiré materials arise when 2D crystal layers are stacked at relative twists. Their almost periodicity and multiscale behavior make these materials particularly mathematically appealing. We will describe the challenges in establishing a framework to study (phonon/vibrational) wave propagation in these materials, and explain how to overcome them.

- Series
- Math Physics Seminar
- Time
- Friday, November 1, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Matthew Powell – Georgia Tech – powell@math.gatech.edu

We will discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization. In this talk, we will discuss the notion of discrepancy and present current and ongoing work establishing novel upper bounds of the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Liu [2023] and Jitomirskaya-Powell [2022].

- Series
- Math Physics Seminar
- Time
- Friday, October 18, 2024 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Clough 280
- Speaker
- Tobias Reid – Georgia Tech – tobias.ried@gatech.edu

The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical estimate on the number of bound states for Schrödinger operators. In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our new proof highlights a natural but overlooked connection of the CLR inequality with bounds for maximal Fourier multipliers from harmonic analysis and leads to a variational problem that can be reformulated in terms of a variant of Hadamard’s three-lines lemma. The solution of this variational problem relies on some interesting complex analysis techniques. (Based on joint work with T. Carvalho-Corso, D. Hundertmark, P. Kunstmann, S. Vugalter)

- Series
- Math Physics Seminar
- Time
- Wednesday, April 10, 2024 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Ruoci Sun – School of Mathematics, Georgia Tech – rsun309@gatech.edu

**Please Note:** Available via zoom at: https://gatech.zoom.us/j/98258240051

This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), which are introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I can establish the explicit expression for general solutions on the torus and on the real line in my work [hal-04227081].

- Series
- Math Physics Seminar
- Time
- Wednesday, April 3, 2024 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skyles 006
- Speaker
- Joan Gimeno – Universitat de Barcelona – joan@maia.ub.es

**Please Note:** Available online at: https://gatech.zoom.us/j/98258240051

We develop a method to construct solutions of some (retarded or advanced) equations.
A prime example could be the motion of point charges interacting via the fully relativistic Lienard-Wiechert potentials (as suggested by J.A. Wheeler and R.P. Feynman in the 1940's). These are retarded equations, but the delay depends implicitly on the trajectory.
We assume that the equations considered are formally close to an ODE and that the ODE admits hyperbolic solutions (that is, perturbations
transversal to trajectory grow exponentially either in the past or in the future) and we show that there are solutions of the functional equation close to the hyperbolic solutions of the ODE.
The method of proof does not require to formulate the delayed problem as an evolution for a class of initial data. The main result is formulated in an "a-posteriori" format and allows to show that solutions obtained by non-rigorous approximations are close (in some precise sense) to true solutions. In the electrodynamics (or gravitational) case, this allows to compare the hyperbolic solutions of several post-newtonian approximations or numerical approximations with the solutions of the Lienard-Weichert interaction.
This is a joint work with R. de la Llave and J. Yang.

- Series
- Math Physics Seminar
- Time
- Wednesday, March 13, 2024 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Shivan Mittal – Department of Physics, The University of Texas at Austin – shivan@utexas.edu

**Please Note:** Available online at: https://gatech.zoom.us/j/98258240051

Consider the following question of interest to cryptographers: A message is encoded in a binary string of length n. Consider a set of scrambling operations S (a proper subset of permutations on n bits). If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations look like a random permutation on all the bits? This question asks for the convergence time for a random walk on the permutation group. Replace the binary string with a quantum state and scrambling operations in S with Haar random quantum channels on two bits (qudits) at a time. Broadly speaking, we study the following question: If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations (quantum channels) look like a Haar random channel on all qudits? This question asks about the convergence time for a random walk on the unitary group. Various protocols in quantum computing require Haar random channels, which motivates us to understand the number of operations one would require to approximately implement that channel.

More specifically, in our study, we add a connected-graph structure to scrambling operations (a step on the random walk), where qudits are identified by vertices and the allowed 2-qudit scrambling operations are represented by edges. We develop new methods for lower bounds on spectral gaps of a class of Hamiltonians and use those to derive bounds on the convergence times of the aforementioned random walk on the unitary group with the imposed graph structure. We identify a large family of graphs for which O(poly(n)) steps suffice and show that for an arbitrary connected graph O(n^(O(log(n))) steps suffice. Further we refute the conjectured O(n log(n)) steps for a family of graphs.

- Series
- Math Physics Seminar
- Time
- Wednesday, February 14, 2024 - 13:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Matthew Powell – School of Mathematics, Georgia Tech – powell@math.gatech.edu

**Please Note:** Available on zoom at:
https://gatech.zoom.us/j/98258240051

We shall discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization.

In this talk we will introduce the notion of the transport exponent, explain its stability, and explain how logarithmic upper bounds may be obtained in the quasi-periodic setting for all relevant parameters. This is based on joint work with S. Jitomirskaya.

- Series
- Math Physics Seminar
- Time
- Thursday, November 2, 2023 - 16:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005 and online at https://gatech.zoom.us/j/99225468139
- Speaker
- Pavel Lushnikov – Department of Mathematics and Statistics, University of New Mexico – plushnik@math.unm.edu

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.