Calendar

Posted August 16, 2025
Last modified August 21, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 232

Moisés Gómez-Solís, Louisiana State University
Laura Kurtz, Louisiana State University
Organizational Meeting

Posted August 30, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 232

Hari Narayanan, Louisiana State University
Introduction to Spectral Theory and Schrödinger Operators

This talk concentrates on the spectral theory of Schrödinger operators with a view toward modern research. The huge literature in this field was spawned by non-relativistic quan- tum mechanics and has led to rich advances in pure spectral theory and applications. After a brief review of finite dimensions, the talk first develops abstract spectral theory of self-adjoint operators in Hilbert space with some emphasis on classical ideas of harmonic analysis, namely spectral resolutions induced by symmetry groups. Then we introduce continuous and discrete Schrödinger operators with electric and magnetic potentials and some of the standard theorems. We treat periodic, quasi-periodic, and ergodic operators, in decreasing detail. The treatment of periodic operators will emphasize the connections to commutative algebra centering around the Fermi and Bloch algebraic or analytic varieties. This is followed by analysis in physical, momen- tum (dual), configuration, and reciprocal space; and a brief look at ergodic and quasi-periodic operators.

Posted September 10, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 136

Shalini Shalini, LSU
Gowri Priya Sunkara, LSU
The Elvis Problem with Convex Bifunctions/The Minimal Time Function Under More General Assumptions

The Elvis problem models a time optimal control problem across two regions having a common interface Sigma, each with a constant convex velocity set. We generalized this framework by introducing convex bifunctions of the form F_i(r, v) which are convex, lower semicontinuous, and proper. The minimization problem is min [ F_0(r_0, y - x_0) + F_1(r_1, x_1 - y) ], over r_0 > 0, r_1 > 0 and y in Sigma. The bifunctions F_i are jointly convex in (r, v) and convex analysis is used to derive optimality conditions. Under general assumptions on the target set S and the system dynamics, we show that the minimal time function is a proximal solution to a pair of Hamilton–Jacobi inequalities. Uniqueness is established via two distinct types of boundary conditions. We also introduce a new propagation result, which characterizes proximal sub gradients of the minimal time function in terms of normal cones and a boundary inequality condition. Furthermore, we provide necessary and sufficient conditions for the Lipschitz continuity of the minimal time function near S. In particular, a Petrov-type modulus condition is shown to guarantee such continuity. Our results extend earlier results to a broader class of time dynamics, even within non-Euclidean settings.

Posted September 18, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 136

Matthew McCoy, Louisiana State University
Introduction to Ergodic Theory, Chebyshev Polynomial Expansions, and Schur Complements

This talk will serve as an introductory talk to Ken Beard's talk on 9/30: Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Trilayer Graphene. We will introduce relevant concepts in spectral theory.

Event contact: Laura Kurtz

Posted September 18, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 136

Ken Beard, LSU
Momentum Space Algorithm for Electronic Structure of Double-Incommensurate Trilayer Graphene

Although recent experimental results seem to indicate the existence of flat bands for twisted trilayer graphene (TTG), at present there is no convergent algorithm for approximating the density of states (or other desirable observables). We attempt to address this by using kernel polynomial approximation with an ab initio momentum-space tight-binding model. The unique challenge in the case of TTG is the lack of a periodic moir\'{e} supercell. This is addressed by modifying the truncation to account for more complex Umklapp scattering.

Event contact: Laura Kurtz

Posted October 5, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 136

Bart Rozenweig, Ohio State University
Borel Summability in Quantum Theory

Borel summation is a canonical summation technique which associates to a divergent power series an analytic function, for which the power series is its asymptotic expansion. This talk gives an overview of asymptotic expansions and the fundamental results on Borel summability, before surveying two major applications of the theory: first, in building actual solutions out of divergent formal power series solutions of ODEs and PDEs; and second, in making sense of divergent Rayleigh-Schrödinger perturbation expansions in quantum mechanics. Along the way, we will touch upon some key aspects of “resurgence theory”, a paradigm for the application of Borel summation ideas in quantum field theory.

Posted October 10, 2025

Informal Analysis Seminar Questions or comments?

3:30 pm – 4:30 pm Lockett 136

Long Teng, LSU
Nodal Sets of Harmonic Functions

In this talk, we study the size of nodal sets of harmonic functions. We introduce the frequency function N(r), which quantifies the growth rate of a harmonic function and plays a crucial role in understanding its zero set. I will first define this frequency function and show its monotonicity property. Then, using this tool, we establish that the (n−1)-dimensional Hausdorff measure of the nodal set is bounded above by C(n)N, where C(n) depends only on the dimension. This result highlights how quantitative unique continuation connects analytic growth properties of harmonic functions to the geometric complexity of their nodal sets.

Event contact: Laura Kurtz