Contributed Talks: Titles and Abstracts
Kenneth Beard (Louisiana State University)
On the nuances of discretizing continuum electrical conductivity
Abstract: We discuss the challenges that arise when formulating a discrete analogue to periodic electrical conductivity in the continuum. In particular, we examine when the effective conductivity of a lattice electrical network is equal to the effective operator of an associated lattice Z-problem. We accomplish this by adapting tools from the abstract theory of composites, algebraic potential theory, and Schur complement theory. This is based on Sect. VI D of our JMP paper "Effective operators and their variational principles for discrete electrical network problems", which is joint work with Anthony Stefan and Dr. Aaron Welters at Florida Institute of Technology and Dr. Robert Viator now at Denison University.
Adam Christopherson (Ohio State University)
Weak-type regularity of the Bergman projection on Hartogs domains
Abstract: For power-generalized Hartogs triangles in ℂ3, we show that the Bergman projection satisfies a weak-type estimate at the upper endpoint of boundedness but not at the lower endpoint. Our work complements related results obtained recently for rational Hartogs triangles in ℂ2 and the punctured unit ball in ℝ3.
This work is joint with K.D. Koenig.
Jesus Cruz-Lugo (Baylor University)
On the stability of the Dirichlet problem for weakly leliptic systems in the plane
Abstract: As opposed to Legendre–Hadamard elliptic systems, the Lp-Dirichlet problem for weakly elliptic systems in the upper-half space may not necessarily be well-posed. Remarkably, when the latter BVP is well-posed we show that it remains so after small perturbations of the given weakly elliptic system. This is joint work with Marius Mitrea.
David Daniel (University of Texas at Dallas)
Mathematical modelling of yellow fever transmission dynamics with stability analysis
Abstract: Yellow fever remains endemic in some parts of the world (Nigeria included), despite the availability of a potent vaccine to curtail the spread of the disease. This necessitates continuous research on the disease's transmission dynamics and its control. Consequently, a deterministic epidemiological model for yellow fever transmission dynamics within the human mosquito population is considered in this work. The model equilibrium solutions are obtained and the conditions for their stability are established. The basic reproduction number R0 for the model is derived. The model is solved numerically using the Runge Kutta of order four scheme. Simulation of our results shows that the disease will continue to be present (no matter how small) in our society as long as immunity conferred by vaccination is not lifelong and there are new births, thus justifying the need for continuous vaccination against yellow fever.
Matthew Faust (Texas A&M University)
An Ambartsuman-type inverse problem for periodic graphs
Abstract: We study Floquet isospectrality of the zero potential for the discrete periodic Schrödinger operator acting on functions on the n-dimensional square lattice. It is well-known that for the square lattice, the zero potential has no nonzero real floquet isospectral potentials. It is in folklore that there exist non-zero complex solutions. As part of an REU led by Wencai Liu, Rodrigo Matos, and the speaker, we provide explicit solutions whenever one of the periods is even using combinatorial methods.
Changyan Shi (University of Texas Rio Grande Valley)
Bilinear structure and KP reductions to CSSI equation (slides)
Abstract: In this presentation, we construct breather and rouge wave solutions to one generalized coupled nonlinear Schrödinger (NLS) equation, so called coupled Sasa–Satsuma (CSS) equation by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and the Hirota's bilinear method. First, we show that the CSS equation can be bilinearized into a set of six equations. Then, starting from 13 bilinear equations in the KP-Toda hierarchy, together with the discrete KP equation, the CSS equation is derived. As a by-product, we provide its multi-breather and general rogue wave solutions. This is a joint work with my advisor, Dr. Baofeng Feng, and Dr. Chengfa Wu and Mr. Guangxiong Zhang at Shenzhen University, China.
Zhaosheng Feng (University of Texas Rio Grande Valley)
Parabolic Systems of Aggregation Formation in Bacterial Colonies
Abstract: The goal of this talk is to study a fourth-order nonlinear parabolic system with dispersion for describing bacterial aggregation. It shows that unlike the model without dispersion, a bacterial cluster can move, which allows us to consider dispersion as some kind of control for bacterial colony. We demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without dispersion.
Himali Gammanpila (Texas Tech University)
Stability analysis and a priori error estimation for Nitsche-type CIP/GP-CutFEM multi-phase flow
Abstract: Two-phase flows are encountered in various industrial applications and natural phenomena. Since the interface is significantly thin in two-phase flow, it can be treated as a discontinuity in the flow field where localized surface tension forces act. Defining the interface implicitly means that elements may be intersected by the interface, and the aforementioned discontinuities may occur inside them. When using the finite element method (FEM) with polynomial shape functions, these discontinuities cannot be explicitly represented. Therefore, many PDE solvers employ a discontinuous function, especially in the context of fluid dynamics problems. These methods utilize discontinuous functions to distinguish different domains and ensure no extrinsic contributions are incurred when utilizing an arbitrary discontinuity. An extension of FEM utilizing discontinuous functions is CutFEM or Extended FEM (XFEM), which allows for reproducing arbitrary discontinuities inside elements by providing an enhanced shape function basis. In this study, a Nitsche-type extended variational multiscale method for two-phase flow is suggested, specifically for discontinuous pressure. To ensure a stable formulation across the entire domain, Continuous Interior Penalty (CIP) based variational multiscale terms are supported by appropriate face- oriented ghost-penalty terms. These terms are introduced to sufficiently control the enrichment value of the solution fields, thereby ensuring the stability of the formulation. The numerical analysis of the proposed CutFEM starts by providing the bilinear form that satisfies an inf-sup condition with respect to a suitable norm. The inf-sup constant is independent of how the boundary cuts the underlying mesh. Furthermore, energy-type a priori estimates are proved to be independent of the local Reynolds number.
Joan Morrill Gavarro (RWTH Aachen)
A hydrodynamic formulation for a nonlinear Dirac equation (slides)
Abstract: We derive a hydrodynamics formulation for a modified Dirac equation with a nonlinear mass term. The nonlinearity has the same homogeneity as the classical Dirac equation but is only Lipschitz continuous. We prove a global existence result for a regularised equation. The nonlinear Dirac equation admits a clean and symmetric split into the left and right-handed spinor components. It is formulated using Clifford algebra tools. This is joint work with Prof. Dr. Michael Westdickenberg (RWTH Aachen University).
Jon Harrison (Baylor University)
A discrete analog of quantum unique ergodicity on circulant graphs
Abstract: A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao (2023). They show that for large graphs there is an orthonormal basis of Eigenfunctions of the adjacency matrix such that quantum probability measures of the eigenfunctions puts approximately the correct proportion of their mass on subsets of the vertices that are not too small. We investigate this property on families of circulant graphs with prime order. We see that the property holds for an orthonormal Eigenfunction basis but fails if the basis is required to be real. The equivalent result for a real basis holds for the Cayley graphs of quasirandom groups. This is work with Clare Pruss at Baylor University, as part of her honors thesis.
Zulaihat Hassan (Auburn University)
Global existence of classical solutions of chemotaxis systems with logistic source and consumption or linear signal production on ℝn
Abstract: The current talk is concerned with the global existence of classical solutions for the following three primary chemotaxis systems with a logistic source on ℝn,
(1) ut=Δu-χ∇∙ (u∇v)+u(a-bu), x∈ℝn, τvt= Δv-uv, x∈ℝn,
(2) ut=Δu-χ∇∙ (u∇v)+u(a-bu), x∈ℝn, τvt= Δv-λv+μu, x∈ℝn,
and
(3) ut=Δu-χ∇∙ (u∇v)+u(a-bu), x∈ℝn, 0= Δv-λv+μu, x∈ℝn,
where χ is a nonzero number and a, b, λ, μ, τ are positive constants. We provide sufficient conditions for the global existence and boundedness of classical solutions of (1), (2), and (3) with nonnegative initial functions in a unified way. It follows that nonnegative classical solutions of (1), (2), and (3) exist globally and stay bounded in one- and two-dimensional settings for any chemotaxis sensitivity χ. Moreover, we show that the methods developed for the study of (1)-(3) can be adapted to the bounded domain case without much effort. Several existing results for (1), (2), and (3) on bounded domains are then improved.
Lawford Hatcher (Indiana University)
The hot spots conjecture with mixed boundary conditions
Abstract: Rauch's hot spots conjecture states that the first non-constant Neumann eigenfunction of the Laplacian on a bounded domain in ℝn attains its local extrema only on the boundary of the domain. We consider an analogous problem concerning the first eigenfunction with mixed Dirichlet–Neumann boundary conditions. We will show why these two problems are closely related to each other and then present a result when the domain is a triangle or a region bounded by the graph of a piecewise smooth function.
Xiaoqi Huang (LSU)
Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds (slides)
Abstract: We will discuss improved Strichartz estimates for solutions of the Schrödinger equation on compact manifolds with nonpositive curvature which is related to the results of Burq, Gérard and Tzvetkov (2004). The proof is based on obtaining lossless estimates on relatively small time intervals which may depend on the frequency, we shall also discuss the flat tori as a sample case. This is based on joint work with Matthew Blair and Christopher Sogge.
Omar Hurtado (University of California, Irvine)
Localization and unique continuation for non-stationary Schrödinger operators on ℤ2
Abstract: We extend methods from the breakthrough paper of Ding–Smart (2020) which showed Anderson localization for certain random Schrödinger operators on the lattice ℤ2 via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum.
Yongming Li (Texas A&M University)
Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance (slides)
Abstract: In this talk, we will discuss dispersive and local decay estimates for a class of matrix Schrödinger operators that naturally arise from the linearization of focusing nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slowdown of the local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. Some applications for the linearized equation for the 1D focusing cubic Schrödinger equation will be discussed.
Sakshi Malhotra (University of Texas at Dallas)
Finite time stability of a sweeping process for an elastoplastic system with stress-controlled loading
Abstract: Based on the ideas of Adly, Attouch, and Cabot on finite time stabilization of dry friction oscillators and the results from "Finite time stability of Polyhedral Sweeping Processes with Application to Elastoplastic systems" by Gudoshnikov, Makarenkov and Rachinskii we establish a theorem on finite-time stabilization of differential inclusions with a polyhedral constraint, where the shape of the moving constraint changes with time (the constraint of the form C(t)). We then employ the ideas of Moreau [in "New Variational Techniques in Mathematical Physics" (Centro Internaz. Mat. Estivo (CIME), II Ciclo, Bressanone, 1973), Edizioni Cremonese, Rome, 1974, pp. 171–322] to apply our theorem to a system of elastoplastic springs with a stress-controlled loading. We further use the theorem on an elastoplastic model to fully illustrate its practical application.
Tal Malinovitch (Rice University)
Directional Ballistic transport for partially periodic Schrödinger operators
Abstract: In this talk, we will consider Schrödinger operators on ℝd or ℤd with bounded potentials V that are periodic in some direction and compactly supported in others. Such systems are known to produce "surface states" that are confined near the support of the potential. Specifically, we will focus on the transport properties of these states – in other words, the rate at which these states spread in different directions. Roughly speaking, we say that a state exhibits ballistic motion if it spreads linearly in time (x ∼ t- in some sense). We show that, under very mild assumptions, a class of surface states exhibits what we describe as directional ballistic transport, consisting of a strong form of ballistic transport in the periodic directions and its absence in the other directions. Furthermore, in some models, we show that a dense set of surface states exhibit this surface ballistic transport property. In this talk, I will briefly review our main results and some of the tools used in this work. This is joint work with Adam Black, David Damanik, and Giorgio Young.
Ellie Matter (Baylor)
Cluster formation in iterated mean field games
Abstract: We look at a simple first-order Mean Field Game that gives players incentive to congregate. With a short enough time horizon, this type of game has a unique Nash Equilibrium given an initial distribution of players. Since the game is played only for a short time, we consider iterating the game, each new iteration starting at the final distribution of the previous game. We prove that after sufficiently many iterations, the players do congregate in tighter and tighter clusters and show where these clusters form.
Hewan Meles Shemtaga (Auburn University)
Chemotaxis models and Keller–Segel systems on compact metric graphs
Abstract: Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this talk we will discuss a pair of logistic type Keller–Segel systems of reaction-advection-diffusion equations modeling chemotaxis on networks. The first part of the talk concerns well-posedness of Keller–Segel systems on arbitrary compact metric graphs. In the second part of the talk, we will focus on asymptotic stability, instability, and bifurcation of constant steady state solutions. This is joint work with Wenxian Shen (Auburn) and Selim Sukhtaiev (Auburn).
Joseph Miller (University of Texas at Austin)
On the effective dynamics of Bose–Fermi mixtures (slides)
Abstract: In this talk, I will be discussing recent work "On the effective dynamics of Bose–Fermi mixtures" with Esteban Cardenas and my advisor Natasa Pavlovic. In this work, we describe the dynamics of a Bose–Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyze the mean-field approximation of the many-body Schrödinger dynamics and prove emergence of a coupled Hartree-type system of equations. We obtain rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identify a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical; this is one of the main contributions of the present article. In this regime, the bosons are much lighter and more numerous than the fermions. We then prove convergence to a coupled Vlasov–Hartee system of equations with an explicit convergence rate.
Alejandro Quintero-Roba (Baylor University)
The Riemann–Hilbert problem for Krall orthogonal polynomials
Abstract: In this talk, we give background on the Riemann–Hilbert problem (RHp) for orthogonal polynomials (OP) and its versatility in finding general properties of specific OP families such as the ordinary differential equation the family satisfies. We also discuss the Krall–Legendre OP sequence, which is known for satisfying a fourth-order differential equation. Then, we formulate the RHp for the Krall–Legendre OP, prove the existence and uniqueness of its solution, and show the method to obtain the first-order matrix ODE, and the second-order scalar ODE for the Krall–Legendre OP, as a first approach to finding the fourth-order scalar ODE from the Riemann–Hilbert formulation, as our final goal.
Miraj Samarakkody (Texas Tech University)
Closed p-elastic curves in spheres of 𝕃3
Variational problems involving curves with energy densities influenced by their curvature are common in mathematical physics. This talk focuses on the existence of closed p-elastic curves in two distinct spaces: the hyperbolic plane ℍ2 and the de-Sitter 2-space ℍ21, for any real number value of p. Our findings indicate that in the hyperbolic plane ℍ2, closed p-elastic curves exhibiting non-constant curvature exist when p>1, while, in the de-Sitter 2-space ℍ21, such curves only exist when p<0. Additionally, we establish that under these specific conditions for the energy parameter p and in both types of spaces, for any pair of relatively prime natural numbers (n,m) that meet the criteria m<2 n<√2 m, a closed p-elastic curve with non-constant curvature can be found. These curves exhibit m-fold symmetry and have a winding number equal to n.
Brian Simanek (Baylor University)
Discrete m-functions with doubly-palindromic continued fraction coefficients
Abstract: We will consider Jacobi matrices whose diagonal and off-diagonal entries are eventually periodic sequences. We show that if each period can be written as the concatenation of two palindromes, then there is a special property satisfied by these matrices, which is most easily expressed in terms of the associated m-functions. This generalizes a known result about the continued fraction expansion of real numbers. Joint work with Hunter Handley.
Yaofeng Su (Georgia Tech)
Some new results of open dynamical systems
Abstract: Open dynamical systems study the statistics of the first hitting time when trajectories escape through a subset, referred to as a "hole," in the phase space. I will discuss my recent work on open hyperbolic systems, addressing three main aspects:
This is joint work with Prof. Leonid Bunimovich.
Pedro Takemura Feitosa Da Silva (Baylor University)
A new class of Herz-type spaces and boundary problems
Abstract: In the past two decades the theory of Herz spaces has seen numerous developments, yielding a large family of Herz-type scales which are particularly relevant from the perspective of Hamonic Analysis. The main point of this talk is to describe a new brand, referred to as Grand Herz spaces, which unifies much of the existing body of work. Notably, this new brand fits within the framework of Generalized Banach Function Spaces, thus paving the way for the treatment of boundary value problems in this inclusive functional analytic setting. As an illustration, we present a well-posedness result for the Dirichlet Problem with boundary data taken in Grand Herz spaces. This is joint work with Marius Mitrea.
Tracy Weyand (Rose–Hulman Institute of Technology)
Calculating the number of spanning trees of a quantum graph from its spectrum
Abstract: Kirchhoff's Matrix Tree Theorem provides a way to calculate the number of spanning trees of a discrete graph from the spectral determinant of the combinatorial Laplacian acting on that graph. We extend this result to quantum graphs. We derive a formula that allows one to calculate the number of spanning trees of a connected equilateral metric graph in terms of the spectral determinant of the Laplacian acting on the graph (with Neumann–Kirchhoff vertex conditions). This is accomplished using a previous result that relates the spectral determinant of Laplace operators on discrete and quantum graphs. We then show that this result holds under small perturbations of edge length. This talk is based on joint work with Jon Harrison.
Mahishanka Withanachchi (Laval University)
Lebesgue constants in local Dirichlet spaces
Abstract: This study delves into the analysis of partial Taylor sums Sn, n≥0, as finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical disc algebra setting, these operators are known as Lebesgue constants, with their precise norm remaining unresolved. However, our focus shifts to the weighted Dirichlet spaces 𝓓w, where we accurately determine the norm of Sn. This exploration involves three distinct norms on 𝓓w, each providing unique values for the norm of Sn as an operator on 𝓓w. Notably, these findings stand in sharp contrast to the classical disc algebra. Moreover, we extend our investigation to Cesaro means σn on local Dirichlet spaces, aiming to precisely determine their norm for the three introduced metrics.
Lebesgue constants in local Dirichlet spaces are vital for guiding the selection of optimal finite-dimensional approximations in numerical solutions of partial differential equations with Dirichlet boundary conditions in mathematical physics.
Lili Yan (University of Minnesota)
Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation
Abstract: In this talk, we consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions n≥2. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on second-order linearization, multivariate finite differences, as well as the stability of the light-ray transform. This is based on joint work with Ru-Yu Lai.
Mengxuan Yang (University of California, Berkeley)
Semiclassical analysis of twisted TMDs: exponentially flat and trivial bands
Abstract: Recent experiments discovered fractional Chern insulator states at zero magnetic field in twisted bilayer MoTe2 and WSe2, which are different types of twisted transition metal dichalcogenides (TMDs). In this talk, using Floquet theory and construction of quasi-mode, I will present some mathematical studies of band properties when the twisting angles are small, including: absence of flat bands, trivial topology of lower bands and exponential flatness of lower bands. This is a joint work with Simon Becker.
Xiaowen Zhu (University of Washington)
Topological edge spectrum along curved interface
Abstract: Topological insulators (TI) are a class of 2D materials which behave like insulators in their bulk but support robust states along their edges. One of the key property of TI that is expected to be true is the robustness of the property above w.r.t. the shape of the edge. In this talk, we will discuss how does shape of edge influence the property of TI above. In particular, we will both give a general, intuitive condition for this property to hold, and provide a counter-example otherwise. We also show why in practical situation, experiments may provide misleading results on TI. This work is based on a joint work with Alexis Drouot.
Yuzhou Zou (Northwestern University)
A Gutzwiller trace formula for semiclassical Schrödinger operators with conormal potentials
Abstract: We discuss ongoing work (joint with Jared Wunsch and Mengxuan Yang) which concerns extending the Gutzwiller Trace Formula from the case of smooth potentials to the case of potentials with conormal singularities. In the smooth case, the formula expresses an eigenvalue-counting function of a Schrödinger operator as a sum of certain dynamical quantities over periodic Hamiltonian trajectories. In the conormal case, a consideration of a WKB ansatz for the Schrödinger propagator suggests the sum should incorporate dynamical information about Hamiltonian trajectories which reflect at the site of the singularity. We will discuss the variational formulation required to make sense of the dynamics of such trajectories, as well as the further work we expect to need in order to complete the proof.
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