>>    Home
>>    Publications
>>    Talks
>>    Teaching
>>    Seminar

  • August 28, 2020
    Giorgio Young
    Department of Mathematics, Rice University
    Location and Time: Zoom, 2pm-3pm
    Title:Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows

    Abstract: We consider the Cauchy problem for the KdV hierarchy -- a family of integrable PDEs with a Lax pair representation involving one-dimensional Schr\"odinger operators -- under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency. This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions.This is joint work with Milivoje Lukic.


  • September 4, 2020
    Yang Zhou
    Department of Computer Science and Software Engineering, Auburn University
    Location and Time: Zoom, 3pm-4pm (special time)
    Title: Resilient Multiple Graph Learning

    Abstract: Rapid development of crowdsourced websites and information technology enables us to collect massive amounts of graph data, which are also known as networked data, ranging from biological, bibliographical, knowledge, and social networks, to communication, electrical, geographic, and transportation networks. Multiple graph data analysis has become a powerful tool for gaining insights and deriving innovations into our increasingly connected world. Real-world graph data are typically noisy due to massive disinformation injected by malicious parties and users. Unfortunately, graph learning models, especially deep learning models, are highly sensitive to small perturbations of their input intended to result in analysis failures. Given the need to understand the vulnerability and resilience of graph data analysis, two questions arise: (1) Are multiple graph learning models sensitive to adversarial perturbations over intra-graph and inter-graph interactions? (2) Can we propose impelling defense techniques to offer sufficient protection to multiple graph learning models against adversarial attacks? In this talk, I will introduce problems, challenges, and solutions for characterizing and understanding and learning vulnerability and resilience of multiple graph learning under adversarial attacks. I will also discuss our recent work on adversarial attacks over multiple graph learning. I will conclude the talk by sketching interesting future directions for resilient multiple graph learning.


  • September 11, 2020
    Tom VandenBoom
    Department of Mathematics, Yale University
    Location and Time: Zoom, 2pm-3pm
    Title: Bilateral periodicity of stationary solutions of the Toda flow

    Abstract: Classically, the Toda flow is a spectrum-preserving flow on symmetric tridiagonal Jacobi matrices described by a simple system of ODEs. Asymptotically, this flow tends to a diagonalization of the initial condition, and the stationary points of the flow are precisely diagonal matrices. The same system of ODEs extends naturally to a hierarchy of flows on Jacobi operators acting on the integer lattice, whose stationary solutions are much more interesting: their parametrizing sequences are almost-periodic. In this talk, we will investigate how "almost" periodic these solutions can be; in particular, we will show that, for certain Toda flows, stationary Jacobi operators with periodic off-diagonal must have periodic main diagonal. (Joint work with Lance Saddler, Tony Zeng, and Richard Zhou.)


  • September 18, 2020
    Jake Fillman
    Department of Mathematics, Texas State University
    Location and Time: Zoom, 2pm-3pm
    Title: Multidimensional Schrödinger Operators Whose Spectrum Features a Half-Line and a Cantor Set

    Abstract: We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe--Sommerfeld criterion for sums of Cantor sets which may be of independent interest. [Joint work with David Damanik (Rice) and Anton Gorodetski (UC Irvine)].


  • September 25, 2020
    Selim Sukhtaiev
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Zoom, 2pm-3pm
    Title: Anderson localization for disordered trees

    Abstract: In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves in random media. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.


  • October 2, 2020
    Selim Sukhtaiev
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Zoom, 2pm-3pm
    Title: Anderson localization for disordered trees

    Abstract: In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves in random media. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.


  • October 9, 2020
    Selim Sukhtaiev
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Zoom, 2pm-3pm
    Title: Anderson localization for disordered trees

    Abstract: In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves in random media. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.


  • October 16, 2020
    Song Chen
    Department of Mathematics, University of Wisconsin at La Crosse
    Location and Time: Zoom, 2pm-3pm
    Title: A neural network approach to sampling based learning control for quantum system with uncertainty.

    Abstract: Robust control design for quantum systems with uncertainty is a key task for developing practical quantum technology. In this paper, we apply neural networks to learn the control of a quantum system with uncertainty. By exploiting the auto differentiation function developed for neural network models, our method avoids the manual computation of the gradient of the cost function as required in traditional methods. We implement our method using two algo- rithms. One uses neural networks to learn both the states and the controls and one uses neural networks to learn only the controls but solve the states by finite difference methods. Both algorithms incorporate the sampling-based learning process into the training of the networks. The performance of the algorithms is evaluated on a practical numerical example, followed by a detailed discussion about the advantage and trade-offs between our method and the other numerical schemes.


  • October 23, 2020
    Andrew Christlieb
    Department of Computational Mathematics, Science and Engineering, Michigan State University
    Location and Time: Zoom, 2pm-3pm
    Title: Kernel based methods as a path to ‘Explicit’ unconditionally stable O(N) methods

    Abstract: In this talk we introduce the idea of a kernel based derivative. The kernel based approach expresses a differential operator in terms of a convolution integral. High order is achieved through successive convolution. The approach can be pared with both explicit multi stage or explicit multi step methods. The approach does not involve iteration, but rather it is a direct update. The method is provably O(N) and Von Neumann analysis shows that the method is unconditionally stable for linear problems. For non-linear problems, the method behaves unconditionally stable, but we have not proven it thus far. We have developed these methods for the Wave equations, Hamilton Jacobi equations, Degenerate Advection Diffusion equations, as well as developed a domain decomposition approach that lets us scale these new methods to large distributed heterogenous computers. Further, we have extended these ideas to mapped grids. In this talk we will introduce the ideas in the context of the Hamilton Jacobi equations and introduce our approach for addressing complex geometry.


  • October 30, 2020
    Xu Yang
    Department of Mathematics, University of California, Santa Barbara
    Location and Time: Zoom, 2pm-3pm
    Title: Seismic tomography, image segmentation and deep learning

    Abstract: Seismic tomography is a scientific field using realistic earthquake data to analyze the inner structure of our Earth. In this talk, we present a natural connection of three-dimensional seismic tomography to image segmentation problems, which we solve efficiently using deep neural networks with a UNet architecture. It is challenging to obtain sufficient valid data to train neural networks, and we overcome it by developing a fast synthetic data generator using multi-scale asymptotic analysis. The accuracy and parallelizability of the proposed algorithm is illustrated by comparing to the spectral element method. Moreover, the developed multi-scale algorithm can be also used to accelerate various standard applications in seismic tomography, including seismic traveltime tomography and full waveform inversion.


  • November 6, 2020
    Aleksey Kostenko
    Department of Mathematics, University of Vienna, Austria
    Location and Time: Zoom, 2pm-3pm
    Title: Generalized Indefinite Strings

    Abstract: In this talk, we review the direct and inverse spectral theory for indefinite strings. As one of our main results we are going to present the indefinite analog of M. G. Krein's celebrated solution to the string density problem. We also plan to discuss its relationships with the conservative Camassa-Holm flow. The talk is based on joint work with J. Eckhardt (Loughborough, UK).


  • November 13, 2020
    Alexis Drouot
    Department of Mathematics, University of Wahsington
    Location and Time: Zoom, 2pm-3pm
    Title: Dynamics in topological insulators

    Abstract: Topological insulators are materials that block conduction along their interior but support extraordinarily robust currents along their boundary. I will first review the bulk-edge correspondence, an index-like theorem that explains the stability of currents. I will then present a research project that aims to quantitatively describe these currents in the semiclassical regime.


  • November 20, 2020
    Erkan Nane
    Department of Mathematics and Statistics, Auburn University
    Location and Time: Zoom, 2pm-3pm
    Title: TBA

    Abstract: TBA