Seminars and Colloquia by Series

The Scattering Problem of the Intermediate Long Wave Equation

Series
PDE Seminar
Time
Tuesday, March 14, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yilun WuUniversity of Oklahoma

The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.

Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory

Series
Stelson Lecture Series
Time
Tuesday, March 14, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wilhelm SchlagYale University

Please Note: Mathematics lecture

 In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. 

By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or  Diophantine  rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of  acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of  finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

The Surprising Robustness and Computational Efficiency of Weak Form System Identification

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 (ZOOM)
Speaker
David BortzUniversity of Colorado, Boulder

Recent advances in data-driven modeling approaches have proven highly successful in a wide range of fields in science and engineering. In this talk, I will briefly discuss several ubiquitous challenges with the conventional model development / discretization / parameter inference / model revision loop that our methodology attempts to address. I will present our weak form methodology which has proven to have surprising performance properties. In particular, I will describe our equation learning (WSINDy) and parameter estimation (WENDy) algorithms.  Lastly, I will discuss applications to several benchmark problems illustrating how our approach addresses several of the above issues and offers advantages in terms of computational efficiency, noise robustness, and modest data needs (in an online learning context).

New approach to character varieties: nilpotent is the new holomorphic

Series
Geometry Topology Seminar
Time
Monday, March 13, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander ThomasU. Heidelberg

The study of representations of fundamental groups of surfaces into Lie groups is captured by the character variety. One main tool to study character varieties are Higgs bundles, a complex geometric tool. They fail to see the mapping class group symmetry. I will present an alternative approach which replaces Higgs bundles by so-called higher complex structures, given in terms of commuting nilpotent matrices. The resulting theory has many similarities to the non-abelian Hodge theory. Joint with Georgios Kydonakis and Charlie Reid.

Macdonald polynomials and the multispecies zero range process

Series
Algebra Seminar
Time
Monday, March 13, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Olya MandelshtamUniversity of Waterloo

Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system.

We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.

Nonlinear waves, spectra, and dynamics in infinite dimensions

Series
Stelson Lecture Series
Time
Friday, March 10, 2023 - 16:00 for 1 hour (actually 50 minutes)
Location
Klaus lecture auditorium 1443
Speaker
Wilhelm SchlagYale University

Please Note: General audience lecture

Waves are ubiquitous in nature. Some wave phenomena are conspicuous, most notably in elastic objects, and in bodies of water. In electro-dynamics, quantum mechanics, and gravity, waves play a fundamental role but are much more difficult to find. Over the past centuries, major scientific breakthroughs have been associated with the discovery of hidden wave phenomena in nature. Engineering has enabled our modern information based society by developing sophisticated methods which allow us to harness wave propagation. Seismic exploration relies on wave scattering in the discovery of natural resources. Medicine depends heavily on wave-based imaging technology such as MRI and CAT scans.

 

Mathematics has played a major role in the understanding of wave propagation, and its many intricate phenomena including reflection, diffraction, and refraction. In its most basic form, the wave equation is a linear partial differential equation (PDE). However, modern science and engineering rely heavily on nonlinear PDEs which can exhibit many surprising and delicate properties. Mathematical analysis continues to evolve rapidly driven in part by the many open questions surrounding nonlinear PDEs and their solutions. This talk will survey some of the mathematics involved in our understanding of waves, both linear and nonlinear.

A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries

Series
CDSNS Colloquium
Time
Friday, March 10, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Emmanuel FleurantinUNC, GMU

https://gatech.zoom.us/j/98358157136 

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River.  Using the Maslov index we identify a subset of the River which is comprised of local minimizers.  The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the talk is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

Lefschetz Fibrations and Exotic 4-Manifolds I

Series
Geometry Topology Working Seminar
Time
Friday, March 10, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Nur Saglam

Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of  exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.

Anderson Localization in dimension two for singular noise, part three

Series
Mathematical Physics and Analysis Working Seminar
Time
Friday, March 10, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Omar HurtadoUC Irvine

Continuing from where we left off, we will go through the proof of the probabilistic unique continuation result in Ding-Smart (2018) for solutions of the eigenequation on large finite boxes in the two-dimensional lattice. We'll briefly discuss the free sites formalism necessary to carry out the multiscale analysis as well, before going through technical lemmas concerning bounds on solutions to our eigenequation on large finite rectangles in the lattice as they propagate from a boundary.

Fast and optimal algorithm for online portfolios, and beyond

Series
Job Candidate Talk
Time
Thursday, March 9, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online via https://gatech.zoom.us/j/98280978183
Speaker
Dmitrii OstrovskiiUSC

In his seminal 1991 paper, Thomas M. Cover introduced a simple and elegant mathematical model for trading on the stock market. This model, which later on came to be known as  online portfolio selection (OPS), is specified with only two integer parameters: the number of assets $d$ and time horizon $T$. In each round $t \in \{1, ..., T\}$, the trader selects a  portfolio--distribution $p_t \in R^d_+$ of the current capital over the set of $d$ assets; after this, the adversary generates a nonnegative vector $r_t \in R^d_+$ of returns (relative prices of assets), and the trader's capital is multiplied by the "aggregated return'' $\langle p_{t}, r_{t} \rangle$. Despite its apparent simplicity, this model captures the two key properties of the stock market: (i) it "plays against'' the trader; (ii) money accumulates multiplicatively. In the 30 years that followed, the OPS model has received a great deal of attention from the learning theory, information theory, and quantitative finance communities.

In the same paper, Cover also proposed an algorithm, termed Universal Portfolios, that admitted a strong performance guarantee: the regret of $O(d \log (T))$ against the best portfolio in hindsight, and without any restrictions of returns or portfolios. This guarantee was later on shown to be worst-case optimal, and no other algorithm attaining it has been found to date. Unfortunately, exact computation of a universal portfolio amounts to averaging over a log-concave distribution, which is a challenging task. Addressing this, Kalai and Vempala (2002) achieved the running time of $O(d^4 T^{14})$ per round via log-concave sampling techniques. However, with such a running time essentially prohibiting all but "toy'' problems--yet remaining state-of-the-art--the problem of finding an optimal and practical OPS algorithm was left open.

In this talk, after discussing some of the arising challenges, I shall present a fast and optimal OPS algorithm proposed in a recent work with R. Jezequel and P. Gaillard (arXiv:2209.13932). Our algorithm combines regret optimality with the runtime of $O(d^2 T)$, thus dramatically improving state of the art. As we shall see, the motivation and analysis of the proposed algorithm are closely related to establishing a sharp bound on the accuracy of the Laplace approximation for a log-concave distribution with a polyhedral support, which is a result of independent interest.

Zoom link to the talk: https://gatech.zoom.us/j/98280978183

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