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Dynamics, Geometry and Groups Seminar

Wednesday, August 3rd, 2022

Time: 2:30 p.m.  Place: Jeffery Hall 422

Speaker: Marco Lenci (Università di Bologna)

Title: Internal-wave billiards in trapezoids and similar tables

Abstract: We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. The phenomenon of internal waves in a fluid occurs in many situations and has been intensively studied by physicists. One of the first experiments, which became paradigmatic, was done in a container shaped like a rectangular trapezoid (with some thickness). For a class of tables including rectangular trapezoids, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). If time permits, we will also discuss the prominent case where the table is an actual trapezoid, studying the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnold tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero. Joint work with C. Bonanno and G. Cristadoro.

Algebra & Geometry Seminar

Monday, April 11th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Emily Cliff (Universite de Sherbrooke)

Title: Moduli spaces of principal 2-group bundles and a categorification of the Freed-Quinn line bundle

Abstract: A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group in the sense of Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed--Quinn line bundle. This line bundle has as its global sections the state space of Chern--Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz--Teichner). This is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

Website details here:

Math & Stats Department Colloquium

Friday, April 8th, 2022

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Runmin Wang (Southern Methodist University)

Title: Statistical Inference for Change Points in High-Dimensional Data

Abstract: Estimation and testing of change points in high-dimensional data have wide applications in many disciplines, such as biological science, economics and finance. In this talk, we introduce a new U-statistic based approach to both problems and show its advantage over several existing methods via theory and simulations. The talk consists of two parts. In the first part, we will introduce a new test based on U-statistics for testing a mean shift in high-dimensional data. The test aims to detect dense alternatives and is tuning parameter free. At the core of our theory, we show weak convergence of a sequential U-statistic based process, and derive the limiting distribution under both the null and alternatives. In the second part, we will discuss a change point location estimator which maximizes a new U-statistic based objective function. Under mild and easily interpretable assumptions, we derive its convergence rate and asymptotic distribution after suitable centering and normalization. A comparison with the popular least squares based approach illustrates the theoretical advantage of ours. A bootstrap-based approach is also proposed to construct a confidence interval with accurate coverage, which is corroborated by simulation results. We shall illustrate our method using a real data example at the end of the talk.

Runmin Wang is an Assistant Professor in the Department of Statistical Science at Southern Methodist University. He got his Ph.D. in Statistics from the University of Illinois at Urbana-Champaign in 2020. His research interests include change point inference, high-dimensional data and time series analysis.

Algebra & Geometry Seminar

Monday, April 4th, 2022

Time: 4:30 p.m.  Place: Online via Zoom (contact Kaveh Mousavand for Zoom link)

Speaker: Matthew Mastroeni (Iowa State University)

Title: Chow rings of matroids are Koszul

Abstract: The Chow ring of an algebraic variety is an algebro-geometric analog of the cohomology ring of a smooth manifold that encodes important information about the intersections between its subvarieties. Feichtner and Yuzvinsky computed a presentation for the Chow ring of a smooth toric variety associated to a matroid (and some other data) which is now called the Chow ring of the matroid. These rings have garnered significant attention in recent years thanks to their role in establishing long-standing conjectures on the combinatorics of matroids, including the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang.

From a commutative algebra standpoint, Chow rings of matroids are very nice graded Artinian Gorenstein rings defined by quadratic relations, and so, a natural conjecture posed by Dotsenko is that the Chow ring of a matroid is always Koszul. In this talk, we will discuss how the combinatorics of a matroid influences algebraic properties of its Chow ring, culminating in recent joint work with Jason McCullough giving an affirmative answer to Dotsenko’s conjecture.

Website details here:

Statistics Seminar

Wednesday, March 30th, 2022

Time: 10:00 a.m.  Place: Online via Zoom (contact Brian Ling for Zoom link)

Speaker: Chuan-Fa Tang (Department of Mathematical Sciences at University of Texas at Dallas)

Title: Taylor's law for semivariance and higher moments of heavy-tailed distributions

Abstract: The power law relates the population mean and variance is known as Taylor's law proposed by Taylor in 1961. We generalize Taylor's law from the light-tailed distributions to heavy-tailed distribution with infinite mean. Instead of population moments, we consider the power-law between the sample mean and many other sample statistics, such as the sample upper and lower semivariance, the skewness, the kurtosis, and higher moments of a random sample. We show that, as the sample size increases, the preceding sample statistics increase asymptotically in direct proportion to the power of the sample mean. These power laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the potential upside ratio, and the Farinelli-Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. In addition, we find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of tail-index based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators.

Curves Seminar

Monday, March 28th, 2022

Time: 11:00 a.m.  Place: Jeffery Hall Room 222, or Online via Zoom (contact Deepanshu Prasad for Zoom link)

Speaker: Deepanshu Prasad

Title: Broken Circuit Complexes for a Matroids

Abstract: I will construct a standard $\mathcal{K}$-basis for the Orlik-Solomon algebra $A(\mathcal{A})$ by using broken circuit module and hence, prove that the Orlik-Solomon algebra is a free $\mathcal{K}$-module.