Two topics in modern nonparametric statistics are highlighted. The first one is missing data. Theory and examples are presented. Results are illustrated via regression with missing predictors and responses. The second topic is devoted to survival analysis, specifically to efficient estimation of the hazard rate function for truncated and censored data. All used statistical notions will be explained.

Geometric Approaches on Persistent Homology

Baris Coskunuzer (UT Dallas) October 15, 2021

Persistent Homology is one of the most important techniques used in Topological Data Analysis. In the first half of the talk, we give an introduction to the subject. Then, we study the persistent homology output via geometric topology tools. In particular, we give a geometric description of the term “persistence”, and discuss its relations to Gromov’s filling radius. This is joint work with Henry Adams. The talk will be non-technical, and accessible to graduate students.

Normal reflection subgroups of complex reflection groups

Carlos Arreche (UT Dallas) October 22, 2021

A complex reflection is an invertible linear transformation of a finite-dimensional complex vector space that has finite order and acts trivially on a complex subspace of codimension one. A complex reflection group is a finite group generated by complex reflections. Replacing “complex” with “real”, one obtains precisely the finite Coxeter groups, which in turn comprise a generalization of Weyl groups of semisimple Lie algebras. Complex reflection groups have many applications, including to the representation theory of reductive algebraic groups, Hecke algebras, knot theory, moduli spaces, low-dimensional algebraic topology, invariant theory and algebraic geometry, differential equations, and mathematical physics. In joint work with Nathan Williams, we study normal reflection subgroups of complex reflections groups (that is, normal subgroups that are also generated by complex reflections). Our approach leads to a refinement of the celebrated theorem of Orlik and Solomon that the generating function for fixed-space dimension over a complex reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of Nathan Williams. This talk only assumes a modest background in abstract algebra and should be accessible to graduate students.

Using Topological Shape Features to Characterize Medical Images: Case Studies on Lung and Brain Cancers

Chul Moon (SMU) October 29, 2021

Tumor shape is a key factor that affects tumor growth and metastasis. This talk presents a topological feature computed by persistent homology to characterize tumor progression from digital pathology and radiology images and examines its effect on the time-to-event data. The proposed topological features are invariant to scale-preserving transformation and can summarize various tumor shape patterns. The topological features are represented in functional space and used as functional predictors in a functional Cox proportional hazards model. The proposed model enables interpretable inference about the association between topological shape features and survival risks. Two case studies are conducted using consecutive 143 lung cancer and 77 brain tumor patients. The results of both studies show that the topological features predict survival prognosis after adjusting clinical variables, and the predicted high-risk groups have significantly (at the level of 0.001) worse survival outcomes than the low-risk groups. Also, the topological shape features found to be positively associated with survival hazards are irregular and heterogeneous shape patterns, which are known to be related to tumor progression.

Scalable DNA-protein binding changer test for insertion and deletion of bases in the genome

Sunyoung Shin (UT Dallas) November 5, 2021

Noncoding regions that do not encode protein are the majority of the genome, e.g., about 99% of the human genome is noncoding DNA. Mutations in the noncoding genome have been crucial to understand disease mechanisms through dysregulation of disease-associated genes. One key element in gene regulation that noncoding mutations mediate is the binding of proteins to DNA sequences. Insertion and deletion of bases (InDels) are the second most common type of mutations that may impact DNA-protein binding. However, no existing methods could be utilized to determine the quantitative effects on DNA-protein binding driven by InDels. We develop a novel statistical test, named binding changer test (BC test), to evaluate the impact of InDels on DNA binding changes using DNA-binding motifs and single sequence modeling. The test predicts binding changer InDels of regulatory importance with an efficient importance sampling algorithm in generating background sequences from an importance distribution more weighting large binding affinity changes. We derive the importance distribution with the optimal tilting parameter. The BC test provides a general statistical framework for any disease types in any species genomes. Simulation studies demonstrate its excellent performance. The application to genome sequencing datasets in human leukemia samples uncovers candidate pathologic InDels by modulating MYC binding in leukemic genomes.

Different Hamiltonians for Painlevé Equations and their identification using geometry of the space of initial conditions

Anton Dzhamay (University of Northern Colorado) November 12, 2021

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique – there are many very different Hamiltonians that result in the same differential Painlevé equation. In this paper we describe a systematic procedure of finding changes of coordinates connecting different Hamiltonian systems. Our approach is based on the notion of Okamoto space of initial conditions and Sakai’s geometric theory of Painlevé equations and is motivated by a similar procedure in the discrete case. As an example, we consider the fourth differential PIV equation and compare Hamiltonians given in the works of Okamoto, Jimbo-Miwq, Filipuk–Zoladek, Kecker, and Its Prokhorov. We explain how geometry makes it easy to find explicit birational change of coordinates transforming one Hamiltonian into another. This approach can be easily adapted to other Painlevé equations as well. This is a joint project with Galina Filipuk and her student Adam Ligeza, and also with Alexander Stokes.

Coxeter Pop-Tsack Torsing

Nathan Williams (UT Dallas) November 19, 2021

Given a finite irreducible Coxeter group W, we use the W-noncrossing partition lattice to define a Bessis dual version of C. Defant’s notion of a Coxeter pop-stack sorting operator. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of this operator and the maximum size of a forward orbit is the Coxeter number of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. This is joint work with C. Defant.

Thermodinamic Formalism in Transcendental Dynamics

Mariusz Urbański (UNT), December 3, 2021

I will present some joint works with Volker Mayer in which we primarily show that for a large class of entire and meromorphic transcendental functions the full geometric thermodynamic formalism holds. Most notably, this means that the transfer operators generated by geometric potentials are well defined and bounded after an appropriate conformal change of Riemannian metric on the complex plane C. We show that these operators are quasi-compact of diagonal type with one leading eigenvalue, which in addition is simple. In particular, the dual operators have positive eigenvalues and eigenvectors that are Borel probability eigenmeasures. The probability measure obtained by integrating these eigenmeasures against leading eigenfanctions of transfer operators are invariant. We show that these measures are equilibrium states of geometric potentials. The primary applications of these theorems capture the stochastic laws such as exponential decay of correlations, the central limit theorem, and the law of iterated logarithm. It also permits us to provide exact formulas (of Bowen’s type) for Hausdorff dimension of radial Julia sets and multifractal analysis.
We will discuss two distinct routes (leading to different though overlapping classes of meromorphic transcendental functions) to get the geometric thermodynamic formalism. One of them is based on Nevanlina’s theory and the other on analogues of integral means spectrum from classical complex analysis of conformal maps.

Some Recent Results On Wave Turbulence: Derivation, Analysis, Numerics and Physical Application

Tran Minh Binh (SMU & MIT), January 28, 2022

Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. Recent mathematical interests on wave turbulence theory have their roots from the works of Bourgain, Staffilani and Colliander-Keel-Staffilani-Takaoka-Tao. In this talk, I will present some of our recent results on wave turbulence theory. In the first part of the talk, I will discuss our rigorous derivation of wave turbulence equations. The second part of the talk is devoted to the analysis of wave turbulence equations as well as some numerical illustrations. The last part concerns some physical applications of wave turbulence theory. The talk is based on my joint work with Staffilani (MIT), Soffer (Rutgers), Pomeau (ENS Paris), and Walton (PhD student at SMU).

Robust Real-Time Monitoring of High-Dimensional Data Streams

Ruizhi Zhang (University of Nebraska – Lincoln), February 4, 2022

Robust change-point detection for large-scale data streams has many real-world applications in industrial quality control, signal detection, bio surveillance. Unfortunately, it is highly non-trivial to develop efficient schemes due to three challenges: (1) the unknown sparse subset of affected data streams, (2) the unexpected outliers, and (3) computational scalability for real-time monitoring and detection. In this work, we develop a family of efficient real-time robust detection schemes for monitoring large-scale independent data streams. For each data stream, we propose to construct a new local robust detection statistic called Lα-CUSUM statistic that can reduce the effect of outliers by using the Box-Cox transformation of the likelihood function. Then the global scheme will raise an alarm based upon the sum of the shrinkage transformation of these local Lα-CUSUM statistics so as to filter out unaffected data streams. In addition, we propose a new concept called false alarm breakdown point to measure the robustness of online monitoring schemes and propose a worst-case detection efficiency score to measure the detection efficiency when the data contain outliers. We then characterize the breakdown point and the efficiency score of our proposed schemes. Asymptotic analysis and numerical simulations are conducted to illustrate the robustness and efficiency of our proposed schemes.

Complete integrability in elementary geometry

Maxim Arnold (UT Dallas), February 18, 2022

In the past two decades we have seen a spark of interest in discrete integrable systems. Many of such systems are defined by iterating a certain geometric construction, with Schwartz’s pentagram map being the best known example. I will report on a few such elementary geometric constructions and discuss their connections to other known examples. The talk is based on joint work in progress with I. Izmestiev, D. Fuchs, S. Tabachnikov, A. Izosimov, etc.

CR geometry of 3-manifolds

Paul Yang (Princeton U.) , February 25, 2022

I plan to describe the geometric and analytic questions about CR geometry in this dimension. CR manifolds arise typically as the boundary of a strictly pseudoconvex domains in a complex manifold. Closely connected with the embeddability criteria are interesting examples of solvability of the CR version of the Yamabe equation. I also plan to describe global invariants that lead to open problems involving fully-nonlinear pde.

Natural Sciences and Mathematics

Mathematical Sciences

Colloquium AY 21-22

Spring 2022

Fall 2021

Missing and Modified Data in Nonparametric Curve Estimation