Mathematical Sciences

School of Natural Sciences & Mathematics

Colloquium

Fall 2021 

Unless otherwise noted, the colloquia will be held on Fridays @ 11:00 am at JO 4.614.

If you have questions about the colloquium please contact Dr. Vishwanath Ramakrishna.

Archive of talks from previous semesters

Date/Room/Host

Speaker/Institution

Title/Abstract

October 8

Sam Efromovich

UT Dallas

Missing and Modified Data in

Nonparametric Curve Estimation

October 15

in JO 3.516

Baris Coskunuzer

UT Dallas

Geometric Approaches on

Persistent Homology

October 22

Carlos Arreche

UT Dallas

Normal reflection subgroups of complex reflection groups

October 29

Chul Moon

SMU Statistical Sciences

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

November 5

 

Sunyoung Shin

UT Dallas

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

November 12

 

Anton Dzhamay

University of Northern Colorado

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

November 19

in JO 3.516

 

Nathan Williams

UT Dallas

Coxeter Pop-Tsack Torsing

December 3

in JO 3.516

 

Mariusz Urbański

U. North Texas

Thermodinamic Formalism in

Transcendental Dynamics

 

 

Missing and Modified Data in Nonparametric Curve Estimation

Sam Efromovich  (UT Dallas) October 8, 2021

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 probality 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.