Mathematics

3/12/2025

Title: "Interpreting the structure of a quadratic regular algebra of global dimension three using its point scheme"

Abstract:  Over the last few decades, work in quantum physics has given rise to algebraic structures which can be viewed as noncommutative analogs of polynomial rings. These algebraic structures are algebras known as Artin-Schelter regular algebras. Classifying and understanding these algebras using modified methods from commutative algebraic geometry is an active area of research in noncommutative algebra. One particular example of such algebras emerged from Poisson geometry as a generalization of a deformation on a Poisson bracket in complex projective three-space. The result of this generalization was a family of AS-regular algebras denoted as R(n,a) for n<=1 and are henceforth referred to as the LS-algebras. In this thesis, we compute the point scheme of the LS-algebra of global dimension three, R(2,a). We then use this scheme to classify the algebra and demonstrate that R(2,a) can be expressed as a twist by a graded automorphism of an Ore extension of the polynomial ring on two variables.  

3/5/2025

Title: Spectral comparison results on metric and discrete graphs

Abstract: In recent years, various authors have derived spectral comparison results in different settings including Euclidean domains and metric graphs. In this talk we review the results on metric graphs and provide a generalization to the discrete setting. We also discuss possible extensions to infinite graphs and applications in inverse spectral theory. Along the way, we study so-called local Weyl laws which are of independent interest. This talk is based on joint work with P. Bifulco (Hagen) and C. Rose (Potsdam).

2/26/2025

Title: The (metric) Space of Fractals

Abstract. In this talk we present the definition of the metric space of fractals and the Hausdorff metric. We should see how to obtain the Sierpinski triangle as the limit of a certain sequence constructed via an iterated function system. The existence of the limit of that sequence is guaranteed by the Banach Contraction Principle. The talk is based on the book “Fractals Everywhere” by Michael Barnsley.

2/12/2025

Title: Improved small sample inference for the Generalized Pareto distribution through a Monte Carlo adjustment to the signed root of the log-likelihood ratio statistic.

Abstract: Estimation methods for the Generalized Pareto distribution have been well studied.  While the maximum likelihood method may yield adequate results for cases in which the shape parameter falls between -.5 to 5 and large sample sizes, very little has been done for small sample sizes.  Small sample sizes can occur when fitting the exceedances over a threshold.  We study an adjustment which centers and scales the signed root of the log-likelihood ratio statistic.  The Monte Carlo adjustment is easily derived and does not require complex calculations that are often required to condition on ancillary statistics.  One-sided inference and confidence intervals for shape, scale, and m-year return levels are compared to profile likelihood and delta methods.  Considerable improvement is shown for small samples and demonstrated with examples.

1/22/2025

Title: Coalescing solutions to Langevin diffusions and their applications

Abstract: We can construct the inverse image of a strong solution to Langevin diffusion as a random boundary, and introduce a coalescing solution to the same diffusion by running backward and reflecting it at the random boundary.  It allows us to propose a set-valued intertwining dual of the Langevin diffusion and formulate a Lambda-linked coupling which extends Pitman's 2M-X theorem for multi-dimensional Langevin diffusions.  In particular we use the Lambda-linked coupling for Monte Carlo applications and examine an adaptive stopping mechanism when the drift coefficient is monotone.

11/20/2024

**11 a.m.**

Title: Manin's Universal Quantum Groups under 2-cocycle Twists

Abstract: We examine 2-cocycle twists of a family of infinite-dimensional Hopf algebras, known as Manin’s universal quantum groups, denoted by aut(A), which Manin showed, universally coact on connected graded quadratic algebras, A. In this talk, we consider aut(A) under a more general setting, namely, when A is a finitely generated algebra subject to m- homogeneous relations and show how aut(A) can be twisted by 2-cocycles. This is joint work with H. Huang, V. C. Nguyen, C. Ure, K. B. Vashaw, and X. Wang under an AIM SQuaRE grant.

11/13/2024

**11 a.m.**

Title: Labeled Graph C*-Algebras and Desingularization
Abstract: Given a directed graph E and a labeling L, one forms the la-
beled graph C*-algebra by taking a weakly left-resolving labeled space
(E;L; B) and considering a universal generating family of partial isome-
tries and projections. In this talk we will introduce concepts in the area
of labeled graph C*-algebras and define what is meant by singularities
in a labeled graph. In addition, we will discuss some ideas on how
to define a process in which one can build a "larger" desingularized
labeled space.

Title: Measuring Diversity in Classifier Ensembles

Abstract: With the advent of classifier ensembles came an interesting observation: when the individual predictors of a classifier ensemble make differing predictions, the ensemble’s performance may increase.  Many researchers understand the induction of diversity in these ensembles on an intuitive level.  However, there is no universally accepted measure for the diversity of a set of classifiers.  In this talk, we’ll discuss a few proposed measures of classifier diversity and look at a few experiments to illustrate the relationship between these measures and ensemble performance.  

Title: An introduction to finite element methods

Abstract: The finite element method (FEM) is a numerical approximation technique that is used extensively for engineering to approximate solutions of partial differential equations.  In this presentation, the FEM will be introduced and applied to a simple model problem.  Some details about the derivation, implementation, and convergence will be discussed.

An Introduction to Spatial Statistics and Kriging

Abstract: From weather forecasting to mining, geographical information systems (GIS) have become one of the mainstays of spatial prediction. This talk will introduce the underlying “engine”, called kriging, of GIS prediction, from the basic spatial model formulation to some of the more exotic multivariate spatial-temporal models, and present relevant examples.  

Title: Introduction to Calculus of Variations

Abstract: Calculus of Variations concern with finding the optimal solution to functionals of certain forms.  In this talk we will visit several famous classical problems, some have demonstrations to help analyzing the best solution.

Title: Bose-Einstein condensation of an ideal gas

Abstract: A Bose-Einstein condensate is a state of matter that arises, under certain conditions, in a gas consisting of bosons and represents an exotic quantum phenomenon. It was theoretically predicted by Bose and Einstein in 1924 but was long regarded as a mathematical curiosity with no practical applications. However, since the experimental observation of such a condensate in 1995, Bose-Einstein condensation has become a field of significant research interest. Although the phenomenon is thought to be well understood from a physical perspective, its mathematically rigorous description remains incomplete. In this talk, we discuss a mathematically precise treatment of Bose-Einstein condensation in the simplified case of an ideal Bose gas confined to a box.

Title: Introduction to Deep Reinforcement Learning

Abstract:Deep Reinforcement Learning (DRL) is an exciting field that merges reinforcement learning with deep neural networks, enabling agents to learn through dynamic interactions with their environment. Unlike traditional methods that rely on static datasets, DRL allows agents to adapt and maximize rewards over time. This seminar will cover the fundamental concepts of DRL, including agents, actions, rewards, and policies. We’ll also explore case studies, like DRL’s success in mastering Atari games, highlighting how these methods drive advancements in fields such as robotics, game AI, and autonomous systems.

Title:  On some Banach Function Spaces

Abstract:  In the first part of the talk, I will present a short introduction to the theory of Banach function spaces. The second part will be devoted to Cesaro spaces and their non-trivial generalization to spaces on general measure spaces. 

Title: Graduate students’ role in the Emporium and SI leaders

Abstract: A talk will overview the graduate students’ role in the Emporium and SI leaders. Q&A session will follow the presentation.

Title: A complete characterization of monotonicity equivalence for continuous-time Markov processes

Abstract: A research article and abstract titled above can be found at https://arxiv.org/abs/2408.10840.

Instead we cover background materials, starting from Markov chains and continuous-time Markov processes on discrete state spaces. Then we introduce two notions of monotonicity for them, and discuss why we care about them in the context of exact Markov chain Monte Carlo (MCMC) sampling algorithms.