Mathematics

9/9/2025

Title: "The Banach Fixed Point Theorem"

Abstract: The Banach Fixed Point Theorem will be presented together with a proof, its history and some applications.

9/2/2025

Title: "Logistic Regression is All You Need."

Abstract: Even though a database contains millions of reports, medical incidents of adverse reactions for example, detecting previously undiscovered associations is often compared to ``finding needles in a haystack.'' For many years such detections have been an important research area in pharmacovigilance. The earlier investigation by DuMouchel proposed the use of empirical Bayes to adjust signal levels, providing elegant solutions to the problem of signal detections. However, it has been known for significant biases due to ``hidden confounding.'' An application of logistic regression model for drug-event combinations has been suggested by Hauben et al., as a promising approach. Genkin et al., devised algorithms to find the maximum a posteriori (MAP) estimates. However, due to a large dimension it has not been practical to analyze the characteristics of posterior distribution such as percentiles for credible intervals by Monte Carlo methodology. The talk communicates the following ideas:

1. A logistic regression can be viewed as a simple encoder model consisting of bijective embedding of one-hot representations and a single layer transformer.

2. Langevin algorithm uses stochastic differential equations (SDE) for Monte Carlo simulations. We present a recent development of the construction of strong stationary time and the exact sampling algorithm using Langevin diffusions.

8/26/2025

Title: "The Tradition of the Tennessee Tech Mathematics Department"

Abstract: As times and values change, it's always important to remember the culture and history that we came from. It's from this that sets the positive trajectory for our future and determines our forward-thinking goals. The objective of this talk is to get all of us, past and present, to reflect upon the positive impact that the Tennessee Tech Mathematics Department has had on our heritage. The end result being a well-balanced perspective that provides us all with the motivation required to achieve our desired goals. 

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.