# Seminar Series

The mathematics department hosts mathematics seminar speakers.  Both internal and outside speakers are welcome to give one hour talks or a series of talks on topics related to mathematics for the seminars. The level of the talks range from undergraduate level mathematics to presentations on mathematical research by visiting mathematicians and faculty.

If you would like to attend these talks or make a presentation yourself, please contact Dr. Jason Graham.

#### Spring 2024 Seminars:

• Dr. Cheng-Han Pan , Western New England University

Monday, March 18

4:00 - 5:00 p.m., LSC 126

Monstrous Examples in Differentiability

Abstract:  “Monsters! An outrage against common sense. An arrogant distraction, and of little use to the subject.” What made H. Poincaré so tilted? In calculus, we know that a continuous function is not differentiable at those sharp corners. To an extreme, a nowhere-monotone continuous function is a “very rugged” fractal curve. Indeed, K. Weierstrass constructed a nowheremonotone continuous function that is nowhere differentiable, and it clearly did not please H. Poincaré and his French gang. In this talk, we will walk through some historical remarks on monster functions and present K. Ciesielski’s simple construction of a differentiable one, that is, a nowhere-monotone everywhere-differentiable function. We call these paradoxically “rugged” and “smooth” functions differentiable monsters.

#### Spring 2023 Seminars:

• Dr. Kristin Camenga, Juniata College (flyer)

Wednesday, March 29

3:00 - 4:00 p.m., LSC 329

Who are the people in your neighborhood numerical range?

Abstract: The numerical range of a matrix is a set of numbers that results when a matrix is multiplied in a certain way by all unit vectors. Classically, we work with matrices that have complex numbers as entries and we can graph the corresponding numerical range on the complex plane. The numerical range is known to be convex (have no holes or dents) and contains the eigenvalues of the matrix. I first learned about the numerical range at a workshop in 2011, where I started working with some collaborators. In this talk, I will introduce the numerical range and some of the questions that we’ve investigated.  I will also share the story of how our research has unfolded, both the various collaborators who have motivated and enriched the process and the lessons I’ve learned about doing math research.

#### Spring 2017 Seminars:

• Marc Gotliboym, Stevens Institute of Techonology

Exploring Conway's Game of Life on Various Compact Topologies

Tuesday, February 21

3:00 p.m., LSC 133

Conway's Game of Life is a beautiful Turing-complete structure. Generally it is played on a plane; we explore its behavior on various other spaces, noticing sundry configurations which act differently in different spaces. We investigate the results of varying the division of a given compact topology into cells. In particular, on the torus we find some interesting patterns, and focus on determining the evolution of a single line of live cells. We prove the existence of certain patterns, construct and apply various tools from group theory and category theory to help investigate, and pose some deeper conjectures for future exploration.

• Dr. Timur Akhunov, Binghamton University

Hypoellipticity of degenerate Laplacians

Tuesday, February 14

3:00 p.m., LSC 133

Pierre-Simon marquis de Laplace in his 5 volume Traité de mécanique céleste unified mathematical physics by placing calculus front and center. Among Laplace's chief innovations was an introduction of a differential equation that now bears his name. Fast forward to the 20th century. Laplacian is crucial to understand the properties of light, heat, sound and atomic phenomena. Moreover, Einstein has pushed us to understand the Laplacian in the curved world, where some of the mathematicians' favorite toys no longer play as advertised. Come hear the story, where calculus, geometry and physics come together. At the very least, you will find out what the title of the talk means.

#### Fall 2016 Seminars:

• Alex Borselli, Lehigh University

Permutation Groups and CM Fields

Tuesday, September 13

4:00 p.m., LSC 133

The study of CM Fields arose from a generalization of the theory of complex multiplication in elliptic curves. A CM field is a totally imaginary number field K, which is a quadratic extension of a totally real field K0. As totally imaginary number fields, K can be imbedded in ℂ. So, complex multiplication on ℂ induces an automorphism of K. The Galois group of a degree m Galois extension of ℚ is always a transitive permutation group, which is just a subgroup of the symmetric group on m elements, Sm. When looking at Galois groups of CM fields, because of the induced automorphism by complex conjugation, these will be transitive permutation groups with even order centers. In this talk, I will discuss permutation groups and give a basic introduction to CM fields. Then, I will describe some important types of transitive permutation groups, called minimally transitive permutation groups, and their analog under the consideration of even order centers due to their importance with respect to CM fields.

#### Spring 2016 Seminars:

• Dr. Sofya Chepushtanova, Wilkes University

Persistent Homology and Its Alternative Vector Representation

Tuesday, March 15

11:00 a.m. in LSC 329

Many data sets can be viewed as a noisy sampling of an underlying topological space. Topological data analysis aims to understand and exploit this underlying structure for the purpose of knowledge discovery. A fundamental tool of the discipline is persistent homology (PH), which captures underlying data-driven, scale-dependent homological information. A useful representation of this homological information is a PH barcode or persistence diagram (PD). There has been considerable interest in applying PH to the analysis of data in a variety of applications, including image webs, signal analysis, neuroscience, and biological aggregations. One application of PH, characterizing information in hyperspectral movies, i.e., sequences of hyperspectral data cubes evolving in time, is presented for illustration.

The space of PDs can be given a metric structure allowing a given diagram to be used as a statistic for the purpose of comparison against other diagrams. A PD can be converted to a persistence image (PI), stable with respect to small perturbations in the inputs. The PI is a vector representation allowing the application of vector-based machine learning tools. Several machine learning techniques, including support vector machines, are applied to persistence images for classification and feature selection tasks on a synthetic data set. To further illustrate the PI technique, classification is performed on a data set arising from a discrete dynamical system called the linked twist map.

• Dr. David Perkins, Misericordia University

Isaac Newton's Original Discovery of the Generalized Binomial Theorem

Thursday, February 11

4:00 p.m. in LSC 133

Expanding binomials like (a + b)^2 poses no problem for most of us. But until Isaac Newton put his mind to it in 1665, no one knew that binomials like (a + b)^(1/2) or (a + b)^(-3) could expand just as meaningfully. His discovery became a central tool in the development of calculus.

In this presentation, I will not present any modern proofs of his theorem. Instead, I will narrate how Newton describes his own discovery. Only algebra skills and a keen sense of pattern are required in order to follow his path.

#### Fall 2015 Seminars:

• Dr. Ami Radunskaya, Pomona College

DDEs, DCs and Doses: Mathematical approaches to designing cancer vaccines

Sponsored by the University of Scranton Math Club

Monday, November 30

4:30 p.m. in LSC 133

Dendritic cells (DCs) are a promising immunotherapy tool for boosting an individual's immune response to cancer.  In this talk, we develop a mathematical model using differential equations and delay-differenital equations (DDEs) to describe the interactions between dendritic cells, other immune cells and tumor cells.

In order to design an efficient treatment strategy, clinicians need to answer three questions: How much? How often? Who will respond?  Our model, along with mathematical tools from control theory and dynamical systems, can be used to suggest answers to these questions.  This work is just one example of possible collaborations between mathematicians and researchers in other disciplines: a few other examples will be introduced, illustrating the synergy between modeling challenges and mathematical discoveries.

This talk is designed for a general math audience: no knowledge of immunology is assumed.

Biographical Sketch:  A California native, Professor Radunskaya received her Ph.D. in Mathematics from Stanford University under the supervision of Prof. Donald Ornstein.  She is a faculty member of the Department of Mathematics at Pomona College in Claremont, California specializing in ergodic theory, dynamical systems, and applications to various "real-world" problems.  Some current research projects involve mathematical models of cancer immunotherapy, designing time-release tablets, and studying stochastic dynamical systems in order to understand how people balance.  Professor Radunskaya believes strongly in the power of collaboration and that everyone can learn to enjoy mathematics.  She has been a strong supporter of women in mathematics: she will be the next President of the Association for Women in Mathematics, and she has taken on leadership roles in several national mentoring programs.  She is the President of the EDGE Foundation, and co-director of the EDGE (Enhancing Diversity in Graduate Education) program, which won a "Mathematics Program that Makes a Difference" award from the American Mathematics Society in 2007.  Professor Radunskaya was awarded an Irvine Fellowship for Excellence in Faculty Mentoring in 2004, she delivered the Falconer Lecture at MathFest in 2010, and she received a Wig Award for Excellence in Teaching in 2012.

• Mackenzie Wildman, Lehigh University

A Gaussian Markov Alternative to Fractional Brownian Motion for Pricing Financial Derivatives

Tuesday, October 13

1:00 p.m. in LSC 403

Replacing Black-Scholes’ driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the ﬁnancial market. I will discuss the development, testing, and implementation of a simpliﬁed alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes’ underlying process, Brownian motion, with the Dobri´c-Ojeda process. I will also discuss recent developments in applications to the stochastic heat equation. This is joint work with Daniel Conus.

• Sarah Raye Dumnich, Lehigh University

Wavelets and the Dilation Measures

Tuesday, September 29, 2015

1:00 p.m. in LSC 133

Adapting a signal’s resolution allows one to process only the relevant details for a particular task. Burt and Adelson introduced a multiresolution pyramid that can be used to process a low-resolution image first and then selectively increase the resolution when necessary. Mallat and Meyer formalized multiresolution analysis (MRA), which set the groundwork for the construction of orthogonal wavelets. The dilation equation is an equation that arises naturally when using an MRA to construct a wavelet basis. One way to understand the dilation equation is through a measure theoretic approach. By constructing a solution to this dilation equation for measures, we are able to uniquely determine a corresponding wavelet basis.

In this talk, I will talk about what a wavelet basis is, how an MRA relates to a wavelet basis, and give a brief introduction to the dilation equation for measures.

#### Spring 2015 Seminars:

• David Perkins, Misericordia University

(1/2)! = sqrt(pi)/2

Tuesday, May 5, 2015

2:30 p.m. in LSC 133

People have known for centuries that there are n! ways to permute n objects. For example, there are 7! = 7x6x5x4x3x2x1 = 5040 ways to scramble the letters in my last name PERKINS (the cutest being SPINKER). In this context, a number like (1/2)! makes no sense; how could we scramble half of a letter? Swiss mathematician Leonhard Euler is renowned for finding reasons to ask (and then answering) questions like, "What does (1/2)! equal?" In this talk, we will see how he answered this particular question, using the generalized binomial theorem (itself the answer to a puzzling question), integration, and clever pattern-seeking. Suitable for all who have studied Calculus 2.​

• Elizabeth Wright, NSA

Defeating the German Enigma

Tuesday, March 31, 2015

2:30 p.m. in LSC 133

The Enigma was a formidable device used by the Germans during WWII used to encrypt military documents.  In this talk, we will explore the history of the Enigma, and the mathematics involved in breaking its encryption, which saved countless lives during the war.  There may also be an opportunity to see a real Enigma from the period on display.

• Dr. Klaus Volpert, Villanova University

On the Mathematics of Income Inequality

Tuesday, March 3, 2015

2:30 p.m. in LSC 133

The Gini-Index based on Lorenz Curves of income distributions has long been used to measure income inequality in societies. This single-valued index has the advantage of allowing comparisons among countries and within one country over time. However, being a summary measure, it does not distinguish between intersecting Lorenz curves, and may not detect certain societal and economic trends over time.

We will discuss a new two-parameter model for the Lorenz curves, essentially the product of two Pareto distributions. This allows us to split in the Gini-index in two. We will present theoretical and empirical evidence for this model, discuss its mathematical properties, and its potential to discern hidden trends in recent income data for the United States.

Klaus Volpert is associate professor of mathematics at Villanova University. He won the University's Lindback Award for Excellence in Teaching in 2009 and the EPaDel's Crawford Award in 2011. Early studies in his native Germany and the 1989 PhD from the University of Oregon were in pure mathematics (algebraic topology), but he has more recently been interested in problems in applied mathematics, specifically at the intersection with finance and economics.

Outside of mathematics, his greatest joy is making music with his family and friends.

• Dr. Esen Saltürk
Linear Codes and the Counting Problem

Thursday, February 12, 2015

2:30 p.m. in LSC 133

The theory of error-correcting codes began in 1948, with the seminal work of Shan-non. In the early stages of the discipline, codes were deﬁned only over ﬁnite ﬁelds. Since 1994, after the seminal work of Hammons and his collaborates, a complete study of codes over ﬁnite rings began. Within the study of codes, determining the number of codes with given parameters has been one of the most important problems of combi-natorial coding theory. Since 1948, the number of sub-codes of a linear code has been studied. This problem was completely solved for codes over ﬁnite ﬁelds by the well known Gaussian coefficients. Recently, the counting problem over ﬁnite chain rings has been solved. This talk will describe the solution of the counting problem in the study of error-correcting codes over ﬁnite chain rings.

#### Fall 2014 Seminars:

• Dr. Kurt Bryan, Rose-Hulman Institute of Technology

Making Do With Less: The Mathematics of Compressed Sensing

Tuesday, September 16, 2014

2:30 p.m. in LSC 133

Suppose a bag contains 100 marbles, each with mass 10 grams, except for one defective off-mass marble.  Given an accurate electronic balance that can accommodate anywhere from one to 100 marbles at a time, how would you find the defective marble with the fewest number of weighings?  You may well have thought about this kind of problem before and know the answer.  But what if there are two bad marbles, each of unknown mass?  Or three or more?  An efficient scheme isn't so easy to figure out now, is it?  Is there a strategy that's both efficient and generalizable?

The answer is "yes," at least if the number of defective marbles is sufficiently small.

Surprisingly, the procedure involves a strong dose of randomness.  It's a nice example of a new and very active research area called "compressed sensing" (CS), that spans mathematics, signal processing, statistics, and computer science, and has many surprising applications.  In this talk I'll explain the central ideas, which require nothing more than straightforward linear algebra.  I'll then show some applications, including how one can use this to build a high-resolution one-pixel camera!

• Dr. Casey Diekman, New Jersey Institute of Technology

Mathematical Modeling of Circadian Clocks and Binocular Rivalry

Tuesday, October 21, 2014

2:30 p.m. in LSC 133

Rhythms can be found nearly everywhere in biology and are fundamental to brain function. In this talk I will discuss two examples of biological oscillations that operate on very different time scales: daily (circadian) rhythms such as the human sleep/wake cycle, and the alternations in visual perception that occur every few seconds when the two eyes are presented with dissimilar images (binocular rivalry). For each of these phenomena, I will describe how mathematical modeling and dynamical systems theory can be used to make experimentally testable predictions.​​​

• Dr. Yusra Naqvi, Muhlenberg University

Coxeter Groups and Tessellations

Tuesday, November 18, 2014

2:30 p.m. in LSC 133

Coxeter groups arise as symmetries of regular polytopes and as reflections in kaleidoscopic arrangements of mirrors. This gives us a way to understand these groups visually, in terms of tessellations of planes, spheres and hyperbolic disks. In this talk, we will take a look at the connections between the algebra and the geometry of Coxeter groups. These ideas can also be extended to visualize more complicated algebraic structures.​

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