## Fall 2021 – Summer 2022 – Abstracts

Wednesday, September 22, 2021. **Mathematics Colloquium****.**

**Dr. Brandon Hanson, ***Department of Mathematics & Statistics, University of Maine.*

**“Sums and products and combinatorial geometry (oh my!)”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall (with refreshments at 3:00 pm)

Abstract: One of the fundamental themes in number theory is the incompatibility of addition and multiplication. As he did so often, Paul Erdos made a wonderfully simple conjecture which beautifully describes this incompatibility, called the Sum-Product Conjecture. Along with Endre Szemeredi, he proved a first estimate toward the conjecture in 1983. In 1997, Gyorgy Elekes introduced ideas from combinatorial geometry that made short work of the best known estimates for the Sum-Product Conjecture and since then two areas have been intimately connected. I plan on introducing the combinatorial background, surveying the bridges between the two areas, and highlighting some recent developments. The talk should be both leisurely and accessible.

Wednesday, October 20, 2021. **Mathematics Colloquium****.**

**Dr. Tyrone Crisp, ***Department of Mathematics & Statistics, University of Maine.*

**“The algebra of pulling things apart”**

3:15 – 4:15 pm, Room 100, Donald P. Corbett Business Building

Abstract: In a first course on abstract algebra we learn how addition and multiplication of numbers and matrices, composition of functions, and many other “putting together” operations, can be viewed as instances of the abstract notion of binary operations on sets. This talk concerns a less well-known branch of algebra dealing with “pulling apart” operations, such as writing an integer as a sum of smaller integers, or decomposing a set into a disjoint union of subsets. The fact that there is typically more than one way to pull something apart means that the axiomatic study of operations like this is a little more subtle than for the “putting together” operations. In this talk I will introduce an algebraic structure—Hopf algebras—that can be used to study putting-together and pulling-apart operations of many different kinds. I hope that the discussion will be accessible to anybody who knows what a matrix is.

As an example, I will present a Hopf algebra related to symmetries and colorings of finite graphs that was discovered in joint work with former UMaine student Caleb Hill.

Wednesday, October 27, 2021. **Graduate Seminar****.**

**Serge Maalouf, ***Mathematics MA student, University of Maine (advisor: Stechlinski)*

**“Fundamentals of the Calculus of Variations”**

3:15 – 4:05 pm, to be held virtually.

Abstract: One of the early accounts of the calculus of variations was presented by J. Hadamard in the book *Leçons sur le calcul de variations* published in 1910. Earlier works on the topic include the works of Volterra, Pincherle and Bourlet. The main purpose of the calculus of variations is to find extrema of functionals in a similar fashion to what is done in calculus, e.g., by finding the points where the derivative vanishes. A classical example is the Brachistochrone problem, aka Bernoulli’s curves of quickest descent, which consists of finding the parametric planar path that minimizes the free sliding time of a bead. The calculus of variations is the starting point of other theories and principles like the finite element method in mechanics, the Rayleigh-Ritz method and Pontryagin’s maximum principle in optimal control.

In this seminar, I will introduce the main concepts of the calculus of variations. Functionals will be defined first, then the study of their linearity and continuity will follow. Subsequently, the Gâteaux variations and Fréchet derivatives of functionals will be introduced alongside their properties. The necessary condition on the variation of a functional to obtain an extremum will be derived. Furthermore, the Euler-Lagrange multiplier theorem will be presented and proven for optimization problems with constraints. Example applications to functionals defined over the spaces of real valued functions will be shown, e.g., different forms of the Euler-Lagrange equation, and the solution to the Brachistochrone problem will follow from these case studies.

Wednesday, November 3, 2021. **Mathematics Colloquium****.**

**Dr. Yeongseong Jo, ***Department of Mathematics & Statistics, University of Maine.*

**“The Riemann zeta function toward the automorphic L-function”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall

Abstract: The Riemann zeta function is one of special cases of harmonic series that we are familiar with. The history of the Riemann zeta function goes all the way back to Euler. Strikingly, this series at certain points implies interesting phenomena in number theory such as the prime number theory. However, lots of properties have still not been explained. Dirichlet subsequently generalized it to a L-function (L-series) attached to so-called Dirichlet characters. Beside the degree one L-function outlined above, Hecke initiated the study of L-functions associated with certain modular forms possessing a degree two Euler product. It turns out that all these objects are reconciled with what is known as automorphic L-functions. After a tour of global L-functions, I present Rankin-Selberg and Langlands-Shahidi methods, which originated from investigating the Fourier coefficients of automorphic forms and Eisenstein series. This eventually gives birth to non-archimedean local factors that I mainly studied. Toward the end, I show that these L-factors constructed by two different approaches are the same in a few cases. Time permitting, I describe a little bit about archimedean local L-factors given by the product of Gamma functions.

Wednesday, November 10, 2021. **Mathematics Colloquium****.**

**Dr. Peter Stechlinski, ***Department of Mathematics & Statistics, University of Maine.*

**“Nonsmooth Analysis”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall

Abstract: A variety of problems from engineering and the life sciences exhibit a mixture of continuous and discrete behavior. Examples range from pharmaceutical manufacturing and power systems to chronic diseases and social contagions. Such nonsmooth behavior can invalidate standard methods, which typically require derivative information, including popular tools for optimization and equation solving. This talk is concerned with calculating (generalized) derivatives of nondifferentiable functions, with aims of replicating standard tools and theory from the smooth case. After an introduction to this area of work, the discussion will focus on a relatively new approach to this problem that is based on lexicographical ordering (in the spirit of alphabetical ordering). This approach has many attractive properties, including computational relevancy, and is applicable to a wide range of mathematical problems and applications of interest.

Wednesday, November 17, 2021. **Mathematics Colloquium****.**

**Dr. Byungjae Son, ***Department of Mathematics & Statistics, University of Maine.*

**“Semipositone Steady State Reaction Diffusion Equations”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall

Abstract: The study of nonlinear reaction diffusion equations is of great importance in various applications such as nonlinear heat generation, combustion theory, chemical reactor theory and population dynamics. This talk introduces semipositone steady state reaction diffusion equations and discusses the existence, multiplicity and uniqueness of positive solutions for various classes of the reaction term.

Monday, December 6, 2021. **Mathematics Graduate Seminar****.**

**Mackenzie Whittaker, ***Mathematics MA student, University of Maine.*

**“Brun’s Theorem”**

3:00 – 3:50 pm, 108 Jenness Hall

Abstract: Work on famous number theory problems such as Goldbach’s conjecture and the twin prime conjecture often led to other interesting results, as it did for Viggo Brun. In this talk we aim to prove his theorem, his sieve, as well as related results that lead to the creation of both. Other topics discussed will be the prime number theorem, the strong twin prime conjecture, and how they relate to Brun’s work. There will also be a brief introduction into sieve theory, with examples such as the Sieve of Eratosthenes and the inclusion/exclusion principles behind it.

Wednesday, December 8, 2021. **Mathematics Colloquium****.**

**Dr. Brandon Lieberthal, ***Department of Mathematics & Statistics, University of Maine.*

**“Human mobility networks and infectious disease epidemics”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall

Abstract:

In a globalized and highly connected world, predicting the trajectory of disease epidemics, whether vector-borne and direct transmitted, must consider environmental and socioeconomic factors in tandem. Epidemic outbreaks on human metapopulation networks are often driven by a small number of superspreader nodes, which are primarily responsible for spreading the disease throughout the network. Superspreader nodes typically are characterized either by their locations within the network or by their habitat suitability for the disease, described by their reproduction number (R0). In this colloquium I discuss my ongoing research into the prediction and detection of superspreaders, based on natural and socioeconomic factors, and how human decision making drives the spread of disease epidemics. In this context we discuss two case studies: the mosquito-borne Zika virus in Colombia, and the ongoing COVID-19 epidemic in the United States.

Wednesday, March 30, 2022. **Mathematics Colloquium****.**

**Dr. Jack Buttcane, ***Department of Mathematics & Statistics, University of Maine.*

**“A meandering through number theory”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall (with refreshments at 3:00 pm)

Abstract:

The topic of number theory pulls in methods from all areas of modern mathematics. I will try to motivate the use of techniques from algebra and analysis by considering the problem of the largest prime divisor of a polynomial.

Wednesday, April 6, 2022. **Mathematics Colloquium****.**

**Dr. Casey Pinckney, ***Department of Mathematics & Statistics, University of Maine.*

**“Independence Complexes of Finite Groups”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall (with refreshments at 3:00 pm)

Abstract:

Understanding generating sets for finite groups has been explored previously via the generating graph of a group, where vertices are group elements and edges are given by pairs of group elements that generate the group. We generalize this idea by considering minimal generating sets for subgroups of finite groups. These form a simplicial complex, which we call the independence complex, whose vertices are nonidentity group elements and whose faces of size k correspond to minimal generating sets of size k. In this talk, we highlight some of our main results, which are as follows. We provide constructive algorithms and enumerative results for the independence complexes of cyclic groups whose order is a squarefree product of primes, finite abelian groups whose order is a product of powers of distinct primes, and a particular nonabelian class of semidirect products. In the process we introduce a new tool called a combinatorial diagram to construct the independence complexes for this last class of groups. We include visualizations of several independence complexes created by algorithms we have written using GAP and Polymake.

Wednesday, April 13, 2022. **Mathematics Colloquium****.**

**Dr. Franziska Peterson, ***Department of Mathematics & Statistics, University of Maine.*

**“TBA”**

3:15 – 4:15 pm, Hill Auditorium, Barrows Hall (with refreshments at 3:00 pm)

Abstract:

Students need to develop a unique set of data literacy and reasoning skills to work with real-world data, make decisions about constructing evidence to support a claim about their data, and interpret a vast variety of graphs and quantitative information. These skills are interdisciplinary and draw from mathematics, statistics, science practices, language, and technology, to name a few. Students often study these skills separately in their disciplines and fail to make cross-disciplinary connections. Drawing cross-disciplinary connections to reason coherently about data as evidence and interpret the evidence is often overlooked in school curricula. In middle school mathematics, for example, students first focus on the analysis of data (measures of central tendency and spread) and the formulation of statistical questions that anticipate variability. In the middle school sciences, when students are carrying out investigations to gather data to support a claim with evidence, there is no emphasis on questions being ‘statistical’. Here, students need to state a ‘claim’ which can be in form of a declarative statement or a question. Without explicitly drawing a connection to the different terminology between the disciplines, students will most likely fail to realize that they can be referring to the same thing. In this talk, we will explore student data exploration examples, including students’ graph constructions, their reasoning about their evidence to support a claim, and discuss the challenges that students need to overcome.

Tuesday, April 19, 2022. **Number Theory Seminar.**

**Kevin Kwan, ***Columbia University*

**“Moments, Periods & Applications”**

12:00 – 12:50 pm, 421 Neville Hall

Abstract:

We will motivate our talk with an elementary problem, but its solution leads us very naturally to the study of moments of L-functions. We will survey some of the techniques, applications, recent trends, and discuss some of my own work.

Tuesday, April 19, 2022. **Mathematics MA Thesis Defense.**

**Davis MacDonald, ***MA Candidate, University of Maine. (Advisor: Tyrone Crisp)*

**“The Category Of Modules Over Leavitt Path Algebras”**

3:00 – 3:50 pm, 102 Murray Hall

Abstract:

A *Leavitt Path Algebra* is an algebraic object that is constructed from a directed graph, using its vertices and edges. Familiar objects in algebra such as fields, matrix algebras, and algebras of Laurent polynomials arise as Leavitt path algebras, but the definition also encompasses many more exotic objects. Under initial examination, the relations used to generate them seem quite opaque. This talk will show that considering instead the category of modules over these algebras provides clarity to the purpose of these relations.

Monday, May 2, 2022. **Mathematics Graduate Seminar.**

**Mackenzie Whittaker, ***MA Candidate, University of Maine. (Advisor: Jack Buttcane)*

**“The Importance of Primality Testing”**

3:00 – 3:50 pm, 125 Barrows Hall

Abstract:

Large prime numbers are a key component in cryptography. When it comes to generating large primes, it often boils down to testing the primality of some large integer. In this talk we will discuss multiple different primality tests, both deterministic and probabilistic, focusing on the AKS primality test and its interesting properties. Examples to follow all tests discussed.

Tuesday, May 3, 2022. **Mathematics Graduate Seminar.**

**Serge Maalouf, ***MA student, University of Maine. (Advisor: Peter Stechlinski)*

**“Optimal Control Theory: Pontryagin’s Maximum Principle”**

1:00 – 1:50 pm, via videoconference.

Abstract:

Optimal control theory is the extension of the calculus of variations used to find an optimal control for a dynamical system. More specifically, optimal control theory problems consist of finding extrema of functionals subjected to a set of constraints represented by a system of first order ordinary differential equations initial value problems. Optimal control theory problems appear in various fields, e.g., optimization, mathematical economics and classical mechanics to cite only a few. The use of the calculus of variations to solve optimal control theory problems faces limitations and therefore needs to be extended. In this seminar, we will introduce and formally define the classical problem of Bolza in optimal control. Subsequently, we will employ the calculus of variations and show the limitations faced by the variational approaches, after which we will introduce Pontryagin’s maximum principle and use it to solve the double integrator problem.

Thursday, May 5, 2022. **Mathematics Graduate Seminar.**

**Davis MacDonald, ***MA Candidate, University of Maine. (Advisor: Tyrone Crisp)*

**Violating the invariant basis number property**

11:00 – 11:50 am, 106 Neville Hall

Abstract:

A module is a mathematical object defined similarly to a vector space, but with scalars coming from a ring. (A module over a field is simply a vector space.) When the ring of scalars is not a field, a module may exhibit behavior that is very different from what we see in traditional linear algebra. For example, modules may lack the “invariant basis number property” – that is to say, a module can have two bases with different numbers of elements. In this talk we will demonstrate that some modules have this property, as well as construct examples of ways in which it can be violated.

Tuesday, May 10, 2022. **Mathematics Graduate Seminar.**

**Abram Karam, ***MA student, University of Maine. (Advisor: Peter Stechlinski)*

**Generalized derivatives of nonsmooth functions, with an application to eigenvalues of a symmetric matrix**

1:00 – 1:50 pm, via videoconference

Abstract:

Derivative information is a fundamental tool used in equation solving, differential equations, and optimization. However, many real-world problems, such as those found in engineering, exhibit nonsmoothness in the form of discrete events, which often means that derivative information is unavailable or inaccurate. This necessitates a robust theory of generalized derivatives for nonsmooth functions. Generalized derivatives theory forms the first portion of the talk. The second portion of the talk focuses on the application of the theory of generalized derivatives to obtain sensitivity information for eigenvalues of a symmetric matrix. That is, as a matrix of parameters varies, how do the eigenvalues change? The theory can be implemented on a computer in an accurate and automatable way, making it ideal for future generalization and application to nonsmooth problems.

Monday, August 15, 2022. **Mathematics Graduate Seminar.**

**Abram Karam, ***MA candidate, University of Maine. (Advisor: Peter Stechlinski)*

**Second-order derivatives of eigenvalues of a symmetric matrix**

12:00 – 12:50 pm, via videoconference

Abstract:

In our first seminar talk, we discussed a generalization of classical derivatives theory to the setting of nonsmooth functions, allowing us to better model and solve problems which are inherently nonsmooth (such as those found in engineering). We then investigated how this theory of generalized derivatives can be applied to obtain sensitivity information for eigenvalues of a symmetric matrix. In this second talk, we consider second-order (directional) derivatives of eigenvalues of a symmetric matrix. Unlike first-order directional derivatives, second-order directional derivatives have several possible definitions as found in the literature, with varying degrees of applicability or utility. We discuss several definitions, and provide relevant results from the theory of convex functions to motivate their utility to eigenvalue problems. Finally, we state and prove several methods to obtain first or second-order expansions to see how the eigenvalues change as one varies the parameters of a matrix. We conclude the talk with compelling possibilities/motivation for future work.