## Fall 2022 – Summer 2023 – Abstracts

Thursday, August 3, 1:00 – 1:50pm. **Mathematics MA Thesis Defense**

**Miles Chasek**, *University of Maine Mathematics MA student *(advisor: Knightly)

**“A vector-valued trace formula for finite groups”**

Room 104 Jenness Hall.

Abstract: It is well-known that the trace of a square matrix is, on the one hand, the sum of its diagonal entries, and on the other, the sum of its eigenvalues. This equality is perhaps the most basic example of a trace formula, which is the expression of a linear operator’s trace in two different ways. In 1956, Selberg developed a general trace formula that has far-reaching implications in number theory. In this talk, we present a finite group analog of the Selberg trace formula, using only basic results from linear algebra and representation theory. We consider operators in the representation of a finite group *G* induced from a finite-dimensional representation *(τ,W)* of a subgroup *Γ*. For simplicity we will focus on the case where *(τ,W)* is the trivial representation of *Γ*. Some well-known results in representation theory fall out as consequences.

Wednesday, May 10, 2023. **Graduate Seminar.**

**Cameron Morin**, *University of Maine Mathematics MA student* (advisor: Stechlinski).

**“Formulating a nonsmooth epidemic model with imitation dynamics”**

1:00 – 1:50 pm, 421 Neville Hall

Abstract: Epidemic modeling has been used to predict the progression of an infectious disease through a population. Using systems of ordinary differential equations (ODEs), these models are an important tool in strategizing pharmaceutical and non-pharmaceutical medical interventions. However, many epidemic models do not take into account human decision making, which likely plays a crucial role in disease spread. Another important aspect of disease spread that is not accounted for is that of medical resources. Many models tend to assume there are no constraints on medical resources. In this talk, we will discuss background on both epidemiology and game theory before exploring a new epidemic model that takes these factors into account.

Friday, May 5, 2023. **Graduate Seminar.**

**Miles Chasek**, *University of Maine Mathematics MA student* (advisor: Knightly).

**“Pontriagin duality and Fourier inversion”**

3:00 – 3:50 pm, 108 Jenness Hall

Abstract: In this talk, we will examine how Pontriagin duality allows us to extend the Fourier inversion theorem to the more general setting of complex-valued functions on locally compact abelian groups. We define the notion of a locally compact abelian group and its dual group. Then, we will state the Pontriagin duality theorem, define the general Fourier transform, and use it to state the general Fourier inversion theorem. Lots of examples will be given along the way.

Thursday, April 13, 2023. **Graduate Seminar.**

**Jacob Callas**, *University of Maine Mathematics MA student (advisor: David Bradley).*

“**The Saddle Point Method and an Application to Bell Numbers**”

3:00 – 3:50 pm, Jenness 102

Wednesday, April 12, 2023. **Mathematics Colloquium****.**

**Dr. Giovanna Guidoboni, ***University of Maine Engineering.*

**“Mathematical and computational properties of differential equations for fluid flows in deformable domains”**

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

Abstract:

This talk focuses on differential problems describing the flow of a viscous fluid in deformable domains. Such problems include flow in compliant tubes, often adopted for the modeling of arterial blood flow, and flow through deformable porous media, often adopted for the modeling of tissue perfusion. The mixed hyperbolic-parabolic-elliptic nature of these systems guides the study of their well-posedness and the design of efficient numerical methods for their approximate solution. Theoretical and numerical results will be presented, along with their impact on real-world applications. In addition, the need of capturing the simultaneous effects of local and nonlocal factors in human cardiovascular physiology requires the coupling between systems of partial differential equations (PDEs), such as those described above, with systems of ordinary differential equations (ODEs) describing the blood flow in the systemic circulation. Open questions and recent results concerning the PDE/ODE coupling in fluid flow applications will be discussed from the theoretical and computational viewpoints.

Wednesday, April 5, 2023. **Mathematics Colloquium****.**

**Dr. Evan Randles, ***Colby College.*

**“The large-time behavior of heat kernels”**

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

Abstract:

One can approach the spectral theory of an elliptic partial differential operator by first studying the behavior of its heat kernel in small time. Correspondingly, much literature has been devoted to the study of heat kernel estimates (and asymptotics) which are useful (or meaningful) only in small time. The corresponding large-time behavior is much less well understood. In this talk, I will motivate the study of the large-time behavior of heat kernels by connecting it to the classical problem in Fourier analysis of understanding the behavior of convolution powers of signed measures. I will then discuss a theory of large-time asymptotics for the heat kernels of higher order partial differential operators with constant coefficients. This theory will focus first on the class of so-called homogeneous operators, where it is relatively easy, and then on a class of inhomogeneous operators where the analysis is more delicate and the results are surprising. This talk is based on recent joint work with Laurent Saloff-Coste.

Wednesday, March 22, 2023. **Mathematics Colloquium****.**

**Chris Stith, ***University of Maine Mathematics.*

**Trapped Surface Formation in General Relativity**

3:15 – 4:15 pm, Room 104, Jenness Hall

Abstract:

Trapped surfaces are a central topic of study in mathematical general relativity. Penrose’s incompleteness theorem (1965) ties the concept of a trapped surface to the study of incompleteness and singularity formation in GR; this theorem cemented its importance in the mathematical GR community. In this talk, we will introduce the concept of a trapped surface as a mathematical structure involved in the study of black holes. In a 500 page manuscript, Christodoulou (2009) provided the first proof of trapped surface formation without any symmetry assumptions; several recent papers have simplified this result and extended it to non-vacuum spacetimes.

Wednesday, March 8, 2023. **Mathematics Colloquium****.**

**Dr. Gil Moss, ***University of Maine Mathematics.*

**“Fermat’s last theorem, L-functions, and the Langlands program”**

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

Abstract:

In this talk we will survey some of the tools and techniques used in the proof of Fermat’s last theorem, including elliptic curves, L-functions, and modular forms. Toward the end, we will describe how the surprising connections between these objects are just a single piece of a global conspiracy called the Langlands program.

Wednesday, February 22, 2023. **Mathematics Colloquium****.**

**Dr. Jane Wang, ***University of Maine Mathematics.*

**“The Illumination Problem”**

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

Abstract:

The illumination problem asks the following: given a room with mirrors for walls, can we always place a single point light source somewhere in the room so that the whole room is illuminated? In this talk, we will survey various solutions to this problem given different constraints on the rooms (e.g. polygonal or not). In doing so, we will draw lots of pictures, consider many strange-looking rooms, and see how modern research at the intersection of geometry and dynamics can help answer this classical question.

Friday, December 16, 2022. **Graduate Seminar****.**

**Crockett Lalor, ***University of Maine Mathematics MA student (advisor: Neel Patel).*

**“ SHOCKing Results in Traffic: Roadblocks and Fan Shocks“**

3:00 – 3:50 pm, on Zoom

Abstract:

In this talk we will continue our investigation of traffic flow with a few more interesting examples motivated by real life events.

FEATURING:

-My mom driving back from the Berkshires

-Strange and inexplicable cop behavior

We will solve some differential equations, but also a REAL LIFE MYSTERY — the resolution to which explicates an important philosophical principle. We will improve on our previous model by extrapolating from driver behavior in a car-following system of ordinary differential equations to aggregate traffic behavior, and compare and contrast outcomes. Is the crazy way people drive in New York City in fact enlightened? You will be SHOCKED by our answer to this question and many more. The joke is that we will be looking at the way shocks propagate through traffic flow — where do they come from? Where do they go? While we will not find the answers to all of life’s persistent questions, we will provide (somewhat) definitive answers to one or two. Attend this Zoom talk from the comfort of your own home, office, or car if you’re feeling festive!

Wednesday, December 7, 2022. **Mathematics Colloquium****.**

**Dr. Brandon Hanson, ***University of Maine Mathematics.*

**“The unreasonable effectiveness of polynomials in the mathematical sciences”**

3:15 – 4:15 pm, Room 101, Neville Hall (with refreshments at 3:00 pm)

Abstract:

Polynomials are ubiquitous objects in math, and provide a fantastic playground for testing hypotheses and formulating conjectures. They are the nicest functions we know of and have all sorts of regularity properties coming from different mathematical perspectives — analysis, geometry, algebra, number theory, combinatorics, etc. These regularity properties make polynomials a fantastic device for tackling problems which at first glance have nothing at all to do with polynomials — an approach sometimes called the polynomial method. In this, sure to be fantastic talk, I will discuss about the origins of the polynomial method and sketch how it is used to solve a number of interesting questions in various disciplines.

Friday, December 2, 2022. **Graduate Seminar****.**

**Crockett Lalor, ***University of Maine Mathematics MA student (advisor: Neel Patel).*

**“Why are all these cars parked on 495? Or, on the origin of road rage”**

3:00 – 3:50 pm, 104 Jenness Hall

Abstract:

In this talk, we’ll delve into the mysterious behavior of people in cars driving down the road. Why are they doing that strange thing? What is wrong with all these people? Through the power of partial differential equations we will find order in chaos, make sense of nonsense, understand the incomprehensible. By squinting our eyes, we change the discrete into the continuous and study the density of traffic in several situations familiar to the everyday motorist.

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

**Dr. Tim Browning, ***IST Austria.*

**“Polynomials over ℤ and ℚ: counting and freeness”**

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

Abstract:

Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk I’ll discuss recent efforts to understand the frequency of integer solutions to cubic polynomials, before turning to the much more evolved picture over the rational numbers.

About the speaker:

Professor Browning is a number theorist at the Institute of Science and Technology, Austria. He studies the integers using a mixture of analytic, geometric, and algebraic methods. He has been featured in several Numberphile videos:

Wednesday, November 16, 2022. **Mathematics Colloquium****.**

**Dr. Aden Forrow, ***University of Maine Mathematics.*

**“An Introduction to Likelihood-free Inference”**

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

Abstract:

A key challenge in modern statistical inference is developing algorithms to handle increasingly complex scientific data. For models in contexts from cosmology to psychology and neuroscience, standard analytical tools hit unsurmountable computational obstacles. Over the past two decades, a wide range of algorithms have been proposed for learning parameters of these models in computationally feasible ways, often under the heading of approximate Bayesian computation or likelihood-free inference. In this talk, I will present the core likelihood-free inference problem, building up by way of traditional maximum likelihood estimation and likelihood-based Bayesian inference with Markov chain Monte Carlo. We will investigate the surprisingly challenging question of determining how well a likelihood-free algorithm performs, finding at the end that appropriately evaluating performance can yield significant gains in computational efficiency.

Wednesday, November 9, 2022. **Mathematics Colloquium****.**

**Dr. Benjamin Weiss, ***Unum.*

**“Valuing Insurance Companies or How I learned to Stop Worrying and Love Quartic Splines”**

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

Abstract:

A recent change in accounting law has added a lot of new work for insurance companies to value their assets and liabilities for SEC reporting. We’ll give an introduction to financial valuations, and a brief history of the laws. This will lead us to needing to connect dots nicely (using linear algebra, and numerically stable algorithms). I expect interested undergraduates who have taken linear algebra would be able to follow the talk; I expect most faculty to know all the math, but be surprised at some of the history and applications.

Dr. Weiss works on the actuarial team at Unum, the country’s leading employer benefits insurer, in Portland. Unum employs many UMaine alumni (math majors, business majors, and many others too!). He would be happy to speak with students before and after the talk about employment opportunities.

Monday, November 7, 2022. **Algebra Seminar****.**

**Victor Carmona, ***University of Seville.*

**“Stabilization of algebras and functor calculus”**

3:00 – 3:50 pm, 102 Jenness Hall

Abstract: From basic algebraic geometry, we know that infinitesimal neighborhoods and tangential data are controlled by nilpotent rings. In this talk, we will explain how this phenomenon is pretty general by explaining that it works for operadic algebras. Also, we will discuss what is the connection with Goodwillie calculus of functors.

Wednesday, October 26, 2022. **Mathematics Colloquium****.**

**Elena Salguero, ***University of Seville, Spain.*

**“The Dynamics of a Stokes Flow”**

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

Abstract:

The Navier-Stokes equations are partial differential equations describing the dynamics of a viscous fluid. They have proved to be very useful for their accuracy in modeling a large number of scientific phenomena: ocean currents, motion of stars, blood flow, etc. Their mathematical analysis is also of great importance, as the behavior of smooth solutions in three dimensions is still unknown. This question constitutes one of the seven most important open problems in mathematics. Despite the complexity of NS equations, we can study a wide range of related problems making certain assumptions over the flow. In particular, we will discuss the dynamics of two viscous fluids whose particles have negligible acceleration compared to viscous forces: the so-called Stokes or creeping flow regime.

Monday, October 17, 2022. **Number Theory Seminar****.**

**Petar Bakic, ***University of Utah.*

**“Theta correspondence and Arthur packets: the Adams conjecture”**

3:00 – 3:50 pm, 102 Jenness Hall

Abstract: In his 1989 paper, Adams conjectured that the theta correspondence should exhibit functorial behavior with respect to Arthur packets. The conjecture is known to be true when the rank of the target group is sufficiently large. However, it is also known to fail in a number of small-rank examples. Describing all the situations in which the conjecture holds thus becomes an interesting problem.

The talk will begin with a brief overview of the main ingredients: the local theta correspondence and local Arthur packets. After formulating the conjecture, I will present new results which provide an answer to the above question. This is joint work with Marcela Hanzer.