## Fall 2023 – Summer 2024 – Abstracts

Friday, April 19, 2024. **Research Seminar.
**

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

**Sumsets, covering lemmas, and probability**

3:00 – 4:00pm, 421 Neville Hall

Abstract: The fundamental object of additive combinatorics is the sumset A+B={a+b:a∈ A,b∈ B} where, say, A and B are finite subsets of an abelian group. One of the basic questions is to learn about the nature of A from information about A+A, or alternatively to learn about other quantities like A+A+A. One of the fundamental tools (ubiquitous throughout math) is a covering lemma, which covers A by a set of translates of some nicer set B. Covering lemmas are often measure theoretic in nature and this leads to probabilistic analysis. With this, I will try to motivate the Lovasz Local Lemma from probability.

Friday, April 12, 2024. **Research Seminar.
**

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

From modular forms to automorphic representations

From modular forms to automorphic representations

*3:00 – 4:00pm, 421 Neville Hall*

Abstract: This is an expository talk that is supposed to provide some motivation for studying modular forms, and then outline the generalization of modular forms to automorphic forms, with maybe some discussion of L-functions. The goal is to use this global framework to motivate studying the local representation theory, perhaps even for its own sake.

Wednesday, April 10, 2024. **Mathematics Colloquium.
**

**Dr. Rose Morris-Wright**, Middlebury College

Hyperbolic geometry in group theory

Hyperbolic geometry in group theory

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

Abstract: Groups are abstract algebraic constructions closely related to the idea of symmetry. Geometric group theory is the subject of active research in mathematics that uses tools from graph theory, hyperbolic geometry and topology to study infinite groups. In this talk we’ll look at some examples of groups and their corresponding geometric spaces, and see how hyperbolic geometry might shed light on the properties of these groups. No prior knowledge of group theory or hyperbolic geometry will be assumed.

Friday, April 5, 2024. **Research Seminar.
**

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

Cutting sequences and continued fractions

Cutting sequences and continued fractions

*3:00 – 4:00pm, 421 Neville Hall*

Abstract: Consider a square grid in the plane, with vertical grid lines labeled by “a” and horizontal grid lines labeled by “b”. If L is any straight line, the sequence of vertical and horizontal lines that it crosses gives a bi-infinite sequences of a’s and b’s (e.g. …abbabbba…) called the cutting sequence of L. A natural question is then: what are the possible cutting sequences of lines in the plane? This question can also be asked about trajectories on other tilings. For example, we could seek to understand the cutting sequences of hyperbolic lines on the hyperbolic plane tiled by ideal triangles.

In this talk, we will survey some known results about cutting sequences. We will see that in both of these above circumstances, the cutting sequence of a Euclidean/hyperbolic line is closely related to the continued fraction expansion of the slope/endpoints of the line.

Wednesday, March 27, 2024. **Mathematics Colloquium.
**

**Dr. Robert Franzosa**, University of Maine

I Saw Total Eclipses of the Sun on February 26, 1979, and on February 26, 1998. Was That a Coincidence?

I Saw Total Eclipses of the Sun on February 26, 1979, and on February 26, 1998. Was That a Coincidence?

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

Abstract: In this talk I will answer this question, examining the notion of eclipse cycles. Along the way I’ll introduce the elements of the moon’s orbit around the Earth that are significant in the occurrences of eclipses, as well as the mathematics of continued fractions estimates, a tool that assists in identifying eclipse cycles.

Wednesday, March 20, 2024. **Mathematics Colloquium.
**

**Dr. Karol Bacik**, MIT

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

Is random walk a good model of walking?

Is random walk a good model of walking?

Abstract: Chaotic motion of an aimless (and possibly intoxicated) pedestrian is often used as an anecdotal motivation for the mathematical definition of a random walk. However, can we take this analogy at face value and use the theory of stochastic process to describe human locomotion, all jokes aside? In this talk, I will introduce a new kinetic theory for active flows to show that the motion of pedestrians crossing a busy concourse can be indeed understood as coupled random walks. I will focus in particular on the phenomenon of lane formation, where two groups moving in opposite directions (e.g. on a crosswalk) spontaneously segregate into contraflowing lanes. The new kinetic theory gives insight into the physical origin of lane formation and predicts some new dynamical phenomena, such as tilted and curved lanes. These new predictions have been confirmed in controlled experiments with human participants, which I will also discuss in my talk. Pretty videos of people wearing little paper hats will be provided.

Wednesday, March 6, 2024. **Mathematics Colloquium.
**

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

Single-cell trajectory inference

Single-cell trajectory inference

*3:15 – 4:15pm, Neville 101*

Abstract: Modern experimental biology generates vast amounts of data on expression of genes at single-cell resolution. Translating that data into knowledge of the underlying dynamic processes requires extensive mathematical modeling and statistical analysis. We will introduce this trajectory inference problem, beginning with the biological background and then working our way through the mathematics from simple ordinary differential equations up to open questions on the minimal information required to reconstruct a stochastic differential equation.

Friday, March 1, 2024. **Research Seminar.
**

**Dr. Aden Forrow**,

*University of Maine*

**Causality, kernels, and optimal transport**

*3:00 – 4:00pm, 421 Neville Hall*

Abstract: Following on our colloquium on causal inference in practice, I will discuss causal inference in theory, particularly Kallus’s conception of generalized optimal matching. We’ll see propensity scores, Hilbert spaces, convex optimization, and bias-variance tradeoffs, pondering along the way how to balance mathematical elegance, statistical efficiency, and practical constraints.

Wednesday, February 2028, 2024. **Mathematics Colloquium.
**

**Lauren Forrow**, Mathematica

A Perfect Match: Designing rigorous causal studies of public policies

A Perfect Match: Designing rigorous causal studies of public policies

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

Abstract: Do physician incentive programs reduce health care costs? Does on-the-job training help adults with disabilities find and retain jobs? Does a new reading curriculum improve proficiency for English learners? All of these questions have important consequences for public policy; all of them also require a specific set of statistical techniques designed to disentangle causal links from the messy, complex web of the real world. This talk will briefly sketch the premises of causal inference, the statistical discipline that addresses causal questions, before focusing in more depth on the design phase of an observational (unrandomized) study, specifically techniques for selecting matched comparison groups. The talk will conclude with an application to the recent evaluation of the Medicare Care Choices Model, a program that offers eligible Medicare beneficiaries the option of receiving a mixture of palliative and curative health care services.

Friday, February 23, 2024. **Research Seminar.
**

**Dr. Neel Patel**,

*University of Maine*

**Bifurcation Points and Applications to PDE**

*3:00 – 4:00pm, 421 Neville Hall*

Abstract: Bifurcation theorems are a consequence of the implicit function theorem and are tools for finding non trivial steady states of differential equations. I will present the Crandall-Rabinowitz bifurcation theorem from their 1971 paper and show how it can be used to find equilibrium solutions for fluid boundaries with surface tension.

Wednesday, February 21, 2024. **Mathematics Colloquium.
**

**Mathematics Faculty**, University of Maine

Teaching workshops

Teaching workshops

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

Abstract: Four of our mathematics professors will present efforts they have been involved in to improve mathematics teaching leading up to university.

Wednesday, February 14, 2024. **Mathematics Colloquium.
**

**Dr. Julian Rosen**, Upstart Network

**Assembling weak learners into strong ones**

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

Abstract: Boosting is a machine learning technique for building a high-accuracy model from a series of lower-accuracy sub-models. Each sub-model is designed make up for some of the shortcomings of the previous ones. A specific form of this technique, gradient boosting, is effective in a wide range of situations. In this talk, I explain the math behind a widely-used algorithm called Extreme Gradient Boosting (XGBoost). I will also talk about some of ways we use this algorithm in my work at Upstart.

Wednesday, February 7, 2024. **Mathematics Colloquium.
**

**Dr. Salimeh Sekeh**, University of Maine

**Can AI Models Be Highly Performant, Efficient, And Robust Simultaneously?**

*3:30 – 4:30pm, Hill Auditorium, Barrows Hall*

Abstract: Performance is not enough when it comes to AI learning-enabled systems including deep neural networks (DNNs); in real-world settings, computational load or efficiency during training and adversarial security are just as or even more important. Often there are critical trade-offs to consider when prioritizing one goal over the others. In this work, we propose to concurrently target Performance, Efficiency, and Robustness, and ask just how far we can push the envelope on simultaneously achieving these goals. To this end, our research advances two lines of approaches: (1) Continual learning (CL) and knowledge transfer, we propose a new probabilistic framework to monitor information flow through layers in DNNs for sequential tasks and CL techniques and its impact on learning performance. (2) Data summarization, our algorithm follows the intuition that samples that are highly susceptible to noise strongly affect the decision boundaries learned by DNNs, which in turn degrades their performance and adversarial robustness. By identifying and removing such samples, we demonstrate increased performance and adversarial robustness while using only a subset of the training data, thereby improving the training efficiency. Through our experiments, we highlight our models’ high performance across multiple Dataset-DNN combinations and provide insights into the complementary behavior of our methods alongside existing adversarial training approaches to increase robustness and efficiency of AI learning systems.

Wednesday, January 31, 2024. **Mathematics Colloquium.
**

**Dr. Sophia Bragdon**,

*ERDC Cold Regions Research and Engineering Laboratory*

**Leveraging AI and Mathematics to improve buried object detection**

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

Abstract: Automatic target recognition algorithms have broad applications in military applications and non-governmental applications. In this talk, I will introduce some mathematical tools (deep learning, physics-informed neural networks, Bayesian classifiers, and signal processing) that are used within different facets of developing automatic target recognition (ATR) algorithms that leverage mathematics to improve the outcome. Specifically, I will discuss research that has been conducted at the U.S. Army Engineer Research and Development Center, Cold Regions Research and Engineering Laboratory (CRREL) in Hanover, NH, which focuses on the development of ATR algorithms for detection and identification of buried objects using thermal images. After introducing the mathematical tools, I will discuss how these tools are used in the different steps of the ATR algorithm developed at CRREL. I will also give a brief high-level overview of CRREL and discuss summer internship opportunities with my group.

Friday, January 26, 2024. **Research Seminar.
**

**Dr. Justin Trias**,

*Imperial College London*

**Towards a theta correspondence in families for type II dual pairs**

*3:00 – 4:00pm, 116 Neville Hall*

Abstract: The classical local theta correspondence for p-adic reductive dual pairs defines a bijection between prescribed subsets of irreducible smooth complex representations coming from two groups (H,H’), forming a dual pair in a symplectic group. Motivated by new perspectives in the local Langlands correspondence for modular representations, Alberto Minguez extended the theta correspondence for type II dual pairs (i.e. when (H,H’) is made of general linear groups) to the setting of representations with coefficients in algebraically closed fields of characteristic l as long as the characteristic l does not divide the pro-orders of H and H’. More recently, the work of Emerton and Helm extended the local Langlands correspondence to families of representations, that is over coefficient rings, with compatibility to both classical and modular local Langlands for general linear groups. We explain how to build a theta correspondence in families, i.e. with coefficients in rings like Z[1/p], for type II dual pairs that is compatible with reduction to residue fields of the base coefficient ring, where central to this approach is the integral Bernstein centre. We translate some weaker properties of the classical correspondence, such as compatibility with supercuspidal support, as a ring morphism between the integral Bernstein centres of H and H’ and interpret it for the Weil representation. This ring morphism between the Bernstein centres brings a richer structure than a simple compatibility of supercuspidal supports and allows to ask new geometric questions for the theta correspondence: we prove that this map is surjective i.e. it is a closed immersion between the associated affine schemes. In particular our result implies a theta correspondence between characters of the Bernstein centre over any coefficient field of characteristic not p. This is joint work with Gil Moss.

Wednesday, December 6, 2023. **Mathematics Colloquium.
**

**Justin Dimmel**,

*University of Maine*

**The SunRule: An Interactive Mathematical Sculpture**

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

Abstract: We report on the design and fabrication of the SunRule, an interactive mathematical sculpture that is installed in Webster Park. The SunRule is a physical realization of a geometric definition of multiplication. Its novel design uses the sun’s parallel rays to geometrically construct products. The sculpture consists of a bronze wall that is partially wrapped around the edge of a ruled bronze disk. The disk and orthogonal wall are affixed to a granite plinth. The SunRule is designed to manipulate beams of light that shine through one of seven narrow windows that are cut out of the wall. A bronze-cast ball-and- socket joint allows the disk to be tilted (to change the multiplier) or swiveled (to change the multiplicand), and these actions vary the apparent length of the sun beam that is projected onto the disk (the product). We consider how activities that connect math, art, and the outdoors can create new opportunities for learning and teaching.

Wednesday, November 29, 2023. **Mathematics Colloquium.
**

**Simen Bjorvand**,

*NTNU*

**Model Predictive Control – Optimization based Control**

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

Abstract: Model Predictive Control is a control algorithm used in Process Control Industry, which has become popular for its ability to handle constrained multivariable systems. It is an optimization-based control algorithm where the control objective, model and operational constraints are embedded as a Mathematical Program called an Optimal Control Program. The Optimal Control program is solved online at each sampling instant so efficient algorithms for solving them are necessary for Real-Time implementation. In this talk we will discuss what Model Predictive Control is, how to set up the Optimal Control Problems and what optimization techniques can be used for Real-Time implementation.

Friday, November 17, 2023. **Mathematics MA Thesis Defense.
**

**Cameron Morin**,

*UMaine Mathematics MA student (advisor: Stechlinski)*

**Nonsmooth Epidemic Models with Evolutionary Game Theory**

*4:00 – 4:50pm, Room 421, Neville Hall*

Abstract: Utilizing systems of ordinary differential equations (ODEs), epidemic modeling is crucial for predicting infectious disease progression and devising interventions. However, many current models neglect the influence of human decision making and the limitations of medical resources. In this presentation, we introduce a novel epidemic model, called the Be-SEIMR model, that addresses these oversights using nonsmooth functions. After the model’s introduction, we will conduct a comprehensive analysis of the Be-SEIMR model, including a stability analysis and sensitivity analysis. This is accomplished using generalized derivatives theory, which allows us to extract crucial insights into a disease’s potential within a population. From this, we identify the most critical aspects of the disease that contribute to its spread, and pinpoint the medical interventions that are most effective in halting its progression.

Friday, November 17, 2023. **Research Seminar.
**

**Dr. Matthew Hernandez**,

*University of Maine*

**Fluid and Plasma Splashes**

*3-4pm, Room 227, Neville Hall*

Abstract: I’ll be talking about “fluid and plasma splashes,” or why it’s hard to rigorously prove the free boundary of a fluid can eventually become self-intersecting, and what the structure of the magnetic field is when this happens for a plasma.

Wednesday, November 15, 2023. **Mathematics Colloquium.
**

**Dr. Neel Patel**,

*University of Maine*

**Bubbles in Groundwater**

*3:15 – 4:15pm, Room 100, Neville Hall*

Abstract: Underground reservoirs of water, called aquifers, are common in the state of Maine. Often, when breaking ground, gas or oil bubbles can contaminate the water. Partial differential equations (PDE) and Fourier analysis are powerful mathematical tools for describing and analyzing the behavior of fluid dynamics underground, i.e. porous media flows. Moreover, the mathematical theory developed using PDE can be applied to other porous media flow settings such as glaucoma in the eye, chromatography and petroleum extraction. In this talk, we will discuss why the dynamics of fluid bubbles is more challenging than an infinite interface system and how the Fourier transform can be used to prove stability of certain bubble geometries.

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

**Dr. Stephanie Dodson**,

*Colby College*

**Traveling waves and cardiac arrhythmias: Investigating the onset of abnormal heart rhythms with mathematical models**

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

Abstract: Within cardiac tissue, regular cardiac function is characterized by coherent, periodic traveling waves of electrical activity that drive heart beats. When this process goes awry, the ensuing abnormal and irregular rhythms are known as arrhythmias. Arrhythmias are common and can be life-threatening. Therefore, it is crucial to understand physiological conditions that promote and deter arrhythmia onset. Regions of local heterogeneous tissue are known to initiate arrhythmias, either by directly interfering with wave propagation or acting as a rogue pacemaker and driving oscillations in the neighboring tissue. In this talk we’ll introduce mathematical models of cardiac cells and tissue and use these models to investigate the arrhythmia initiation mechanism of wave reflection.

Wednesday, November 1, 2023. **Mathematics Colloquium.
**

**Dr. Leah Sturman**,

*Bowdoin College*

**Comparing Integer Partitions**

*3:30 – 4:30pm, Room 100, Neville Hall*

Abstract: We will discuss inequalities of integer partitions and the methods involved in proving such inequalities. In particular, the Alder-Andrews Theorem, a partition inequality generalizing Euler’s partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to *d*-distinct partitions of *n*, was proved successively by Andrews in 1971, Yee in 2008, and Alfes, Jameson, and Lemke Oliver in 2010. While Andrews and Yee utilized *q*-series and combinatorial methods, Alfes et al. proved the finite number of remaining cases using asymptotics originating with Meinardus together with high-performance computing. In 2020, Kang and Park conjectured a “level 2” Alder-Andrews type partition inequality which relates to the second Rogers-Ramanujan identity. Duncan, Khunger, Swisher, and Tamura proved Kang and Park’s conjecture for all but finitely many cases using a combinatorial shift identity. We generalize the methods of Alfes et al. to resolve nearly all of the remaining cases of Kang and Park’s conjecture.

Friday, October 13, 2023. **Research Seminar.
**

**Dr. Brandon Hanson**,

*University of Maine*

**The sum-product problem without too many primes**

*3-4pm, Room 227, Neville Hall*

Abstract: The Erdos-Szemeredi conjecture is that for any finite set of integers A, either A+A or AA is near quadratic in the size of A. Spurred, perhaps by a famous result of Mei-Chu Chang, Szemeredi once asked if the problem gets easier under the assumption that the integers in A have only a few prime factors. We answer this in the affirmative. The ingredients are the satisfying menagerie of elementary combinatorics, harmonic analysis, and number theory, and I’ll try to indicate the proof in three parts, giving each the stage.

Wednesday, October 25, 2023. **Mathematics Colloquium.
**

**Cole Butler**,

*North Carolina State*

**Modeling insect populations and the next generation of genetic pest control**

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

Abstract: Gene drives are any natural or artificial mechanism of propagating a gene into a population. If the gene reduces organism fitness, then gene drives can suppress or even eradicate pest populations in the wild. Gene drives overcome many drawbacks of current pest control strategies and have demonstrated immense potential in laboratory populations. Despite this, there has yet to be a wild release of a gene drive-bearing organism. Technological development and accurate forecasting rely on mathematical models that account for realistic population genetic dynamics. We discuss some of the ways in which these dynamics can complicate gene drive success, including density dependence, dispersal behavior, and resistance. Particular attention is paid to the yellow fever mosquito, Aedes aegypti, and spotted wing drosophila.

Friday, October 13, 2023. **Research Seminar.
**

**Dr. Jane Wang**,

*University of Maine*

**The Oppenheim Conjecture**

*3-4pm, Room 227, Neville Hall*

Abstract: A quadratic form (e.g. consider Q(x,y,z) = x^2 + y^2 – e*z^2) can be thought of as a map from R^n to R. The Oppenheim conjecture, formulated in 1929, gives conditions for when the image of a quadratic form on the integer vectors is dense in R. Over the years, many mathematicians tried to tackle this conjecture using techniques from analytic number theory with limited success. It wasn’t until 1987 that the conjecture was proven by Gregory Margulis using dynamical techniques and ergodic theory. In this talk, we’ll introduce quadratic forms and survey some of the dynamical ideas that go into proving the Oppenheim conjecture. If time permits, we will also discuss how the values of quadratic forms relate to spacing statistics of the eigenvalues of the Laplace operator on tori. This talk should be accessible for a broad audience. No prior knowledge of quadratic forms or ergodic theory will be assumed.

Friday, October 6, 2023. **Research Seminar.
**

**Dr. Neel Patel**,

*University of Maine*

**When Temperature Fronts Collide**

*3-4pm, Room 227, Neville Hall*

Abstract: The surface quasi geostrophic (SQG) equation models atmospheric flows. Mathematically, it is a 2D analogue of the 3D Euler equations (which are 3D Navier Stokes without viscosity friction). When the atmospheric temperature is given by step functions, we can study movement of sharp temperature fronts. In this talk, we will use basic ODE and real analysis to figure out how to make temperature fronts collide and how their curvature changes at the point of impact.

Wednesday, October 4, 2023. **Mathematics Colloquium.
**

**Mathematics and Statistics Faculty**,

*University of Maine*

**Research Symposium**

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

Abstract: Have you wondered what your math professors think about outside of class? What does research in mathematics entail? What kind of projects might you get involved in? Please join us for an afternoon of very short talks, designed for a student audience, where faculty members will present questions they find interesting. The talks will be followed by open discussion where you can follow up with anyone whose presentation caught your attention. Students in search of capstone or thesis ideas are particularly encouraged to attend.