## Fall 2023 – Summer 2024 – Abstracts

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.
**

**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.
**

**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.
**

**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.