## Fall 2019 – Summer 2020 – Abstracts

Friday, August 28, 2020. Mathematics Graduate Seminar.
Kristin Tenney, Mathematics MA student

“Gödel’s Second Incompleteness Theorem and Turing Machines”
3:00 – 3:50 pm, via Videoconference.

This talk will give a brief introduction to the idea of incompleteness as well as the First Incompleteness Theorem. An alternative proof of Gödel’s Second Incompleteness Theorem will be stepped through as well as the impacts the theorems had in the mathematics community. These theorems will be related to Turing Machines and computability and the fundamental idea behind classical computing.

Monday, August 17, 2020. Mathematics Graduate Seminar.
Kristin Tenney, Mathematics MA student

“Gödel’s First Incompleteness Theorem”
3:00 – 3:50 pm, via Videoconference.

This talk aims to provide the audience with both the historical and mathematical background to better understand formalism, omega-consistency, and ultimately Gödel’s First Incompleteness Theorem. An overview of Gödel’s original proof will be presented before an alternative first-order logic proof. This theorem and proof’s substantial impact on the math community will then be briefly discussed.

Friday, April 24, 2020. Mathematics MA thesis defense.

“A Method to Reclaim Multifractal Statistics from Saturated Images”
4:00 – 4:50 pm, via Videoconference.

The 2D Wavelet Transform Modulus Maxima (WTMM) method is a multiscale multifractal formalism suited for the analysis of rough surfaces. This is done by characterizing density fluctuations and spatial correlations within these surfaces. The CompuMAINE lab has developed a patented computational cancer detection method utilizing the 2D WTMM method to help predict disrupted, tumor-associated breast tissue from mammography. The lab has a database of mammograms in which some of the image subregions contain artefacts which are excluded from the analysis, image saturation is one such artefact. To maximize statistical power in our clinical analyses, our goal is therefore to minimize the rejection of image subregions containing artefacts. The goal of this particular project is to explore the effects of image saturation on the resulting multifractal statistics from the 2D WTMM method. Groups of numerically simulated (monofractal) fractional Brownian motion (fBm) surfaces with varying roughness exponents were generated and saturated at the 1%, 5%, 10% and 20% levels. We find that image saturation reduces the range of available statistical order moments relative to an unsaturated image. By assessing the effects of image saturation on the 2D WTMM calculations, we developed a filtering approach where we nearly regained the entire range of statistical order moments thus limiting the impacts of image saturation.

Wednesday, April 22, 2020. Mathematics MA thesis defense.
Isaac Vaccaro, Mathematics MA candidate (advisor: Crisp)

“Gray Codes in Music Theory”
3:00 – 3:50 pm, via Videoconference.

Abstract: In the style of music composition called serialism, it is desirable to find chord progressions that use each chord in a chosen set exactly once. We approach this problem through the mathematical theory of Gray codes: the notion of cycling through a finite set with respect to some minimal change. Connecting reflected Gray codes to the theory of finite groups generated by involutions, we study Hamiltonian paths in Schreier and Cayley graphs to construct Gray codes in music composition.

Monday, April 20, 2020. Mathematics MA thesis defense.
James Poulin, Mathematics MA candidate (advisor: Rosen)

“Calculating Infinite Series Using Parseval’s Identity”
3:00 – 3:50 pm, via Videoconference.

Abstract: Parseval’s identity is an equality from Fourier analysis that relates an infinite series over the integers to an integral over an interval, which can be used to evaluate the exact value of some classes of infinite series. We compute the exact value of the Riemann zeta function at the positive even integers using the identity, and then we use it to compute the exact value of an infinite series whose summand is a rational function summable over the integers.

Friday, April 17, 2020. Mathematics MA thesis defense.
Caleb Hill, Mathematics MA candidate (advisor: Crisp)

“Hopf algebras in the representation theory of combinatorial families of groups”
12:00 – 12:50 pm, via Videoconference.

Abstract: We will recall the difficulty of studying the representation theory of certain interesting semidirect products, and discuss alternative methods to study these representations. We generalize the construction of a positive, self-adjoint Hopf algebra in the representation theory of the symmetric groups to a family of groups of matrices by expressing these groups of matrices as combinatorial semidirect products.

Friday, February 21, 2020. Joint Mathematics/Physics Colloquium.

Dr. Glen Van Brummelen, Quest University/Institute for Advanced Studies
“The Forgotten Man: Astronomy in the Transformative 15th Century”
3:15 – 4:15 pm, Hill Auditorium, Barrows Hall.

Abstract: We know a lot less than we think. The history of mathematics is partly a record of what happened, but uncomfortably more than we might expect, over the decades it has also been a record of what we care about. We will explore one episode that illustrates the “forgotten” history of mathematics, in early 15th-century mathematical astronomy. The forgotten man is Giovanni Bianchini, the CFO from Ferrara who turned his eye away from bookkeeping and toward the heavens. His unusual path to academia provoked a number of revolutions, including among others the birth of the European tangent function. Overlooked in the shadow cast by his now more illustrious successor Regiomontanus, Bianchini’s role in the history of mathematics and science fully deserves a resurrection.

Friday, December 6, 2019. Graduate Seminar.

James Poulin, Mathematics MA student, UMaine. (Advisor: Rosen)
“Calculating Infinite Series Using Parseval’s Identity”
3:00 – 3:50 pm, 421 Neville Hall.

Abstract: Euler famously discovered that $\displaystyle \sum_{n=1}^\infty \frac1{n^2}=\frac{\pi^2}6$. One proof of this identity uses Parseval’s Theorem from Fourier Analysis. The idea is to find a function $f$ whose Fourier transform $\widehat{f}(n)$ is equal to $1/n$. Parseval’s identity then allows us to evaluate the sum by computing an integral involving $f$ over the interval [0,1]. We develop an algorithm that generalizes this proof to compute $\sum\frac1{n^{2A}}$ where $A$ is a natural number. We then present an extension of the algorithm that allows us to replace $1/n^{2A}$ with any non-negative rational function of $n$.

Wednesday, December 4, 2019. Mathematics Colloquium.

Dr. Byungjae Son, Department of Mathematics & Statistics, University of Maine.
“Steady state reaction diffusion equations where a parameter influences the equation as well as the boundary condition”
3:15 – 4:15 pm, Barrows Auditorium

Abstract: In this talk, we study positive solutions to steady state reaction diffusion equations where a parameter influences both the equation and the boundary condition. We discuss existence, nonexistence, multiplicity and uniqueness results for various classes of the reaction term, including a logistic growth model in population dynamics, a model in combustion theory and a semipositone model.

Monday, November 25, 2019.  Mathematics Graduate Seminar.

Jeremy Juybari, Mathematics MA student. (Advisor: Khalil)
“The Effects of Image Saturation on Multifractal Statistics”
4:30 – 5:20 pm, 421 Neville Hall

Abstract: The goal of this particular project is to explore the effects of image saturation on the resulting multifractal statistics from the 2D Wavelet Transform Modulus Maxima (WTMM) method. The CompuMaine lab has a database of mammograms with saturated (distorted) image subregions which are excluded from the lab’s patented computational cancer detection method. Groups of numerically simulated (monofractal) fractional Brownian motion (fBm) surfaces, with varying roughness exponents, were generated and saturated at the 1%, 5%, 10% and 20% levels. We find that image saturation reduces the range of available statistical order moments relative to an unsaturated image. By assessing the effects of image saturation on the WTMM calculations, we are developing an algorithmic strategy to regain about a full statistical order moment for fBms with long-range correlations, thus limiting the impacts of image saturation.

Wednesday, November 20, 2019.  Mathematics Graduate Seminar.

Isaac Vaccaro, Mathematics MA student. (Advisor: Crisp)
“Hamiltonian Paths in the Cayley Graphs of Groups Generated by Involutions”
3:00 – 3:50 pm, 421 Neville Hall

Abstract: A Hamiltonian path or cycle in a graph is a path or cycle that visits each vertex exactly once. In this talk we will demonstrate the existence of Hamiltonian cycles in the Cayley graphs of finite groups generated by involutions with acyclic commuting graphs. This problem is applied to the search for Gray codes in the binary numeral system and the existence of parsimonious chord progressions in Western music theory.

Friday, November 8, 2019.  Mathematics MA Thesis Defense.

Stephanie Ayres, UMaine Mathematics MA Candidate (advisor: Thomas Bellsky).
“Three-dimensional analytical model of the tidal flow in the Damariscotta River Estuary, ME”
3:00 – 3:50 pm, 421 Neville Hall

Abstract: Analytical models of estuaries provide a systematic method to investigate the impacts of individual forcing mechanisms on water level elevations and current velocities. A three-dimensional analytical model, modified from Ensing et al. (2015), was developed for a tidal-dominated, weakly-stratified estuarine system, such as the Damariscotta River Estuary, ME. After performing a perturbation expansion of the non-dimensionalized Navier-Stokes equations in the shallow water limit, the resulting zero-order solution is analyzed to provide insight into the tidal flow of the Damariscotta estuary. Depth is considered a parabolic function of across-channel position, and mean density is prescribed as a function of along-channel position. The model assumes a narrow basin, such that zero-order water level elevation does not vary across channel, and constant eddy viscosity, which is a proxy for mixing. Variations in water level elevations and three-dimensional tidal currents within the estuary will be discussed. Barotropic pressure gradients, friction, and width convergence determine along-channel velocities, while Coriolis (rotation) and lateral gradients of depth-averaged density and higher-order water level drive across-channel velocities. Results of the model compared well to previous studies within the estuary (Lieberthal et al., 2019) and to the Upper Ems estuary (Ensing et al., 2015), which has similar dynamics as the Damariscotta estuary although morphologically diverse in length and width convergence. The model can be applied to study to similar estuarine systems with weak-stratification and tidal-dominated flow. Future work should implement a more complicated friction regime and analyze the first-order problem, which represents residual flow occurring on timescales greater than the tides.

Wednesday, November 6, 2019.  Mathematics Colloquium.

Dr. Sara Chari, Mathematics Department, Bates College.
“Metacommutation of primes in locally Eichler orders”
3:15 – 4:15 pm, Barrows Auditorium

Abstract: We study the metacommutation problem in locally Eichler orders. From this arises a permutation of the set of locally principal left ideals of a given prime reduced norm. Previous results on the cycle structure were determined for locally maximal orders. As we extend these results, we present an alternative, combinatorial description of the metacommutation permutation as an action on the Bruhat-Tits tree.

Wednesday, October 30, 2019.  Mathematics Colloquium.

Dr. Shenhui Liu, Department of Mathematics & Statistics, University of Maine.
“Non-vanishing of automorphic L-functions”
3:15 – 4:15 pm, Barrows Auditorium

Abstract: Automorphic L-functions are among the central objects of study in modern analytic number theory. An interesting topic is to study the non-vanishing of such L-functions. I will talk about two popular (and often successful) methods for establishing non-vanishing results for families of automorphic L-functions: the method of moments and the method of mollification. In particular, I will use L-functions for Hecke—Maass forms for SL(2,Z) as an example to illustrate the use of these methods, and (if time permits) present some applications.

Wednesday, October 23, 2019.  Mathematics Colloquium.

Dr. Keshav Aggarwal, Department of Mathematics & Statistics, University of Maine.
“Sums of Oscillatory Functions and Circle Method”
3:15 – 4:15 pm, Barrows Auditorium

Abstract: An effective method to study the properties of a function is to expand it into a series, like the Taylor series or Fourier series. It often happens that the coefficients appearing in such a series are highly oscillatory. Although, the exact nature of such oscillations is hard to determine, there are methods to estimate the cancellations appearing in the series formed by summing these oscillatory coefficients. We will describe why we care about such functions, and motivate the use of the circle method in this regard.

Wednesday, September 18, 2019.  Graduate Seminar.

Caleb Hill, Mathematics MA student, University of Maine.
“Representations of combinatorial wreath products”
3:00 – 3:50 pm, 421 Neville Hall

Abstract: Representation theory studies the way groups act on vector spaces. In this talk we will discuss a parameterisation of the irreducible representations of certain families of finite groups of combinatorial origin. We will exhibit the difficulty of explicitly classifying the irreducible representations of the individual members of these families, and suggest an alternative strategy for studying the representations of the family as a whole.

Thursday, September 19, 2019.  Mathematics Colloquium.

Dr. Tonghui Wang, Department of Mathematical Sciences, New Mexico State University.
“Family of skew normal distributions, properties, applications, and recent development”
3:15 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:00pm.

Abstract: In this talk, the following topics will be covered: (i) The background, motivation, and definition of the skew normal family are discussed; (ii) Its extensions to multivariate skew normal and multivariate closed skew normal families are investigated; (iii) The skew chi-square distribution and closed skew chi-square distribution are introduced; (iv) Applications and examples are given for illustration of main results; (v) Recent development on this topic is listed.