## Fall 2015 – Summer 2016 – Abstracts

Wednesday, September 16, 2015. Mathematics Colloquium.

Padmavathi Srinivasan, MIT Mathematics PhD Candidate.
“Conductors and minimal discriminants of hyperelliptic curves”
3:30 – 4:20 pm, Neville Hall Room 100. Snacks at 3:15pm.

Abstract: Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.

Wednesday, September 30, 2015. Mathematics Colloquium.
Prof. Amod Agashe, Florida State Mathematics.
“Stark-Heegner/Darmon points”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: The classical theory of complex multiplication gives certain points on elliptic curves defined over quadratic imaginary fields called Heegner points. These points played a crucial role in the resolution of the Birch and Swinnerton-Dyer conjecture when the analytic rank of the elliptic curve is zero or one. Darmon and others have given conjectural generalizations of Heegner points to construct points conjecturally defined over certain fields other than quadratic imaginary fields; these points are called Stark-Heegner points or Darmon points. We will give a brief survey of this area.

Thursday, October 1, 2015. Number theory seminar.
Prof. Amod Agashe, Florida State Mathematics.
“Stark-Heegner/Darmon points on elliptic curves over totally real fields”
1:00-1:50, 421 Neville Hall.

Abstract: We will describe the construction of Stark-Heegner/Darmon points on elliptic curves over totally real fields in a particular case and mention what can be done more generally. The rest of this abstract serves as motivation/background and is a repeat of the abstract for our colloquium talk: The classical theory of complex multiplication gives certain points on elliptic curves defined over quadratic imaginary fields called Heegner points. These points played a crucial role in the resolution of the Birch and Swinnerton-Dyer conjecture when the analytic rank of the elliptic curve (over the rationals) is zero or one. Darmon and others have given conjectural generalizations of Heegner points to construct points conjecturally defined over certain fields other than quadratic imaginary fields; these points are called Stark-Heegner points or Darmon points.

Wednesday, October 7, 2015. Mathematics Colloquium.
Prof. Robert Niemeyer, UMaine Mathematics and Statistics.
“Billiard orbits on the T-fractal billiard with an interesting application”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: This talk will be comprised of two parts: 1) a description of recent results on the T-fractal billiard and 2) discuss an interesting application in the study of a nanowire lying upon a superconductor. Recent results on the T-fractal billiard include the description of a countably infinite family of periodic orbits and the even more interesting example of an irrational singular orbit behaving in the manner of a periodic orbit. This latter phenomenon supports the intuition that the classical topological dichotomy does not hold for the T-fractal billiard, though it holds for each of its approximations. Using results on the T-fractal billiard, we show how a T-fractal perturbation of a nanowire effectively severs a nanowire, from the perspective of an electron, into two pieces.

Wednesday, October 14, 2015. Mathematics Colloquium.
Prof. Joshua Zelinsky, UMaine Mathematics and Statistics.
“Heuristics in elementary number theory”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: There are many open problems in number theory of the form “Is the set of integers with the following property finite or infinite?” One example is “Are there infinitely many twin primes, that is, primes p such that p+2 is also prime?” Another is “Are there infinitely many Mersenne primes, that is, primes which are one less than a power of 2?” This talk will discuss how we can use heuristic approaches to arrive at answers to such questions with high confidence even as we cannot actually prove that our answers are correct. In particular, we will show that by pretending that whether a number has a probabilistic property rather than a fixed property we can make good predictions about the answers to many questions.

Wednesday, October 21, 2015. Mathematics Colloquium.
Profs. Sergey Lvin & Paul Van Steenberghe, UMaine Mathematics and Statistics.
“STABLE OR UNSTABLE? (An Inverted Pendulum Mystery) “
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: A rigid pendulum has two equilibrium positions: top and bottom. Obviously the top position (an inverted pendulum) is unstable while the bottom one (a regular pendulum) is stable. The regular pendulum can become unstable when it vibrates vertically with a certain frequency – that is why it is possible to swing by legs pumping. More surprisingly, more than a century ago it was discovered that when the pivoting point vibrates fast enough, then the inverted pendulum becomes stable. It was first explained only a half century later. However, human beings used this phenomenon for millions of years to balance themselves on two or even on one leg. We will present a more recent elementary explanation of the inverted pendulum mystery. The same approach can be applied to a regular pendulum as well.

Wednesday, October 28, 2015. Mathematics Colloquium.
Prof. Henry Boateng, Bates College Mathematics.
“Mathematical Modeling of Crystal Growth”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: We present an approximate off-lattice kinetic Monte Carlo (KMC) method for simulating heteroepitaxial growth. The model aims to retain the speed and simplicity of lattice based KMC methods while capturing essential features that can arise in an off-lattice setting.

Wednesday, November 4, 2015. Mathematics Colloquium.
Prof. Jacqueline M. Dresch, Clark University Mathematics and Computer Science.
“The Mathematics Behind Gene Regulation”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Since the complete human genome was released over a decade ago, researchers have struggled to understand how 3.2 billion nucleotides encode the instructions for every biological process that takes place throughout a person’s lifetime. Accompanying the expansion in sequence data, new technologies have provided copious amounts of gene expression data, which is detected by the production of either RNA or proteins. This gene expression data along with DNA-sequence information is ideal for mathematical modeling studies. In this talk, I will use Drosophila melanogaster (the common fruit fly) to try to answer some of the questions regarding how DNA-sequences alone can direct gene expression during development. The mathematical modeling approaches I will discuss weave together many familiar concepts from Biology, Mathematics, and Computer Science, and involve techniques from a wide range of areas in applied mathematics, including probability, statistics, differential equations, and numerical analysis. These models have allowed us to predict the expression of unknown genes or variants of genes, revealing the driving forces behind gene regulation and continuing to elucidate the complex grammar of the genome.

Wednesday, November 18, 2015. Mathematics Colloquium.
Prof. David Batuski, UMaine Physics and Astronomy.
“Einstein’s Universe – ‘Precision’ Cosmology and General Relativity at Age 100”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: In this 100th Anniversary Year of General Relativity, a hot research area in Cosmology is finding out what Dark Matter and Dark Energy actually are. These are two components of the Universe that make up 95% of its contents and about which we know so little that the term ‘Dark’ fully applies. (The other 5% is ‘normal’ matter, out of which stars, dust clouds in space, planets, asteroids, people, dandelions, etc., are made – which is generally quite visible and which we understand pretty well).

This talk will outline a bit about General Relativity itself, discuss some of the mathematics of Cosmology, and cover a UMaine observational research project to detect the effects of Dark Matter at the scale of superclusters of galaxies (roughly 100 million lightyears) through the mechanism of ‘weak gravitational lensing’ – a prediction of General Relativity.

Wednesday, December 2, 2015. Mathematics Colloquium.
Prof. Chan Ieong Kuan, UMaine Mathematics and Statistics.
“Sums of nonnegative values of quadratic forms and modular forms”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Sums of specific powers of nonnegative values of quadratic forms of fixed discriminant will be examined. In particular, we will see that summing first powers of such nonnegative values will always give a constant which depends only on the discriminant. Then we will find out summing third powers will also give another constant. However, this doesn’t hold for fifth powers.

In the process of understanding the patterns of such sums, we will come across modular forms.

Wednesday, December 9, 2015. Mathematics Colloquium.
Prof. Daniel Vallieres, UMaine Mathematics and Statistics.
“Have you ever asked yourself what cos(2π/5) is?”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract:  We will give a little introduction to algebraic groups. Along the way, an answer to the question of the title will be provided. This talk will be geared towards advanced undergraduate students who know about trigonometric functions and group theory.

Friday, December 11, 2015. Graduate Seminar.
Michael Mudarri, Mathematics MA student.
“Information Theory, Coding, and Cryptography”
3:10 – 4:00 pm, 421 Neville Hall.

Abstract:  There are many ways of sending information to other people, but some are more efficient than others. In this talk we will discuss this problem as first considered by Claude Shannon, using the concept of entropy. After discussing several examples, we will explain how it relates to encryption, with particular emphasis on RSA.

Friday, January 22, 2016. Graduate Seminar.
Trevor Gionet, Mathematics MA student.
“What do Tomahawk Missiles, the International Space Station and Carbon Sequestration have in Common?
The Kalman Filter: An Introduction and Derivation”

3:00 – 3:50 pm, 421 Neville Hall.

Abstract:  From tracking cruise missiles and navigating the International Space Station to weather forecasting and mitigating climate change, this talk will provide an overview of the Kalman filter its uses and applications. A complete derivation will be presented for the linear model scenario. This will be followed by a look into some of the extensions which concludes with the Local Ensemble Transform Kalman Filter (LETKF), a filter which is currently being studied for its use in forecasting forest variables. Other data assimilation methods will be mentioned to shed light on the advantages of this simple yet powerful approach.

Wednesday, February 3, 2016. Statistics Colloquium.
Prof. Randy Lai, UMaine Mathematics and Statistics.
“Data compression and histograms”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Minimum description length (MDL) principle is a model selection philosophy in which the best description for a given dataset is the one that leads to the largest data compression. In this talk, I am going to introduce the basics of MDL principle. We will explore its connections with Shannon’s coding theory and information theory. An application of MDL in histogram density estimation will be given. It would be argued that the optimal number of bins in a histogram is in the order of $n^{1/3}$, where $n$ is the number of observations.

Friday, February 5, 2016. Graduate Seminar.
Danielle David, UMaine Mathematics MA student.
“Nonabsolute Integration, Part 1”
3:05 – 3:55 pm, 421 Neville Hall.

Abstract: After reviewing the main features and drawbacks of the Riemann and Lebesgue integrals, we describe two equivalent “nonabsolute” extensions of these theories.  One is based on Lebesgue’s approach, while the other is based on Riemann’s.  Both preserve the most desirable properties of each, while at the same time integrating more functions, removing the need for improper integrals, and simplifying the fundamental theorem of calculus.

Wednesday, February 10, 2016. Mathematics Colloquium.
Dr. Sergey Lvin, University of Maine Mathematics and Statistics.
“One-Dimensional Traveling Waves”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: First I will discuss traveling waves on a uniform string: how fast they move, what happens when they reach the end of the string. This will be a review for those who have studied partial differential equations and easy to follow for those who haven’t.

By the way, do three-dimensional sound waves move similarly to one-dimensional waves on a string or not? Why are two-dimensional waves in a bathtub different if it’s full versus if it’s almost empty? Can two traveling waves on a string create a standing wave?

Next, I will turn your attention to waves on a composite string consisting of two different strings connected to each other. What happens when a traveling wave on a lighter string reaches the connection with a heavier string? Will the wave propagate or reflect? Will it change its speed, its shape, or its orientation? What changes when a traveling wave on a heavier (or less tense) string reaches the connection with a lighter (or tenser) string?

I’ll use Calculus and Pre-Calculus concepts to answer these questions.

This is a 3.5-star event.

Wednesday, February 17, 2016. Mathematics Colloquium.
Prof. Ben Weiss, University of Maine Mathematics and Statistics.
“Magic Squares and Inflection Points on Elliptic Curves”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: This talk is based on a paper of a similar title by Ezra (Bud) Brown. The goal of the talk is to introduce some interesting results about elliptic curves and magic squares, and tie these disparate subjects together into a friendly introduction for both.

This is a 1 star talk.

Wednesday, March 23, 2016. Graduate Seminar.
Ayesha Maliwal, UMaine Mathematics MA student.
“Sperner’s Lemma and its Application to Fair Division Problems”
3:05 – 3:55 pm, 421 Neville Hall.

Abstract: All of us know how to divide an object equally between n people. However, does an equal share guarantee envy-free division? A person may prefer to have a smaller piece of the cake with icing on it rather than a large piece without any icing on it. A person may prefer to pay more rent if only they can get the room of their choice. This talk explores how Sperner’s lemma, given by a German mathematician Emanuel Sperner while he was still a research student, guarantees envy-free division under certain assumptions. We will first prove Sperner’s lemma and then delve into its applications to real life situations like dividing goods (for instance, a cake) or bads (for instance, rent) such that each person is content with their share.

Thursday, March 24, 2016. Mathematics Colloquium.
Prof. Solomon Friedberg, Boston College Department of Mathematics.
“Theta functions and covering groups”
3:30 – 4:20 pm, 100 Neville Hall. Snacks at 3:00pm in Neville 231.

Abstract:   We introduce aspects of the classical theory of theta functions, starting from the Jacobi theta function, and then discuss generalizations.  This will lead us to the theory of automorphic forms on covering groups.  We will survey some recent work on this topic, including some open questions.

Friday, March 25, 2016. Graduate Seminar.
Prateek Kunwar, UMaine Mathematics MA student.
“Hypercubic Icing on Doughnuts”
3:05 – 3:55 pm, 421 Neville Hall.

Abstract:   Have you ever wondered how many holed doughnuts you would need if you had the inexplicable, yet quite reasonable desire of having an icing that resembled a 4-dimensional cube? What about a 5-dimensional cube? n-dimensional?  Worry not, whether you are a conscious baker looking to do it with a minimum number of holes or someone who craves a more pretzely doughnut, this talk will give you an insight on how to maximally or minimally embed a hypercubic graph on an orientable surface. The talk will focus on what graph embeddings are and how rotation systems can be used to represent such embeddings.  We will introduce an algorithm that will help us create walks on the graph that enable us to determine what surface a particular graph lives on.

Wednesday, March 30, 2016. Mathematics Colloquium.
Prof. Richard Ehrenborg, University of Kentucky Mathematics.
“(Cyclically) consecutive 123-avoiding permutations”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB).

Abstract: A permutation $\pi=(\pi_1,...,\pi_n)$ is consecutive 123-avoiding if there is no index i such that $\pi_i < \pi_{i+1} < \pi_{i+2}$. Similarly, a permutation $\pi$ is cyclically consecutive 123-avoiding if the indices are viewed modulo n. These two definitions extend to (cyclically) consecutive S-avoiding permutations, where S is some collection of permutations on m+1 elements. We determine the asymptotic behavior for the number of consecutive 123-avoiding permutations. In fact, we give an asymptotic expansion for this number. Strangely enough we obtain an exact expression for the number of cyclically consecutive 123-avoiding permutations. A few results will be stated about the general case of (cyclically) consecutive S-avoiding permutations. Part of these results are joint work with Sergey Kitaev and Peter Perry.

Thursday, March 31, 2016. Number theory seminar.
Prof. Margaret A. Readdy, University of Kentucky Mathematics.
“New perspectives on q-analogue theory”
3:30 – 4:20 pm, 100 Neville Hall.

Abstract: The idea of q-analogues can be traced back to Euler in the 1700’s who was studying q-series, especially specializations of theta functions. Recall a q-analogue is a method to enumerate a set of objects by keeping track of its mathematical structure. For example, in 1916 MacMahon gave a q-analogue of the binomial coefficient by summing over certain 0-1 bit strings using the weighting q to the inversion statistic.

After reviewing some classical q-analogue theory, we will highlight
more recent results including the cyclic sieving phenomenon of Reiner,
Stanton and White, and new work of Cai and Readdy on q-(1+q)-analogues.

Friday, April 1, 2016. Graduate Seminar.
Muhammad Waleed, UMaine Mathematics MA student.
“Bessel Distribution and Associated Inference”
3:05 – 3:55 pm, 421 Neville Hall.

Abstract: The study of probability distributions involving Bessel functions began as early as 1932. In all such studies, continuous distributions were involved. In this talk, we consider a discrete distribution generated by modified Bessel function of the first kind. This distribution originally arises in the work of Pitman and Yor (1982), and later studied in detail by Yuan and Kalbfleisch (2000). We will discuss various structural properties of this distribution, and present its relationship with some well-known probability distributions. Furthermore, we will present an algorithm to generate Bessel distribution, and study Maximum Likelihood Estimation (MLE) of the model parameters. Finally, we will perform simulation studies to examine the performance of the estimates of the Bessel distribution.

Wednesday, April 6, 2016. Mathematics Colloquium.
Dr. Robin Koytcheff, University of Massachusetts Amherst Mathematics and Statistics.
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Knots are tangled closed loops in 3-dimensional space which have been studied mathematically at least since the 19th century. I will first discuss the idea of knot invariants, which can be used to prove that a knot can’t be untangled. Links are like knots, but with multiple strings instead of just one. I will discuss the Gauss linking number, which is an invariant of 2-component links and is arguably simpler than any knot invariant. It detects linking of the two components, but it ignores knotting of either component. I will describe this invariant in elementary terms by counting crossings. I will also describe it as the degree of a map of surfaces. If time permits, I will further describe it in terms of homotopy classes of maps of configuration spaces. This latter description generalizes to an invariant of n-strand links which ignores knotting but conjecturally detects all linking phenomena. This is the subject of recent joint work with Fred Cohen, Rafal Komendarczyk, and Clay Shonkwiler, where we proved a certain analogue of this conjecture.

Friday, April 8, 2016. Graduate Seminar.
Daniel Buck, UMaine Mathematics MA student.
“Transitivity and Periodic Behavior in Chaotic Dynamical Systems”
3:05 – 3:55 pm, 421 Neville Hall.

Abstract: Dynamical Systems is a very active field of mathematics used in a wide variety of scientific disciplines to model any number of real-world phenomena.  Some of the most interesting behavior exhibited by such systems is observed in those described as “chaotic.”  This talk will introduce dynamical systems as a field of study, discuss several distinct definitions of chaos, and explore several interesting results pertaining to chaotic dynamical systems.  In particular, we will show that in a chaotic dynamical system, any finite number of open sets share a periodic orbit.

Wednesday, April 13, 2016. Mathematics Colloquium.
Prof. Joshua Zelinsky, University of Maine Mathematics and Statistics.
“Diophantine Equations, Hilbert’s Tenth Problem, and Mazur’s Conjecture”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: A Diophantine equation is an equation where we are interested in only the integer’ solutions to an equation. Hilbert’s Tenth Problem asked for a general procedure that would say whether a given Diophantine equation had any solutions. We will discuss the history of this problem culminating in Yuri Matiyasevich’s negative resolution of the problem. We will also discuss recent work involving related open problems.

Wednesday, April 20, 2016. Mathematics Colloquium.
Prof. Billy Jackson, University of Maine Mathematics and Statistics.
“From Difference Equations to Differential Equations and Everything Else in Between”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: Of all areas of modern mathematics, differential equations is perhaps one of the oldest, of course going back to the days of Leibniz and Newton. Mathematicians in the latter half of the 20th century also began examining difference equations in detail as these structures became important in numerically solving differential equations with the advent in computing that took place. There are many similarities between discrete and continuous dynamical systems, but there are often many differences as well. For example, solutions to the usual logistic equation can exhibit chaotic behavior in discrete systems while simultaneously not being chaotic in their continuous counterparts.

In 1988, a German mathematician by the name of Stefan Hilger constructed a framework that we now call analysis on time scales which explains the similarities and differences between the two extremes: in addition to this unification aspect, we can now extend the analysis to other dynamical systems as well. For example, systems containing hybrid continuous and discrete components, fractal structures, or non-uniformly spaced domains are all ripe for exploration within this framework that will allow for a single analysis to treat them all.

In this talk, I will present Hilger’s framework and then discuss applications of the work in some of my projects over the years. These include stability analysis, control theory from an engineering perspective, and mathematical ecology models. Most details will hopefully be accessible to those with at least a slight background in differential equations and an adequate preparation in analysis.

Monday, April 25, 2016. Mathematics MA Thesis Defense.
Trevor Gionet, MA Candidate (advisor: Bellsky).
“Improvements to Data Assimilations Strategies in Forestry”
3:30 – 4:20 pm, 100 Neville Hall.

Abstract: An iterative data assimilation strategy and two novel applications will be presented. In particular, an outline of the development of the Kalman filter will be given. Two applications will be thoroughly explored to predict forest stand characteristics. In the first application a Chapman–Richards forecast yield model and observed field data from a collection of fully stocked eastern white pine stands is utilized. The results from this application demonstrate the ease of implementation of the ensemble Kalman filter data assimilation methods, and show the capacity of these methods to accurately forecast future forest stand basal area when compared to other methods. The second application utilized the United States Department of Agriculture’s Forest Inventory Analysis data set for the state of Maine and a spatially coupled variation on the Chapman–Richards Yield Model. The results of this application shows the robust capacity of the described method when dealing with large, spatially explicit data sets.

Wednesday, April 27, 2016. Mathematics Colloquium.
Dr. Alan Weiss, MathWorks, Natick, MA.
“The 3x+1 Problem and Large Deviations”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: The 3x+1 problem addresses the iterates of the following map on positive integers:

x —> x/2 if x is even
x —> (3x+1)/2 if x is odd

The question is, does every positive integer eventually map to 1 by repeated applications of this map? I have nothing to say about this long-standing open question. However, a related random walk arises if we regard x(0) as a large number, and consider the random walk with independent steps in log space:

z(n+1) = z(n) + log(1/2) with probability 1/2
z(n+1) = z(n) + log(3/2) with probability 1/2

The average step of this walk is 1/2(log(1/2) + log(3/2)) = 1/2 log(3/4) ~ -0.1438…, so starting anywhere, with probability 1 this random walk eventually goes below 0. How long does that take? How long does it take starting from a very large number of nearby initial points? How high can the random walk get?

I will discuss these questions and other related ones using the theory of large deviations, which is a study of how rare events relate to certain scalings of random processes. I will assume no knowledge of large deviations or random processes.

This is joint work with Jeff Lagarias.

Thursday, April 28, 2016. Mathematics MA Thesis Defense
“Nonabsolute Extensions of the Lebesgue Integral on the Real Line”
2:00 – 2:50 pm, 208 Neville Hall.

Abstract: Henri Lebesgue’s theory of integration is a turning point of the concept of the integral; Riemann’s concept of areas of rectangles meeting the measure theory of Camille Jordan. While Lebesgue’s theory successfully extends the class of integrable functions, it does not improve upon the Fundamental Theorem of Calculus. We will focus our attention on the nonabsolute extension of the Lebesgue integral given by J. Kurzweil and R. Henstock. The Henstock-Kurzweil integral, a modification of Riemann’s integral, gives us a familiar notation. It integrates all Lebesgue-integrable functions, possesses the convergence properties of the Lebesgue integral, and improves upon the Fundamental Theorem of Calculus. We will also discuss the framework of the Henstock-Kurzweil integral, its equivalence with the Denjoy integral, an extension of the Henstock-Kurzweil integral known as the ultrafilter integral, and the Khintchine integral.

Friday, April 29, 2016. Mathematics MA Thesis Defense
Prateek Kunwar, MA Candidate (advisor: Franzosa)
“Embedding of hypercube graphs on orientable surfaces”
3:00 – 3:50 pm, 227 Neville Hall.

Abstract: After giving a brief introduction to graphs, surfaces and what it means to embed a graph on a surface, we will delve into a particular class of graphs called the hypercube graphs. We will learn about rotation systems and use them to represent embeddings of graphs. Introducing the idea of adjacent changes in rotation systems, which allows us to traverse all possible embeddings, we will prove a general result about the effect of such changes on the embedding. We will develop a rotation system, which we call the ABC rotation system, and use it to give a general result about minimal embeddings of hypercube graphs. We will then talk about various kinds of maximal embeddings of the hypercube graphs, like the “Eulerian walk embedding” and the “big-face embedding” and prove a general result about the “big-face embedding”.

Wednesday, May 4, 2016. Mathematics MA Thesis Defense
Jennifer Fatula, MA Candidate (advisor: Franzosa)
“An investigation of linking number, twist, and writhe, in knots, links, ribbons, and Möbius ribbons”
2:00 – 2:50 pm, 208 Neville Hall.

Abstract: A prominent formula in knot and ribbon theory is White’s formula that looks at the relationship between linking number, twist, and writhe. After a brief background in knot theory we look at knots, links, and diagrams of each and we examine their associated properties, writhe and linking number. We prove that linking number is an invariant of links and that writhe is not a knot invariant since it is dependent on the diagram from which it is defined.  We then extend these concepts to ribbons and diagrams of them.  We examine how linking number and writhe are defined for ribbons and we introduce another property, twist. We demonstrate how linking number, writhe, and twist are related in a particular diagram and we examine how the relationship is affected as a diagram changes. The result is the LTW formula for ribbons, a diagram-based version of White’s formula. We then introduce Mӧbius ribbons and show how linking number, writhe, twist, and the LTW formula carry over to that setting.  Lastly, we look at the integral-based White’s Formula and we discuss how our diagram-based LTW formula relates to it.

Friday, May 6, 2016. Mathematics Colloquium.
Dr. Elizabeth Gillaspy, University of Colorado – Boulder Mathematics.
“K-theory and twisted groupoid C*-algebras”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Abstract: The goal of this talk will be to explain some of what we know about the K-theory of twisted groupoid C*-algebras, and why we should care.

In the first part of the talk, I will define the objects in the title and explain how the K-theory of groupoid C*-algebras can tell us about other mathematical objects, from dynamical systems to string theory. My hope is that this will be a gentle introduction to the topic(s) at hand; no prior familiarity with groupoids or with C*-algebras will be assumed. In the second part of the talk, I’ll discuss the particular question I’ve investigated about the K-theory of twisted groupoid C*-algebras, why I chose it, and the progress that I’ve made so far. Time permitting, I will sketch some of the proofs, so this part of the talk will be more technical than the first part.

Tuesday, May 10, 2016. Graduate Seminar.
Daniel Buck, UMaine Mathematics MA Candidate.
“Two Approaches to the Fibonacci Sequence Modulo m.”
2:00 – 2:50 pm, 421 Neville Hall.

Abstract: In 1960 D.D. Wall published some results concerning periodicity of the Fibonacci sequence when taken modulo m for any natural number m.

In particular, Wall proved that the sequence is simply periodic for any m. Furthermore if m=p for some prime p congruent to 1 or 4 mod 5 then then length of the period divides p-1and if m=p is congruent to 2 or 3 mod 5 then the length of the period divides 2p-2.

Wall used numerous combinatorial techniques to show his results.  In 2012 Gupta, Rockstroh and Su provided some substantially simplified proofs of Wall’s results using splitting fields.

Monday, June 27, 2016. Statistics MA Thesis Defense.