## Abstracts

Wednesday, September 17, 2014.  Pizza Pi Seminar.
Douglas Weathers, UMaine Math MA student.
“Generating Functions: Mathematics You Can Count On”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Pizza at 3:15pm.

A generating function is a mathematical object that encodes information about a sequence of numbers in an accessible, manipulable way. We will build and use this tool over the course of three problems, developing or recalling some mathematical reasoning skills along the way.

Wednesday, September 24, 2014.  Math Colloquium.
Prof. Joshua Zelinsky, UMaine Mathematics.
“The ABC Conjecture, The Mason-Stothers Theorem, and Fermat’s Last Theorem”
3:30 – 4:20 pm, 100 Neville Hall. Snacks at 3:15pm.

The ABC Conjecture is a difficult conjecture in number theory. The conjecture is extremely strong in the sense that the conjecture implies many other open conjectures. The ABC Conjecture also implies a number of theorems whose known proofs are very difficult. This talk will present the motivation behind the ABC conjecture and sketch some of the nice consequences that would follow if we knew ABC.  This talk should be accessible to those with knowledge of calculus.

Wednesday, October 1, 2014.  Math Colloquium.
Nathan McNew, Dartmouth College Mathematics.
“Popular values of the largest prime divisor function”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Consider the largest prime factor of each of the integers in the interval [2,x] and let q(x) denote the prime number which shows up most often in this list.  In addition to using estimates of smooth numbers to investigate the behavior of this function as x tends to infinity, we look at the range of q(x) and see that it misses most of the primes.  We conjecture that the set of these “popular primes” is related to other interesting subsets of the prime numbers.

Wednesday, October 8, 2014.  Math Colloquium.
Prof. S. Jerry Farlow, UMaine Mathematics.
“Mathematics’ Greatest Paradox and other Minor Conundrums”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

There’s more than one way to define a paradox, and this talk will explore examples from Stanley J. Farlow’s new book Paradoxes In Mathematics.

Students and puzzle enthusiasts will find plenty of thought-provoking enjoyment mixed with a bit of painless mathematical instruction among the twenty-eight conundrums. Some of them involve counting, some infinity, and others draw on principles of geometry and arithmetic.

Wednesday, October 15, 2014.  Statistics Colloquium.
Prof. Ramesh Gupta, UMaine Statistics.
“Normality – A Myth: Skew Normal Distribution”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

In this presentation, it will be shown that, in most cases, the data is not normally distributed and is skewed (right or left). To analyze such data, we will introduce a skew normal distribution. Its historical development and applications in various fields, including psychometric, selection models, econometrics, financial market, portifolio selection and reliability problems, will be pointed out. The mathematical properties of the model will be explored and its similarities and dissimilarities with the normal distribution will be investigated. Some situations where the skew normal distribution arises will be mathematically studied. Finally, some key references will be pointed out.

Tuesday, October 21, 2014.  Math Colloquium.
Prof. Scott Taylor, Colby College Mathematics.
“Of Knots and Surfaces”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

One of the best ways of understanding knots in 3-dimensional space is by studying surfaces. In this talk, I’ll give an eclectic survey of old and new results on the relationship between knots and surface, including such gems as Seifert surfaces and bridge surfaces. I’ll explain some recent work which shows how the additivity properties of Seifert genus and bridge number can be generalized to the width of a knot. The additivity of width relies on a process called “untelescoping a bridge surface”. When a knot has a bridge surface that can’t be untelescoped, there is another useful invariant called “bridge distance”. It turns out that upper bounds on the bridge distance of a knot can be obtained from the genus of interesting surfaces with boundary on the knot. Most of the talk should be broadly accessible. Much of this is joint work with Maggy Tomova, Ryan Blair, Marion Campisi, and Jesse Johnson.

Wednesday, October 29, 2014.  Pizza pi seminar.
Dr. Sergey Lvin, UMaine Mathematics.
“How to swing wisely and efficiently”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Pizza at 3:15pm.

How do children achieve large amplitudes on a swing-set? Little kids use an external force (their parents). Older kids pump with their legs. Why does it work? As an educated adult, you can actually do even better!

A topic in differential equations, called parametric resonance, will teach you how to swing like a circus gymnast.

Wednesday, November 5, 2014.  Mathematics Colloquium.
Prof. Alanna Hoyer-Leitzel, Bowdoin Mathematics.
“Relative Equilibria in the 1+N-Vortex Problem”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

Studying the dynamics of vortex configurations with one large vortex and N smaller vortices has applications to physical electromagnetic systems and atmospheric science, as well as being a historical example of a Hamiltonian system. The problem is related to Newton’s n-body problem. The existence and linear stability of relative equilibria configurations of the (1+N)-vortex problem are examined. Such configurations are shown to be critical points of a special potential function, and their linear stability depends on the weighted Hessian of this potential. Algebraic geometry and some numerical methods are used to examine the bifurcations of critical points and stability specifically in the case of N=3.

Wednesday, November 12, 2014.  Math Colloquium.
Prof. George Markowsky, UMaine School of Computing and Information Science.
“The Not-So Golden Ratio”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

The golden ratio, also called by different authors the golden section, golden number, golden mean, divine proportion, and division in extreme and mean ratios, has captured the popular imagination and is discussed in many books and articles.  Generally, the mathematical properties of the golden ratio are correctly stated, but much of what is presented about the golden ratio in art, architecture, literature, and esthetics is false or seriously misleading. Unfortunately, these statements about the golden ratio have achieved the status of common knowledge and are widely repeated. Even current high  school geometry textbooks make many  incorrect statements about the golden ratio. This talk will set the record straight about the golden ratio. It will discuss its mathematical significance as well as some of the most commonly repeated falsehoods about it.

Friday, November 14, 2014.  Thesis Defense. (Advisor: David Hiebeler)
Jack Hill, UMaine Mathematics MA candidate.
“A Population on a Dynamic Landscape Driven by the Voter Model”
1:00 – 1:50 pm, 202 Shibles Hall.

This study examines a single population on a dynamic and spatially explicit voter model driven landscape. Outputs of interest, such as population density, are examined for both a set of computer simulations and a set of systems of differential equations. Comparing the outputs of both models not only provides insight into the accuracy of the mathematical models, but also reveals underlying characteristics of the simulation beyond the scope of the mathematical models.

Tuesday, November 18, 2014.  Thesis Defense. (Advisor: David Hiebeler)
Rachel Rier Carter, UMaine Mathematics MA candidate.
“Moment Differential Equations of a Community-Structured SIS Epidemiological Model”
9:15 – 10:05 am, 421 Neville Hall.

The SIS (Susceptible-Infected-Susceptible) epidemiological model is used to describe the spread of diseases which do not confer immunity to infected individuals. A model in which an SIS epidemic spreads through a population which is subdivided into communities is explored. The dynamics of the proportion of infected individuals per community are described with a set of differential equations, supported by data from computer simulations. Increasing the proportion of within-community interactions increases the variance of the proportion of infected individuals per household, but it has little effect on the variance of the infection level between different populations.

Wednesday, November 19, 2014.  Math Colloquium.
Prof. Tom Bellsky, UMaine Mathematics.
“Forecasting with Kalman filter data assimilation”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

The atmosphere behaves as a nonlinear, chaotic dynamical system, where small differences in initial conditions lead to great differences in future outcomes. Thus, accurately determining the current atmospheric state is a key step in short-term numerical weather forecasting. Data assimilation is a technique that combines model forecasts with observational data to produce a best guess of the current state. This talk will introduce Kalman filter data assimilation techniques under the context of weather prediction and briefly describe results for low-dimensional chaotic systems.

Wednesday, December 3, 2014.  Math Graduate Seminar.
Jennifer Fatula, UMaine Mathematics MA student.
“Linking number, Twist and Writhe in Knots, Links and Ribbons”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

This talk will discuss the motivation behind studying knots, links and ribbons as well as what they are and how equivalence is established.  We will discuss and explore invariants, and define linking number, twist and writhe. We will introduce and prove a discrete version of White’s formula that relates twist, writhe, and link in a ribbon.

Wednesday, December 10, 2014.  Math Graduate Seminar.
Matthew Brenc, UMaine Mathematics MA student.
“Uniformly distributed sequences”
3:30 – 4:20 pm, 421 Neville Hall. Snacks at 3:15pm.

Consider the sequence of points on the unit circle corresponding to all integer multiples of a fixed angle theta. We’ll examine the behavior of such sequences for all possible real values of theta and this will lead to the definition of a uniformly distributed sequence. After further discussion, we will tie everything back together by using the Weyl Criterion (Hermann Weyl [1916]) to prove that for certain values of theta, the resulting sequence of points on the circle is uniformly distributed.

Wednesday, February 4, 2015.  Pizza Pi Seminar.
Prof. Bob Franzosa, UMaine Mathematics.
“The Baseball Simulator – A Team-Statistics Based Program for Recreating MLB Seasons and for Use as a Baseball Analytics Lab”
3:30 – 4:20 pm, 208 Neville Hall. Snacks at 3:15pm.

I will introduce the Baseball Simulator, a baseball simulation program that I wrote in the mid-1980s as a way to have the Red Sox overcome misfortune and win World Series. I will discuss the model baseball game that the Simulator plays and the simplifying assumptions behind it, particularly the goal of developing an accurate recreation employing a minimum number of game statistics. I will briefly introduce the field of Baseball Analytics and some of the history behind it. I will further discuss how the Baseball Simulator can be used to investigate questions of interest in Baseball Analytics.

Wednesday, February 18, 2015.  Mathematics Colloquium.
Ewerton Rocha Vieira, Univ. Estadual de Campinas.
“How to apply homological algebra to obtain bifurcations in dynamical systems (a Conley index approach)”
3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

A challenging question in the study of dynamical systems is that of the existence of global bifurcations. The difficulty in detecting such bifurcation orbits is the fact that one must analyze the dynamical system globally. Topological techniques for global analysis are, therefore, a perfect fit for such an investigation. In particular, the Conley index theory has proven to be quite useful in this role. In this spirit we present the transition matrix, a Conley-index based algebraic transformation that tracks changes in index information under continuation and thereby identifies global bifurcations that could occur during the continuation. In first 25 min of the talk, we introduce singular homology, a pre-requisite for understanding the homological Conley index. And finally, in last 25 min, we will use the Conley index theory to define a transition matrix and explore its properties applied to bifurcation theory.

Wednesday, March 25, 2015.  Pizza Pi Seminar.
Prof. Carrie Diaz Eaton, Unity College Center for Biodiversity.
“Supporting faculty in mathematics and biology education”
3:30 – 4:20 pm, 208 Neville Hall. Pizza at 3:15pm.

At the interface of mathematics and biology emerges not only new research, but also resurging call for education to meet the demands of this research.  Reports suggest foundational courses in both statistics and calculus for the biology of the 21st century, as well as quantitative integration into the curriculum. I will discuss some approaches to determining the content and delivery structure.  Then I will discuss the support those who might teach it.  QUBES (Quantitative Undergraduate Biology Education and Synthesis) hosted a recent summit of mathematical biology educators and representatives from institutions and professional societies to identify urgent areas of need to move quantitative biology education forward – continuous faculty support. QUBES is an NSF funded project designed to advance quantitative biology education, primarily by offering a community of integrated faculty support.  I will discuss the context of the issues facing both the mathematical and biology education community, and discuss QUBES, its founding, and its mission to address the needs of today’s faculty at the interface of biology and mathematics.

Wednesday, April 1, 2015.  Mathematics Colloquium.
Prof. Katharine Ott, Bates College.
“Validated Numerics for Harmonic Analysis and Partial Differential
Equations”

3:30 – 4:20 pm, Hill Auditorium, Barrows (ESRB). Snacks at 3:15pm.

In this talk I will discuss ongoing research at the interface of Harmonic Analysis, Partial Differential Equations and Validated Numerics. A combination of methods from these three areas lead to effective tools for proving:
* Spectral properties of singular integral operators,
* Well-posedness results for boundary value problems in non-smooth domains,
* Sharpness of well-posedness results.
I will demonstrate the approach by discussing the case of the regularity problem for the Lamé system of elastostatics in domains with isolated singularities in two dimensions and with boundary data in L^p, 1<p<∞. The basic underlying principle is to reduce the problem at hand — which is notoriously difficult to be handled purely analytically — to checking whether a certain inequality holds for the values of the parameters involved belonging to a compact set. This last step is handled via Validated Numerics methods. This is joint work with Hussein Awala, Irina Mitrea, and Warwick Tucker.

Wednesday, April 8, 2015.  Mathematics Colloquium.
Prof. Chan Ieong Kuan, UMaine Mathematics.
“Arithmetic Functions and Complex Analysis”
3:30 – 4:20 pm, 208 Neville Hall. Snacks at 3:15pm.

In number theory, one often comes across arithmetic functions such as the divisor function, the sum of divisors function and the Euler phi-function. One can ask how large these functions are on average. In this talk, we will estimate the size of the averages via Dirichlet series and complex analysis. We will also adapt these methods to count the number of integer lattice points on a hyperboloid of one sheet (example: $x^2+y^2 = z^2 + 1$).

Wednesday, April 15, 2015.  Mathematics Colloquium.
Prof. Meredith Greer, Bates College.
“The DEs to your heart”
3:30 – 4:20 pm, 208 Neville Hall. Snacks at 3:15pm.

What brings together seemingly separate upper-level courses in biology and mathematics?  What tools can physics students use as part of understanding the world?  What topics in calculus show students that their efforts have a purpose?  What type of mathematics allows investigation of epidemics, softball pitching, lake ecology, and roller coasters?  One answer to all of the above is: differential equations (DEs).  This talk focuses on DEs in classes and in research projects, at all levels of the undergraduate curriculum.

Thursday, April 23, 2015.  MA Thesis Defense. (Advisor: Ben Weiss)
Douglas Weathers, UMaine Mathematics MA candidate.
“Decompositions of the Kuchment-Lvin Polynomials”
3:30 – 4:20 pm, 208 Neville Hall.

While studying the Radon transform in the context of medical tomography, Peter Kuchment and Sergey Lvin encountered an infinite family of differential polynomials $f_{n,lambda}(u)$ where $n$ is a positive integer, $lambda$ is a complex parameter, and $u$ is a smooth function with derivatives $du, d^2u, ldots, d^mu$, etc. It was discovered that (i) if $du - lambda u = 0$, then $f_{n,lambda}(u) = 0$ for all $n$ and that (ii) if $d^2u - lambda^2u = 0$, then $f_{n,lambda}(u) = 0$ for all odd $n$.

This leads to a natural question (iii): if $m ge 3$, does $d^mu - lambda^m u=0$ imply $f_{n,lambda}(u) = 0$ for a progression of $n$ whose common difference is $m$?

We divide $f_{n,lambda}(u)$ into a linear part, quadratic part, etc. according the degree of the terms. With an eye to the algebraic structure of the $f_{n,lambda}(u)$, we then use combinatorics to prove that all parts of the polynomial vanish if $du = lambda u$ and provide an alternate proof of identity (i). We then apply the same techniques to answer question (iii) negatively for the linear part $f^L_{n,lambda}(u)$ of $f_{n,lambda}(u)$ in the case that $lambda neq 0$: If $m ge 3$ is an integer such that $d^mu - lambda^m u = 0$ and $f_{n,lambda}^L(u) = 0$ for an integer $n ge 2$, then it must in fact be the case that $du - lambda u = 0$ or $d^2u - lambda^2 u = 0$.

These results identify the expressions $du - lambda u$ and $d^2u - lambda^2u$ as being key to the decomposition of the linear part of the Kuchment-Lvin polynomials. Schanuel’s conjecture on the transcendence of exponential functions with linearly independent exponents, if true, implies that our result for $f_{n,lambda}^L(u)$ gives the same result for $f_{n,lambda}(u)$.

Monday, May 4, 2015.  Number Theory Seminar.
Prof. David Rohrlich, Boston University.
Quaternionic Artin representations of Q
4:00 – 4:50 pm, 421 Neville Hall.

Wednesday, May 13, 2015.  Graduate Seminar.
Matthew Brenc, UMaine Mathematics MA candidate.
Embedding the Complete Graph in Orientable Surfaces using Rotation Systems
3:30 – 4:20 pm, 421 Neville Hall.

A rotation system on a graph is a specification of the order in which edges appear about each vertex. For any connected graph G, each rotation system on G corresponds to a cellular embedding of G into some orientable surface S, and vice versa. In this talk, we’ll consider a specific rotation system on the complete graph Kn and we will use the Euler characteristic  to determine which surface Kn embeds into under this system.

Wednesday, July 15, 2015.  Thesis Defense. (Advisors: Snyder and Atzema)
Joseph Arsenault, UMaine Mathematics MA candidate.
“Dedekind’s Role in the Development of Infinite Galois Theory”
9:00 – 9:50 am, 421 Neville Hall.

We provide the principal results concerning infinite Galois theory obtained by Richard Dedekind, described from both a modern perspective and from Dedekind’s point of view, as summarized in his 1901 article “On the Permutations of the Field of All Algebraic Numbers.”

Monday, August 10, 2015.  Thesis Defense. (Advisor: Khalil)
Brian Toner, UMaine Mathematics MA candidate.
“The Geometric Properties of Cells Exhibiting Huntington’s Disease”
9:00 – 9:50 am, 421 Neville Hall.

Huntington’s disease (HD) is a hereditary genetic neurological disorder which causes neuronal death.  It affects muscular coordination and mental state, and can lead to premature death.  HD is unique in the respect that in addition to the neurons, it affects gene expression in other tissues, including brain, blood and peripheral tissue.  It is thought that HD’s destructive behavior has an impact at the nuclear level, specifically the morphology of the chromosomes.

During the interphase, chromosomes tend to position themselves in a non-random way which allows them to perform their functions in tandem with the rest of the nucleus.  Gene-rich chromosomes on average are elongated and occupy regions which are closer to the centroid of the nucleus. Since disease states such as cancer and laminopathies have been shown to impact the positioning and shapes of interphase chromosomes, it was hypothesized that similar traits may occur in HD.  This thesis explores the effect HD has upon the architecture of the cell nucleus from human samples.

Data for this analysis was collected from blood samples from three groups of participants: individuals with early HD, moderate HD, and a control group.  Samples were stained using florescence markers and imaged using the cryoFISH method, a high-resolution Fluorescence In Situ Hybridization microscopy applied to cryosections which were approximately 150nm thick. DNA florescence probes were used to mark the nuclei, chromosomes 4, and 19 for one set of cells, chromosomes 5, and 22 for the other set of cells. The images collected for the experiment were segmented and analyzed using a combination of machine learning, thresholding, and detection algorithms.  The geometrical analysis revealed measurable changes in the structure and organization of chromosomes.  The progression of HD causes overall elongation of the nucleus.  Chromosome 4 becomes smaller and more compact in moderate HD and larger and more elongated in moderate HD.  Chromosome 5 is larger and is more often positioned at the periphery of the nucleus. Chromosomes 19 and 22 become more compact and are positioned near the centroid more often.  These changes in chromosome structure and organization could provide a new avenue for diagnosis of HD.