## Abstracts

Thursday, July 31, 2014. **Thesis defense.** (Advisor: Robert Franzosa)**Sophia Potoczak, ***UMaine Math MA candidate.***“A survey of graph embeddings into compact surfaces”**

10:00 – 10:50 am, 421 Neville Hall.

A prominent question of topological graph theory is “What type of surface can a nonplanar graph be embedded into?” This thesis has two main goals. First to provide a necessary background in topology and graph theory to understand the development of an embeddding algorithm. The main purpose is developing and proving a direct constructive embedding algorithm that takes as input the graph with a particular order of edges about each vertex. The embedding algorithm will not only determine which compact surface the graph can be embedded into, but also determines the particular embedding of the graph in the surface. The embedding algorithm is then used to investigate surfaces into which trees and a class of the complete bipartite graphs can be embedded. Further, the embedding algorithm is used to investigate non-separating graph embeddings.

Monday, July 28, 2014. **Thesis defense.** (Advisor: Natasha Speer)**Shahram Firouzian, ***UMaine Math MA candidate.***“Graduate Students’ Mathematical Knowledge for Teaching the Derivative and Applications of the Derivative”**

11:00 – 11:50 am, 108 Neville Hall.

Previous studies have indicated that effective mathematics teaching relies on teachers’ knowledge of both students’ thinking and mathematical content. Very little is known about the kind of knowledge teachers draw on when teaching general ideas related to the derivative in calculus or applied derivative problems. Graduate teaching assistants play important roles in the teaching of calculus however, very little is known about their knowledge for teaching. Using a well-established framework for elementary school teachers’ mathematical knowledge for teaching, mathematics graduate teaching assistants’ (GTA) mathematical knowledge for teaching the derivative and applied derivative problems was examined. Interview tasks targeted different domains of knowledge GTAs might access when talking about the concept of the derivative and solving applied derivative problems and while engaged in teaching-related work such as examining student solutions. Findings suggest that GTAs draw on mathematical knowledge for teaching when discussing derivative and applied derivative problems but that they also used knowledge of science. When presenting applied derivative examples, discussing sample student work from applied derivative problems, it was noticed that GTAs drew on something referred to as their integrated mathematics and science knowledge for teaching. It was also noticed that experienced GTAs have more organized and more accessible knowledge for teaching compared to novice GTAs. Implications for research and for novice college mathematics instructor professional development will also be discussed.

Monday, June 23, 2014. **Number theory lecture.****Joshua Zelinsky, ***University of Maine.***“Upper and lower bounds in integer complexity”**

11:00 – 11:50 am, 421 Neville Hall.

For an integer , Let be the minimum number of 1s needed to represent a positive integer as a product or sum of 1s without regard to the number of parentheses. For example, the equation shows that . This notion was first defined by Selfridge who asked what one can say about the asymptotic growth rate of . He noted that one has the easy inequality and asked if is asymptotic to this lower bound. This talk will focus on recent progress in answering Selfridge’s question.

Monday, June 23, 2014. **Number theory lecture.****Chip Snyder, ***University of Maine.***“On some Constructions using Marked Ruler and Compass”**

10:00 – 10:50 am, 421 Neville Hall.

We examine some geometric constructions using marked ruler and compass and outline a proof that a reguar 11-gon can be constructed using these tools.

Monday, June 23, 2014. **Number theory lecture.****Jonathan Sands, ***University of Vermont.***“Zeta functions of orders in quaternion algebras”**

9:00 – 9:50 am, 421 Neville Hall.

Thursday, May 15, 2014. **Number theory lecture.****Benjamin Weiss, ***University of Maine.***“Probabilistic Galois theory”**

2:15 – 3:00 pm, 421 Neville Hall.

Thursday, May 15, 2014. **Number theory lecture.****Pericles Dokos**.**“Automorphisms of Lubotzky-Phillips-Sarnak graphs”**

11:00 – 11:50 am, 421 Neville Hall.

Thursday, May 15, 2014. **Number theory lecture.****Florian Luca**, *Dartmouth College.***“Irregular primes”**

10:00 – 10:50 am, 421 Neville Hall.

Thursday, May 15, 2014. **Number theory lecture.****Elliot Benjamin**.**“Some New Fields That Satisfy the 2-Class Field Tower Conjecture for Imaginary Quadratic Number Fields with 2-Class Group of Rank 4”**

9:00 – 9:50 am, 421 Neville Hall.

In this talk I will show that there are new families of infinitely many imaginary quadratic number fields k with 2-class group of rank 4 such that k has infinite 2-class field tower. This lends support to the conjecture that all imaginary quadratic number fields k with of rank 4 have infinite 2-class field tower.

Wednesday, April 30, 2014. **Math Colloquium.****Prof. Nigel Pitt**, *UMaine Mathematics.***“The Circle Problem”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15pm.)

A very classical problem in number theory considers a large circle centered at the origin in the Euclidean plane, and asks how many points with integer coordinates are inside it. It isn’t very difficult to see that the answer should be approximately the area A of the circle, but it is much harder to understand the size of the error. We will consider two approaches to this question, which is known as the circle problem. The first is an elementary geometric argument, due to Gauss, which shows that the error is no larger than for some constant C. The second, due to Sierpinski, uses harmonic analysis to show that the error is no more than for any . Along the way we will discuss the Poisson summation formula.

The first part of this talk will be accessible to a general audience. The second part requires more experience to understand all the details but the general ideas should be understandable to anyone who has seen some real or complex analysis.

Monday, April 28, 2014. **Thesis defense.** (Advisor: André Khalil)**Derrick Cox**, *UMaine Mathematics MA candidate.***“Computational Analysis of Mammograms Using the Metric Space Technique”**

4:00 pm, 227 Neville Hall.

The mathematical formalism called the metric space technique is used to quantitatively compare the complexity of mammogram data. The technique uses one-dimensional output functions to find differences in the morphological data inherent to the mammogram images. The tool is used to analyze 1044 mammograms with a resolution between 43.5-50 micrometers per pixel. Every mammogram sub-image is compared to a standard image generating standardized metric spaces. Also, the sub-images are compared to the same locality in the mammogram of the opposite breast to form a ‘relative’ metric space. The distribution of the complexity values are generated, and analyzed. As an additional approach, this study stresses the importance of studying the output functions themselves on a threshold by threshold basis. From the distributions generated, we develop a probabilistic model, with the capability of classifying arbitrary mammogram sub-images. The probabilistic model is applied to whole mammograms as a means to identify regions previously identified as mass tissue by a trained radiologist.

Wednesday, April 23, 2014. **Math Colloquium.****Prof. Ben Weiss**, *UMaine Mathematics.***“The card game Set“**

3:30 – 4:20 pm, 110 Little Hall. (Snacks at 3:15pm.)

*SET* is a fun pattern matching game involving picture cards with various shapes and colors. We will use the patterns in the game to discuss such mathematical concepts as lines in non-real space, sum-free subsets and other combinatorial and probabilistic constructions.

Friday, April 18, 2014. **Thesis defense.** (Advisor: Eisso Atzema)**Amber Hathaway**, *UMaine Mathematics MA candidate.***“Emmy Noether’s Theorem on Integral Invariants in the Context of the Calculus of Variations”**

3:30 – 4:20 pm, 421 Neville Hall.

Stated informally, Emmy Noether’s theorem asserts that if a system has a symmetry property, then one can obtain a corresponding conservation law for the system. For example, the Lagrangian of a ball thrown into the air, that is, the difference between the ball’s kinetic and potential energies, will be the same regardless of the time at which the ball is tossed into the air. Noether’s theorem assures that a conservation law for this system exists and provides the means to find it. In this talk, a brief biographical sketch of Emmy Noether will be given, followed by an overview of the history of the calculus of variations and of general relativity to provide context for Noether’s theorem. A formal statement and proof of Noether’s theorem will be given, as will an example to demonstrate her theorem.

Wednesday, April 16, 2014. **Pizza Pi.****Dr. Eisso Atzema**, *UMaine Mathematics ***“Classifying Quadrilaterals”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15 pm.)

In the definitions of Book 1 of Euclid’s Elements one can find a rudimentary classification of the quadrilaterals. By and large, the same classification is still taught today. Over time, however, various changes from Euclid became commonplace and at least one new type of quadrilateral was introduced. In this presentation, we will look at the history of the classification of quadrilaterals from the mid-16th century through the 19th century. In close connection with this, we will also have a look at the history of the notion of the general or “irregular” quadrilateral, i.e. the class of quadrilaterals generally not included in the usual classifications of the quadrilateral.

This talk will be accessible to anyone who has completed high school mathematics.

Wednesday, April 9, 2014. **Math Colloquium.****Prof. Justin Sukiennik**, *Colby College***“From Transcendental Numbers to Diophantine Approximation”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15 pm.)

Before mathematicians knew about countability, Joseph Liouville proved that transcendental numbers existed in 1844. He used an approximation method that was later refined by several mathematicians whose research led to a new field called Diophantine approximation. This field eventually led to discoveries about Diophantine equations. In 1955, Klaus Roth made the final best possible refinement. In this talk, we will investigate Liouville’s initial result, examine the existence of transcendental numbers, and the progress and history made to extract the best possible result.

Wednesday, April 2, 2014. **Math Colloquium.****Shawn Firouzian**, *UMaine Mathematics MA student***“Knowledge for Teaching Applied Problems in Calculus”**

3:30 – 4:20 pm, 110 Little Hall. (Snacks at 3:15 pm.)

Previous studies have indicated that effective mathematics teaching relies on teachers’ knowledge of both students’ thinking and mathematical content. Very little is known about the knowledge of science that teachers use when teaching application problems in mathematics. I will present information about mathematicians teaching applied derivative problems. We will look at some math problems and consider the kind of knowledge teachers draw on when teaching applied concepts in calculus.

Wednesday, March 26, 2014. **Math Colloquium.****Prof. Andrew Knightly**, *UMaine Mathematics***“Representation theory and the trace formula”**

3:30 – 4:20 pm, 421 Neville Hall. (Snacks at 3:15 pm in 419 Neville.)

In the summer of 2013, Carl Ragsdale received research support from the math department for writing a paper based on his UMaine MA thesis. His main result is an explicit calculation of matrix coefficients of a special type of irreducible representation of the group of two-by-two invertible matrices over the p-adic numbers. In this talk I will briefly motivate the problem as coming from a nonabelian analog of Fourier theory. Then I will explain, in the much simpler context of finite groups, what representations are, and how their matrix coefficients can be used. This will involve a rudimentary version of the trace formula (which generalizes Poisson summation). This talk should be accessible to those familiar with basic group theory and linear algebra.

Wednesday, March 19, 2014. **Math Colloquium.****Prof. Neil Comins**, *UMaine Physics & Astronomy***“So You Think You Know Tides”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15 pm.)

Most people who learn about tides are taught that they are due to the difference in the gravitational force from the Moon across the Earth: The oceans on the side of the Earth closest to the Moon are pulled most by it; hence they are pulled upward to create a high tide under the Moon. The oceans on the side of the Earth farthest from the Moon feel less gravitational attraction from it than does the Earth beneath them, so these oceans are “left behind,” meaning they pull away from the Earth opposite the direction of the Moon, thereby creating a second high tide on the opposite side of the

Earth from the first high tide.

In this talk, using just algebra, I will explain how tides really work.

Wednesday, February 26, 2014. **Math Colloquium.****Prof. Dawn Nelson**, *Bates College Mathematics***“Identification Numbers and the Mathematics behind Them”**

4:45 – 5:40 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 4:30 pm.)

In today’s digital age books, food, financial transactions, and even people are identified by numbers. Major problems can result if these numbers are transmitted or stored incorrectly: Imagine your hard earned paycheck being deposited in someone else’s bank account. Many techniques have been designed to identify errors in transmission and record keeping. In this talk we will discuss several *check digit schemes,* their strengths, their weaknesses, and the mathematics behind them. We will start with schemes based on modular arithmetic used for UPCs and credit card numbers. We will conclude with a scheme used by the German Bundesbank that is based on permutations and dihedral groups.

Wednesday, February 26, 2014. **Math Colloquium.****Prof. Thomas Bellsky**, *Arizona State University***“Stability of localized structure for a semi-arid climate model”**

3:00 – 4:00 pm, 100 Neville Hall.

This talk will discuss the interaction of pulses in coupled reaction-diffusion systems, and the application of such systems in modeling the stability of vegetative patterns in semi-arid climates. For a particular family of fast-slow, weakly-damped reaction-diffusion systems, we rigorously derive laws of motion for multi-pulses. We establish the existence of a manifold of quasi-steady multi-pulse solutions and identify a ‘normal-hyperbolicity’ condition, which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result rigorously demonstrates the stability of the manifold of pulse solutions.

Wednesday, February 19, 2014. **Pizza Pi Seminar.****Kevin Roberge**, *UMaine Mathematics Instructor***“Puzzling Penrose Pathologies”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Pizza at 3:15 pm.)

In the spirit of my everyday erratic investigations this presentation combines Turing machines, Penrose tilings, cantor sets and noncommutative geometry in a gluttonous indulgence of mathematical variety.

The first half of the talk will be accessible to a wide audience, as it will feature historical background, lots of neat images, and basic computations that require nothing beyond high school mathematics. Thus the first half will be * 1 star (high school level). The second half, in the tradition of academic colloquia, will feature an abrupt spike in abstraction and become ***** 5 star (grad level).

Wednesday, February 12, 2014. **Mathematics Colloquium.****Seth Albert**, *UMaine Math graduating senior***“The Renaissance Actuary”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15 pm.)

The actuarial career used to be the best-kept secret for Math majors. Now, more is required of actuaries and of STEM majors in every career. From my summer experience at Unum to the success of Nate Silver’s 538 blog, from Moneyball to teaching Calculus, this talk will describe what is needed beyond the technical expertise in today’s world.

Wednesday, January 29, 2014. **Mathematics Colloquium.****Prof. Catherine Buell**, *Bates College Mathematics***“Involutions and Fixing the World: Symmetric Spaces”**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Snacks at 3:15 pm.)

Symmetric spaces are studied in both mathematics (through algebraic and geometric theory) and physics (in the study of integrable systems and quantum theory as well as applications to carbon nanotubes). These spaces have unique structure determined by an involution (an order-two automorphism) on a group. During the talk the audience will be introduced to involutions and fixed points, discover various symmetries in the plane and in matrix groups, and learn current results and open questions in the field.

Wednesday, January 22, 2014. **Pizza Pi Seminar.****Jon Janelle**, *UMaine MST student***Modeling a Zombie Outbreak Using Systems of Ordinary Differential Equations**

3:30 – 4:20 pm, Hill Auditorium, 165 Barrows Hall (ESRB). (Pizza at 3:15 pm.)

Zombies, the flesh-eating undead terrors with which we are all familiar, have become a fixture in contemporary pop culture. According to MSNBC economics columnist Jon Ogg, zombie-related movies, TV shows, books, video games, and a host of other goods generated an estimated latex mathbf{e^x}$, and x that came from medical imaging”

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

We will introduce an infinite set of previously unknown differential identities for certain elementary functions, Including trigonometric and hyperbolic sines and cosines, exponential and linear functions. These identities resemble the binomial formula and they initially appeared as a byproduct of our medical imaging research. Some of the results are published in The American Mathematical Monthly in August-September 2013, some are new.

Students are welcome to attend.

Wednesday, September 25, 2013. **Pizza Pi Seminar.****Brian Toner**, *UMaine Mathematics MA student***“Math that Learns, the Mathematical Principles of Neural Networks and Machine Learning”**

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

A brief overview of Neural Networks, Machine Learning and the

mathematical machinery behind them.

Wednesday, September 18, 2013. **Mathematics ****Colloquium.****Prof. Robert Franzosa**, *UMaine Mathematics***“You Cannot Beat Bob In The Triangle Game Implies That A Beating Heart Cannot Respond In A Continuous Manner To A Stimulus Applied With Continuously Varying Strength And Timing”**

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

We will explore a game played on a triangular grid and see how properties of the game lead to the 2-dimensional Brouwer Fixed Point Theorem. Then we will see some interesting consequences of the Brouwer Fixed Point Theorem including the You-Are-Here Scenario and the No-Retraction Theorem. Finaly, one of the consequences will be applied to a simple heart-beat model to prove that a beating heart cannot respond in a continuous manner to an applied stimulus.