Title: Quantum Computing and Grover's Algorithm

Time: Tuesday, March 11th at 4:40 - 5:35pm

Place: duPont 104

Refreshments at 4:20pm.

Title: Them Infamous Exploding Dots

Time: Tuesday, March 18th at 12:20 - 1:15pm

Place: Hillel Multipurpose Room

Refreshments providedSchedule for Winter 2014

Title: A DOZEN PROOFS THAT 1 = 2: A Misguided Review of Mathematics

Time: Tuesday, March 18th at 4:40 - 5:30pm

Place: Leyburn Northern Auditorum

Title:

Time: Wednesday, March 19th at 4:40 - 5:35pm

Place: duPont 104

Refreshments at 4:20pm.

Title: Folds and cuts: mathematics and origami

Time: Tuesday, April 1st at 4:40 - 5:35 pm.

Place: Hillel Multipurpose room

Refreshments at 4:20pm

Wenda Tu ('14)

Title: Finding cycles in the kth power digraphs over integers modulo a prime and over Gaussian integers modulo a Gaussian prime

Time: Wednesday, April 2nd at 3:35 - 4:30 pm.

Place: duPont 104

Refreshments/Reception at 3:15

W&L Math Department

Time: Tuesday, February 25th at 4:40pm

Place: duPont 104

Refreshments at 4:20 in duPont

Abstract: You're not alone; it's happened to all of us. When the moment presented itself, you balked. You stuttered. You stammered. And all you could come up with was x^2 + 2x + 3. That's the best you could do. And now, youre lying in bed, and just as you drift off to sleep, it comes to you: the perfect polynomial. If only you had thought of it at the right moment you would have been the life of the party! You would have put that bully in his place! You would have secured that promotion! Don't let this happen to you! In this talk, we'll discuss some tricks to help you quickly decide whether a polynomial is irreducible, and some other tricks to help you come up with an irreducible or reducible polynomial on a moment's notice. It will just be left to you to discern which one is called for in your particular social situation!

Ph.D. student at the University of Texas - Austin

Time: Wednesday, January 8th at 4:40pm

Place: duPont 104

Refreshments at 4:20 in duPont

Abstract: We begin by investigating the well-known Fibonacci sequence and observing that there are many surprising relationships between algebraic identities involving the golden ratio, and recurrence relations involving the Fibonacci sequence. We demonstrate that such relationships are no coincidence - many more nontrivial examples stem from a much broader class of recurrence relations.

Time: Thursday, October 3 at 4:40pm

Place: duPont 104

Refreshments at 4:20pm in duPont

Student Summer Research Presentations - Parent's Weekend

Colin Hagenbarth

Alyssa Hardnett

Time: Friday October 4 at 3:30

Place: duPont 104

Refreshments at 3:15 in duPont

Title: True or False?

Time: Thursday, October 17 at 4:40pm

Place: duPont 104

Refreshments at 4:20 in duPont

Abstract: Various statements will be presented and, with the help of the audience, we will attempt to determine whether each statement is true or false. Monetary rewards will be given for the best participants from the audience.

Carrie Finch, W&L Mathemematics Professor

Time: Thursday, Nov. 7 at 4:40pm

Place: duPont 104

Abstract: Come and learn about summer research opportunities in math and the physical sciences.

Time/Place: Friday, Nov. 8th, 2:30-4pm in Baker 207

Liz is a mathematics professor at Haverford College in Connecticut and Susan is an actuary.

All are invited to come and talk about graduate school in math or what it's like to be an actuary.

Time: Tuesday, November 12 at 4:40pm

Place: duPont 104

Refreshments at 4:20 in duPont

Abstract: Toeplitz matrices are square matrices which are constant along its diagonals. Recently, Ye and Lim showed that every matrix is the product of Toeplitz matrices. I will outline their proof. I addition, I will discuss work of one of my students who extended this result and showed that the set of all square matrices is the product of algebras of Toeplitz matrices. This talk will be very general and is definitely suited for both faculty and students.

Time: Wednesday, November 20 at 3:35pm

Place: duPont 104

Refreshments at 3:15 in duPont

Abstract: Can every conceivable mathematical statement, at least in principle, be either proved or disproved? Here "in principle" means that we ignore any constraints on time, resources, or ingenuity. This question was posed by David Hilbert around 1920. Hilbert also proposed a strategy, now known as Hilbert's Program, to obtain a "yes" answer to this question. In modern terms, he sought to prove the existence of a computer program which would take as an input any mathematical statement, and after some finite number of steps, tell us (correctly) whether the statement is true or false. Such a program actually exists in limited contexts; two famous examples are Presburger Arithmetic (basically number theory without multiplication) and the theory of the ordered field of real numbers. However, Kurt Goedel proved in the 1930s that the general form of Hilbert's Program is doomed: there exist simple mathematical statements about natural numbers which can neither be proved nor disproved. Moreover, this phenomenon persists even if we augment our mathematical arsenal by including new fundamental assumptions (axioms) about mathematical objects.

Title: The Elephant is a Rope: Metaphor and Mathematics

Time: Thursday, February 28 at 3:30pm

Place: Northern Auditorium, Leyburn Library

**Abstract: **George Lakoff and Rafael Nunez have sketched a linguistic/cognitive science description of the origins of elementary notions of mathematics through the mechanism of metaphor. Extending their idea to higher mathematics, a story emerges when we consider how metaphor changes over time. The talk will focus on these notions as a lens through which to view the history and development of mathematics.

John McCleary is Professor of Mathematics on the Elizabeth Stillman Williams Chair of Mathematics at Vassar College. His areas of research include topology and geometry and their histories.

Title: Diophantine equations with generalized Fibonacci numbers

Time: Thursday, March 21 at 4:40pm

Place: Robinson Hall, Room 6

*Refreshments in Robinson Hall 2 at 4:20pm

Praphat Fernandes (Emory University)

Title: Geometry of Graph Braid Groups

Time: Thursday, March 28 at 4:40pm

Place: Robinson Hall, Room 6

*Refreshments in Robinson Hall 2 at 4:20pm

**Abstract:** Given a set number of robots on a fixed system of rails, one can consider the set of all their possible configurations on the system of rails. Additionally one can consider their set of configurations geometrically, so that configurations which are geometrically close, correspond to configurations which are intuitively close on the system of rails. We will then use this space of configurations to define the graph braid groups, and use the geometry of these spaces to study the geometry of the group.

Title: How far can you see in the woods?

An introduction to the geometry of numbers.

Time: Wednesday, April 3 at 4:40pm

Place: Robinson Hall, Room 6

Refreshments in Robinson Hall 2 at 4:20pm

Time: 4:40-5:35, Thursday, September 20

Place: Robinson Hall, Room 6

Refreshments at 4:20 in Robinson Hall 2

Abstract: In this talk we investigate the factorization of trinomials of the form x^n+cx^{n-1}+ d in Z[x]. We then use these results about trinomials to prove results about the factorization of polynomials of the form x^n+c(x^{n-1}+...+x+1) in Z[x].

Time: 4:40 - 5:35, Tuesday, September 25

Place: Robinson Hall, Room 6

Refreshments at 4:20 in Robinson Hall 2

Abstract:

The Roanoke Valley Reef is a satellite reef of the Hyperbolic Crochet Coral Reef, a project of the Institute for Figuring in Los Angeles. The Hyperbolic Crochet Coral Reef has been exhibited in museums and art galleries around the world, including the Natural History Museum of the Smithsonian. The project combines mathematics, the arts, environmental science and other disciplines in an exciting way. The Roanoke Valley Reef is sponsored by Roanoke College and will open for exhibition in Olin Gallery in January 2013. Everyone is welcome to contribute to this unique community art project. This talk will explain the reef project, including an introduction to the mathematics involved.

The Institute For Figuring Hyperbolic Crochet Coral Reef, with urchins by Christine Wertheim and sea slug by Marianne Midelburg. Photo (C) The Institute For Figuring (by Alyssa Gorelick).

Presentation by Colin Bracis (‘03) representative of Mercer, Inc.

Time: 4:40 pm, Monday, October 15

Place: Robinson Hall, Room 6

Refreshments at 4:20 in Robinson Hall 2

Nathan Feldman

Time: 4:30pm, Thursday, November 1

Place: Northern Auditorium, Leyburn Library

Student Summer Research Presentations

Time: 3:35 pm, Friday November 2

Place: Robinson Hall, Room 6

Refreshments at 3:15 in Robinson Hall 2

Michael Dorff, BYU, currently at the MAA Headquarters in Washington DC

Time: 4:40 pm, Wednesday, November 14

Place: Robinson Hall, Room 6

Refreshments at 4:20 in Robinson 2

Abstract: In high school geometry we learn that the shortest path between two points is a line. In this talk we explore this idea in several different settings. First, we apply this idea to finding the shortest path connecting four points. Then we move this idea up a dimension and look at a few equivalent ideas in terms of surfaces in 3-dimensional space. Surprisingly, these first two settings are connected through soap films that result when a wire frame is dipped into soap solution. We use a hands-on approach to look at the geometry of some specific soap films or "minimal surfaces".

Discrete Mathematics, theoretical computer science, additive number theory and ergodic theory

Carrie Finch, W&L Mathematics Professor

Time: 12:40pm , Thursday, November 15

Place: Hillel Multipurpose Room

Time: 12:30 p.m., Wednesday, November 30

Place: Hillel Multi-purpose Room

Lunch will be paid by the Office of the Dean.

**Abstract:** The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2011 to John Milnor of the Institute for Mathematical Sciences, Stony Brook University, New York "for pioneering discoveries in topology, geometry and algebra.

Time: 4:40 p.m., Thursday, December 1st

Place Robinson 6

Refreshments at 4:15 pm

**Abstract:** Those who understand interest theory can be informed borrowers, making intelligent choices about mortgages and other loans, and they can also be wise investors. The students in this class learn now investment grows over time using mathematically precise methods. This course also helps prepare students for the actuarial exam, 2/FM.

In this "Visitor's Guide" talk, several annuity problems will be introduced incluing student loan payments and car loan payments. The last example in this talk is a practice exam problem from actuarial exam 2/FM, which uses put-call parity.

Time: 4:40 pm, October 11

Place Robinson Hall, Room 6

Refreshments at 4;15 in Robinson 2

**Abstract: **Number theory is the elegant branch of pure mathematics devoted to the study of numbers, especially the integers. In this talk, we highlight some of the interesting problems and techniques that arise in elementary number theory. Don't let the name fool you -- this means number theory that does not rely on tools from other mathematical fields.

Time: 4:40 pm, Monday, October 24

Place: Robinson Hall, Room 6

Time: 3:35 pm, Friday October 28

Place: Robinson Hall, Room 6

Time: 3:35 pm, Thursday, November 10

Place: Robinson Hall, Room 6

Kyle Parsons, Washington & Lee University, Class of 2011

Time: 4:00 - 5:00pm, Monday, May 9

Place: Robinson Hall, Room 6

Abstract: Arithmetic progressions in permutations have been studied for over sixty years, often under other names. Our focus is on permutations with progressions of distance 1 and rise 1 or 2. We generalize some of the previous results by examining circular permutations and modular progressions. We also find an n-to-1 correspondence from regular permutations with modular progressions to circular permutations with regular progressions.

Michele Theroux, Class of 2007

Time: 3:35 - 4:30, Thursday, May 12

Place: Robinson Hall, Room 6

Abstract: I will discuss life after graduation, how I became a torpedo analyst working as a contractor for the Navy and what torpedo analysis entails.

Lu Li, Washington & Lee University, Class of 2011

Time: 2:00-3:00 p.m., Friday, May 13

Place: Robinson Hall, Room 6

*W&L math professors will present an overview of the ideas, problems, and theories that are discussed in upper-level math courses to give calculus students and others a glimpse of upper-level mathematics.* *These survey talks should be valuable and accessible to anyone with an interest in mathematics, having a calculus background.*

Paul Bourdon, Washington & Lee University

Time: 3:35 - 4:30pm, Thursday, January 20th

Place: Robinson Hall, Room 6

Jacob Siehler, Washington & Lee University

Time: 3:35 - 4:30pm, Wednesday, February 2nd

Place: Robinson Hall, Room 6

Title: A Visitor's Guide to Math 321-322: Abstract Algebra

Wayne Dymacek, Washington & Lee University

Time: 3:35 - 4:30pm, Thursday, February 17th

Place: Robinson Hall, Room 6

Paul Humke, Washington & Lee University

Time: 3:35 - 4:30pm, Wednesday, March 9th

Place: Robinson Hall, Room 6

Adam Berliner, St. Olaf College

Time: 3:35 - 4:30pm, Thursday, March 24th

Place: Robinson Hall, Room 6

**Abstract: **For centuries, people have been fascinated and even sometimes preoccupied with puzzles and games. In this talk we'll learn some elementary graph theory. Using this along with a small amount of linear algebra and some mathematical savvy, we'll discuss how puzzles such as Magic Squares and Sudoku and games such as Othello can motivate some pretty interesting and deep mathematics.

Nathan Feldman, Washington & Lee University

Time: 3:35 - 4:30pm, Thursday, March 31st

Place: Robinson Hall, Room 6

Wenling Shang, Washington & Lee University, Class of 2011

Time: 3:35 - 4:30pm, Thursday, April 7

Place: Robinson Hall, Room 6

Presentation by Colin Bracis (‘03) and Agata Kasza ('10), representatives of Mercer, Inc.

Time: 4:40 p.m., Thursday, September 16th

Place: Robinson Hall, Room 6

Josie Ryan, Lander University

Time: 3:35-4:30p.m., Wednesday, September 22

Place: Robinson Hall, Room 6

**Abstract: **We will use origami, including the flapping crane and Thomas Hull's PHiZZ units, to demonstrate concepts from graph theory with applications to 2- and 3-colorings of maps. We will discuss the edge colorings of soccer balls and bucky balls and create a coloring of a dodecahedron. We will use Euler's theorem to explain our results. We will look at a proof by Kempe and Heawood (1879 and 1890) that states: If $G$ is a planar graph, then $\chi{G}\leq 5$ and quickly discuss the work by Koch, Appel, and Haken which lowered the number of require colors to four.

Ericson Davis, Logistics Management Institute, W&L Class of 2003

Time: 4:40-5:35 p.m., Thursday, October 7

Place: Robinson Hall, Room 6

Meagan Herald, Virginia Military Institute

Time: 3:35-4:30 p.m., Thursday, October 28

Place: Robinson Hall, Room 6

**Abstract: **A properly functioning immune system is required to maintain a healthy respiratory system. Dysfunctions in this system can cause bacterial colonization and chronic inflammation which in turn lead to damaged respiratory tissue, potential respiratory failure or even death. In order to manage respiratory inflammation and bacterial colonization, therapies need to focus on improving the function and activation levels of innate immune cells and immune secretions. Mathematical biology can be used to identify potential therapeutic targets, focusing experiments and describing the structures underlying complex biological systems.

We will explore two different models to understand the dynamics of bacterial infections in patients with cystic fibrosis. The first model consists of a system of nonlinear ordinary differential equations to describe interactions of the immune response. This model demonstrates, through bifurcation analysis, that nonaggressive bacteria are able to initiate chronic inflammation and suggests that therapies targeted towards restoring innate immune function will be vital to managing inflammatory respiratory diseases. The second model considers the interactions of a quorum sensing bacteria and the consequences of a slow moving mucociliary tract. This system of nonlinear partial differential equations allows us to determine what conditions promote biofilm formation and when the biofilm formations will occur.

Time: 3:35-4:40 p.m., Friday, November 5

Place: Robinson Hall, Room 6

Abstract: Suppose that each of two parties "Alice and Bob" possesses one particle of a two-particle entangled quantum system and that Alice wishes to send a message to Bob via her particle. Roughly speaking, the more entangled their shared system, the greater the possible efficiency of the communication between Alice and Bob using the system. The precise relationship between degree of entanglement and communication efficiency is not understood. We will discuss this relationship.

Abstract: In 1960, W. Sierpinski discovered that k = 1511380746462593381 has the property that k · 2 n + 1 is composite for any positive integer n. Because of his discovery, we call any positive integer k such that k · 2 n + 1 is always composite a Sierpinski number. In 2009, F. Luca and J. Mejıa discovered that there are infinitely many numbers in the Fibonacci sequence that are also Sierpinski numbers. They also found that there are infinitely many numbers in the Fibonacci sequence that are also Riesel numbers - a positive integer k such that k · 2 n - 1 is always composite for any positive integer n. In this talk, we show that there exists infinitely many Lucas-Riesel and Lucas-Fibonacci numbers for the Lucas sequence which is a generalized Fibonacci sequence which begins with 2 and 1.

Dan Baczkowsi, Washington and Lee

Time: 3:30 p.m., Wednesday March 3

Place: Robinson Hall, Room 6

**Abstract**: By "Diophantine equation" we merely mean an equation in which we are only concerned with positive integer solutions. Hence, the goal is to completely solve a particular Diophantine equation or to determine if the equation has finitely many solutions. Diophantine equations involving factorials have predominantly been studied utilizing techniques from the pulchritudinous subject of number theory. Among them are some outstanding problems aging over 130 years. The focus of the presentation will be on the improvements of problems investigated by Florian Luca. Those improvements are joint work with Florian Luca, Michael Filaseta, and Ognian Trifonov.

Paul Bourdon, Washington and Lee

Time: 3:30 p.m., Wednesday March 17

Place: Robinson Hall, Room 6

**Abstract:** By sliding a coin across a dining table to your friend "Bob", you can convey one of two messages: "heads" (which might mean "pass the salt") or "tails" (which might mean "pass the pepper"). However, if you and Bob share two fully entangled "quantum coins" (one in Bob's possession and one in yours), you can send one of four different messages with your quantum coin. When Bob looks at your quantum coin he will see either heads (H) or tails (T), but you can prepare your coin in such a way that when he looks at both your coin and his, he will see exactly that sequence in {HH, HT, TH, TT} that you wish him to see, and thus receive one of four messages from your two-sided quantum coin. This is a simple example of the magic of dense coding, in which the quantum coins might be photons (with, e.g., heads = vertically polarized; tails = horizontally polarized). In general, dense coding involves two parties, customarily called Alice and Bob, with each assumed to possess one of a pair of entangled d-dimensional quantum particles known as qudits. If the pair of qudits that Alice and Bob share is fully entangled, then in theory Alice can convey one of d-squared messages to Bob via her qudit. These messages are prepared/encoded by Alice by applying to her qudit a physical operation modeled by a d x d unitary matrix; moreover, in order that the messages encoded by Alice never be misinterpreted by Bob, the messages must correspond to orthogonal vectors in the state space of the two-qudit system that Alice and Bob share. The mathematics of this situation will be the focus of the talk, with the principal research-level issue being: how is the number of distinct messages that Alice may encode related to the degree of entanglement of the two-qudit system she and Bob use as the basis for their communication?

Lenny Jones, Shippensburg University

Time: 4:30 p.m., Thursday March 25*

Place: Robinson Hall, Room 6

1. Does there exist a positive integer k such that s_n is composite for all integers greater than or equal to 1?

2. If the answer to question 1. is yes, then can we find a smallest such positive integer k?

Lea Lanz, Virginia Military Institute

Time: 3:30 p.m., Wednesday April 7*

P. Jameson Graber, University of Virginia

Time: 3:30 p.m., Wednesday May 5***Abstract:** Many real world engineering problems, for instance the problem of stabilizing an acoustic chamber, can be modeled mathematically and studied from the point of view of mathematical control theory. The basic philosophy of control theory can be summarized in a three step process: (1) model thephysical system using a system of partial differential equations; (2) study the mathematical properties of the system of equations; and (3) alter the model to achieve desired results. In this presentation, I will use the structural acoustic model as an example to illustrate the basic philosophy of control theory. Of the three basic steps of control theory outlined above, step (2) will receive the most attention. I will explain the concept of a well-posed dynamical system and attempt to describe in general terms the tools in analysis used to prove that a dynamical system is well-posed. Additionally I will introduce topics relevant to control, including asymptotic and uniform decay rates of solutions of a dynamical system. This will lead to a discussion of how adding controls can alter the behavior of a dynamical system to obtain optimal results. I will conclude the presentation by describing how controls can be added to optimize stability in the structural acoustic model, and why this is important in real-world applications.

*Refreshments at 3:15 p.m. in Robinson Hall 2

**"When you can't be with the one you love, take a slow bus to Budapest: A combinatorial look at arithmetic progressions in permutations.**

Wayne M. Dymacek, W&L Mathematics Department

Time: 3:30pm, Wednesday, September 23*

Place: Robinson Hall, Room 6

**Careers in Actuarial Science**

Presentation by Mercer

Time: 4:40, Wednesday, October 7*

Place: Robinson Hall, Room 6

**Spectral Pictures, Special Generators, and Infinite Checkerboards**

Nathan Feldman, W&L Mathematics Department

Time: 3:30pm, Wednesday, October 21*

Place: Robinson Hall, Room 6

**W&L Student Presentations**

Sarah Merritt & Cliff Gaddy, Neville Fogarty, and Kelsey Wright

Time: 4:40pm, Friday, October 30*

Place: Robinson Hall, Room 6

**Mathematics and Air Traffic Flow Management**

Christy Spofford Bittle

Time: 3:30pm, Wednesday, November 4*

Place: Robinson Hall, Room 6

**Abstract:** Demand in the National Airspace System is expected to double or triple by the year 2025. During this time, it is unlikely that airpsace infrastructure, such as new runways and airports, will alone be sufficient to meet the increased demand. Current avenues of research look to not only alleviate current congestion problems in the airspace but also to anticipate and address new problems that will arise with this increased demand. I will discuss some of these avenues of research, as well as how my background in mathematics relates to the research being done.

**Cyclotomic polynomials, their gcd, and their resultants**

Greg Dresden, W&L Mathematics Department

Time: 3:30pm, Wednesday, November 11*

Place: Robinson Hall, Room 6

**Abstract: **Recall that the cyclotomic polynomials, which begin x - 1, x + 1, x^2 + x + 1, x^2 + 1, ... , are the irreducible factors of x^n - 1. We show that most of the time we can give a linear combination of two cyclotomic polynomials that is equal to 1, but sometimes we can only get p, a prime. We show how this lemma is related to the resultant.

*Refreshments 20 minutes before the talk begins in Robinson Hall, Room 2

**Solving Graph Isomorphisms via Hyperdimensional Noisewith excursions into evolutionary game theory, computer science, cognitive neuroscience, and dynamical systems theory**

Simon D. Levy, W&L Department of Computer Science and Program in Neuroscience

Time: 3:30pm, Thursday, April 30

Place: Robinson Hall, Room 6

**But I Don't Want to be an Actuary! An Alternative Choice for Math Majors Not Planning on Going into Academia**

Meagan Clement, W&L Class of '02

Time: 4:40pm, Thursday, April 23*

Place: Robinson Hall, Room 6

Dr. Clement earned a Ph.D. in biostatistics from UNC-Chapel Hill in 2008 and currently works for Rho Inc., a contract research organization providing support to the pharmaceutical/biotech industry.

*Refreshments at 4:20 in Robinson Hall 2

**On Sums of Powers of Primes**

Florian Luca, Universidad Nacional Autónoma, De México

Time: 4:40, Tuesday, March 24*

Place: Robinson Hall, Room 6

*Refreshments at 4:20pm in Robinson Hall 2

**Pi Mu Epsilon SpeakerIt was the quickest of times, it was the sameness of times**Steve Abbott, Middlebury College

Time: 4:30, Tuesday, March 17*

Place: Robinson Hall, Room 6

*Refreshments at 4:00 p.m. in Robinson Hall 2

**An Introduction to the Mystery of Fractals**

Zoltán Buczolich, Eötvös Loránd University, Budapest, Hungary

Time: 4:40pm Thursday, February 26*

Place: Robinson Hall, Room 6

*Refreshments at 4:20 p.m. in Robinson Hall 2