Mathematics & Computer Science
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- Problem of the Month
Title: Magic Squares
Abstract: Magic Squares have been around for thousands of years. At their core, they involve solving specific linear equations in nonnegative integers. We let Hn(r) be the number of nxn N-matrices having line sums r, where a line is a row or column and an N-matrix is a matrix whose entries belong to the natural numbers. We look at some specific values of Hn(r), some historic magic squares, and learn to construct them
Bio: Dr. Heidi Hulsizer was born in Pensacola, Florida, but was raised in Missouri. She graduated from Drury University and then did her graduate work at the University of Missouri – Columbia. She is currently teaching at Hampden-Sydney College and her research is in Commutative Algebra.
Title: Red Hat, Blue Hat: An Introduction to Coding Theory
Abstract: What's the optimal strategy for you and your team to use in guessing randomly-selected hat colors in a "mathematical game show"? A hat has been placed on your head, but you don't know the color. Should you try to guess the color, or should you pass? And, how does this question relate to codes?
Broadly speaking, coding theory describes any situation in which information is being transmitted from a source to a receiver over a communication channel. In transmitting or storing and reading messages, there is always a possibility of error; to combat this problem, error-detecting or error-correcting codes are employed. This talk will use "the hat problem" to introduce the foundational ideas of coding theory, give a taste of the mathematics involved, and describe some of the many applications of this branch of mathematics.
Bio: Deirdre L. Smeltzer is professor of mathematics and chair of the Department of Mathematical Sciences at EMU, as well as co-author of a recently-published geometry textbook. Dr. Smeltzer earned a Ph.D. in mathematics from the University of Virginia, where her research focused on algebraic coding theory and other branches of combinatorics.
Title: The Pigeonhole Principle
Abstract: The pigeonhole principle also known as Drichlet's principle states that if n + 1 objects are placed into n boxes, then some boxes contain more than one objects. Unlike many theorems in Mathematics, this principle is "trivial" to state and prove. As trivial as it may sound, the pigeonhole principle has numerous nontrivial applications in many areas of mathematics. We will illustrate the beauty and power of this principle through applications in number theory, graph theory, Ramsey theorems, etc. Believe it or not, we are not going to differentiate nor integrate. Just count.
Bio: Moa Apagodu received his Ph.D. from Rutgers University, The State University of New Jersey, and is now an Assistant Professor of Mathematics at Virginia Commonwealth University. His research area includes Algebraic and Enumerative Combinatorics, Computer Algebra/Algorithmic Proof Methods, and Experimental Mathematics.
Title: Freedom or Death: A Warden's Game
Abstract: All 100 prisoners are summoned to a large room, where the warden tells them they must play a game. If the team of prisoners wins the game, they all go free; if they lose the game, they are all put to death. The rules of the game are simple:
• The warden has written each prisoner's name on a slip of paper. In an adjacent hallway stand 100 closed lockers, and each locker contains exactly one of these slips of paper.
• The first prisoner must enter this hallway by himself, and is allowed to open 50 lockers, but may not move any of the slips of paper.
• If the first prisoner finds his name, he exits the hallway from the opposite end. All locker doors are shut, and the second prisoner is allowed to enter.
• The second prisoner, now, is given 50 chances to find his name.
The process continues until one prisoner fails to find his name. If this happens, the prisoners lose the game, and all 100 prisoners are put to death. The prisoners win, and are set free, if all 100 prisoners are able to find their names.
No collaboration among prisoners is allowed once the game starts, but they are told they may discuss possible strategies beforehand. After ten minutes of hushed and gloomy discourse and much nail-biting, one of the prisoners, a mathematician who had been frantically scribbling on the prison wall, tells the others he has an idea. The smiling mouth of the onlooking warden twitches downward slightly. . . .
Bio: David Clark majored in math and physics at Tufts University, during which time he studied abroad at Oxford for a year (where he fell in love with topology!). He got his PhD in math from UC San Diego and have been at Randolph-Macon College since 2008. His research interests lean in the direction of things that can be knotted, especially strings and surfaces, and their corresponding algebraic structures.
Speaker: Camilla Smith Barnes
Assistant Professor of Mathematical Sciences
Sweet Briar College
Title: Fun with Catalan Numbers
Abstract: The Catalan Numbers are said to be the most famous sequence in mathematics that is still obscure enough that people continue to "rediscover" them. In this talk, we will look at some fun examples of the Catalan Numbers, including counting the number of distinct shuffles of the identity permutation with itself. This example provides motivation for my recent research, in which I enumerate the number of distinct shuffles of any two permutations. Two possible shuffles of the permutations 123 and 321 are, for instance, 123321 and 132231.
Bio: Camillia "Cammie" Smith Barnes joined the Sweet Briar College Department of Mathematical Sciences in the fall of 2009, after completing a Ph.D. at Harvard University. Her dissertation focuses on counting the number of distinct shuffles of two permutations. Barnes grew up in Michigan, completed her graduate work in England and in Massachusetts, and is glad to have moved to Virginia. She is a 2009 National Project NExT fellow and an alumna of Sweet Briar's Junior Year in France program. Her current work is in enumerative combinatorics, with a particular interest in permutation enumeration. Besides doing math, Barnes enjoys reading classic fiction, watching films, cooking, jogging, playing board games and playing the violin.
Speaker: Holly Gaff
Department of Biological Sciences
Old Dominion University
Title: Estimating tick-borne disease risk with an agent-based model
Abstract: Ticks have a unique life history including a distinct set of life stages and a single blood meal per life stage. While some tick species have a single preferred host for each life stage, other tick species will feed on a variety of hosts. All of this makes tick-host interactions more complex from a mathematical perspective. In addition, any model of these interactions must involve a significant degree of stochasticity on the individual tick level. In an attempt to quantify these relationships, we have developed an individual-based model of the interactions between ticks and their hosts as well as the transmission of tick-borne disease between the two populations. Preliminary analysis of disease prevalence as a function of host diversity is presented.
Bio: Dr. Holly Gaff is an Assistant Professor in the department of Biological Sciences at Old Dominion University and is affiliated with the Virginia Modeling, Analysis and Simulation Center. Dr. Gaff earned her Ph.D. In Mathematics at the University of Tennessee, Knoxville, in 1999. Dr. Gaff's research interests have focused mainly on studying the dynamics and control of infectious diseases using mathematical modeling and computer simulation. Most of her research has focused on developing mathematical models for exploring the ecology of vector-borne diseases including Rift Valley fever and tick-borne diseases in the Hampton Roads area. She has had funding for these and other projects from NIH, NSF, DHS, CDC and the VA.
Date: 31 March (Thursday)
Speaker: Jill Tysse
Mathematician at Large
Title: Using Symmetry to Count in Chemistry, Music, and Tic-Tac-Toe
Abstract: From the time when you were an infant and gazed into your parents' eyes, to the time when you made a symmetrical "butterfly" in preschool by splodging paint on some paper and folding it in two, to the times more recently when, in pre-calculus class, your professor helped you to analyze graphs of even and odd functions, you have been interested in two-dimensional symmetries! We will talk about a certain type of counting problem that occurs in pursuits as noble as the counting of chemical isomers and 20th century music composition, and as whimsical as the counting of tic-tac-toe boards and crossword-puzzle grids and show how symmetry is used to help solve this problem.
Bio: Jill Tysse was born and raised in Cork, Ireland, and completed her undergraduate studies in Mathematics and Applied Mathematics at University College Cork. Tysse completed a Master's degree in Mathematics and the Foundations of Computer Science at Oxford University and then earned her Ph.D. from the University of Virginia where her research combined group theory and algebraic combinatorics. She was an Assistant Professor at Hood College in Frederick, MD for two years and was a 2008 National Project NExT fellow. These days, her time is filled with "teaching" and "advising" of a different sort as she recently temporarily retired from academia to stay home and be mom to her two young children.
Date: 7 April (Thursday)
Speaker: Della Fenster
Professor of Mathematics
University of Richmond
Title: Mathematics: A Question of History
Abstract: Former Governor (now Senator) of Virginia Mark Warner inquired, "What is the history of mathematics? Is it about the problems or the people who solve the problems?" Beginning with the former governor's curiosities, this talk introduces the history of mathematics as a vibrant field that considers technical questions within a broad framework that includes issues related to biography, institutional settings, and political dynamics among others.
Bio: Della Fenster is a Professor of Mathematics at the University of Richmond. Her research lies in the history of mathematics, particularly in the late 19th and early 20th centuries. She is the recipient of the University of Richmond Distinguished Educator Award (2003) and the State Council of Higher Education for Virginia Outstanding Faculty Award (2004).