The Mathematics Department at Davidson College hums with mathematical activity. Read below to see how students and alumni work with faculty members to explore and discover mathematics. Below you will find work in the areas of algebra, combinatorics, mathematical and computational biology, and numerical analysis. For more information on faculty scholarly activity, beyond the research with students featured on this page, take a look at the department's newsletter, the Bernard Review. Algebra Professor John Swallow continues research with Andrew Schultz ’02, extending work from Schultz's honors thesis as an undergraduate in the department. Their paper, Automatic realizations of Galois groups with cyclic quotient of order pn, co-authored with Ján Mináč (Western Ontario) (the three are pictured to the right) appeared in the Journal de Théorie des Nombres de Bordeaux as 20 (2008), 419-430. The trio also also co-authored the paper Galois module structure of Milnor K-theory mod ps in characteristic p that appeared as 14 (2008), 225--233 in the New York Journal of Mathematics. In the summer of 2008, Professor Swallow also worked with Jason Ferguson (Duke ’09) for six weeks on Galois cohomology under National Science Foundation funding.
Combinatorics This summer Professor Sarah Mason will supervise the research of Mali Zhang ’11 under the Davidson Research Initiative. The two will analyze questions concerning Scramble Squares, puzzles involving nine 4" by 4" squares. The goal is to place the pieces so that a continuous image appears on each internal edge. An example is pictured to the right and also at Amazon.com. It is interesting to note that the image on Amazon does not represent a valid solution, reflecting that while very easy to play these puzzles can be quite difficult to solve.
Mathematical and computational biology Professor Laurie Heyer continues her collaborative research with students and faculty at both Davidson and Missouri Western State (MWSU) in the intersection of math and biology. In the summer of 2008, they designed, modeled, and constructed a bacterial computer that uses XOR logic to compute a cryptographic hash function, used to authenticate documents. This work, led by professors Laurie Heyer (mathematics) and Malcolm Campbell (biology), along with Todd Eckdahl and Jeff Poet (Missouri Western State), resulted in a provisional patent on their invention of the XOR logic built in DNA.
This was the third year of iGEM (International Genetically Engineered Machine) collaboration for the Davidson-Missouri Western synthetic biology team and the first time for the group to work shoulder-to-shoulder as the team spent one week visiting each other's campuses. The Davidson portion of the team consisted of math majors Kristi Muscalino ’09 and Madeline Parra ’09 as well as Kelly Davis ’11, Erin Feeney ’11, Pallavi Penumetcha ’11. In addition, bioinformatics major Max Win ’10 developed software for visualizing the extensive catalog of biological parts that iGEM teams use in their projects. See the team's web page for more information. James Barron from Hampton University and Karlesha Roland from Spelman College also participated on campus under Davidson Research Initiative funding. The work resulted in another impressive Gold Medal from the 2008 iGEM Jamboree at MIT, an event that included 84 teams from 21 countries. During the 2008-09 academic year Professor Heyer also met each Friday with Shashank Suresh ’11 and Annie Temmink ’11, through the Bio-Math Connections program, in preparation for this summer's research in synthetic biology. Mark Trawick ’10 will work on MAGIC Tool software development project this summer. Numerical analysis and scientific computation During the summer of 2008, Daniel Orr ’08 worked with Dr. Tim Chartier on integrating recent research into an open-source multigrid code of Los Alamos National Laboratory. Meanwhile, Peter Simov ’08 and Dr. Chartier analyzed Netflix ratings data with matrix clusters and adapted sports ranking methods. Peter and Chartier's article March madness to movies, co-authored with Amy Langville (College of Charleston) was accepted for publication by Math Horizons in the spring of 2009. Beginning in the fall of 2008, Erich Kreutzer ’10 continued Peter's work with Dr. Chartier. This led to Erich's presentation of a poster When Movies Compete at the Southeastern Atlantic Mathematical Sciences Workshop in Chapel Hill, as well as a talk by the same title at the Shenandoah Undergraduate Mathematics and Statistics Conference. Erich also presented the paper How fair are BCS ratings? Analysis of Colley methods for sports ranking at the Joint Mathematical Meetings in Washington D.C. In February, Dr. Chartier and Erich along with Max Win ’10 turned their attention to sports ranking. Working in collaboration with Amy Langville (College of Charleston) and her students, the group designed and implemented ratings algorithms to predict the outcome of the NCAA Men’s Basketball Tournament. The team submitted their predictions to ESPN’s annual basketball bracket-picking challenge. Of the over 4.6 million entries this year, the group produced 3 predictions landing in the top 10%; their best method finished in the top 3% of all submissions. The work generated media interest with two Charlotte Observer articles: Feeling smart on your picks? Bad news and Davidson math whizzes show brackets skills. The Davidson group also appeared on News14 in the story Mathematics meets March Madness at Davidson. Professor Chartier’s paper Searching for text in vector space, co-authored with Nicholas Dovidio ’07, appeared in The UMAP Journal as 29 (2008), no. 4, 417-430. The paper derives from Dovidio’s honors thesis at Davidson. The online article An Integer Programming Model for the Sudoku Problem authored by Bartlett and Langville (College of Charleston) with Tim Rankin ’07 and Dr. Chartier appeared in the Journal of Online Mathematics and its Applications8 (2008). Professor Chartier's article Efficiency of Multigrid Algorithms for Head Models on Electroencephalography Simulations, which was co-authored with Tim Rankin ’07 and Ceon Ramon (University of Washington), appeared as International Journal of Pure and Applied Mathematics45 (2008), 349--357. Dr. Chartier also worked with several students this year. Nathalia Paulinelli ’10 implemented and tested algebraic block smoothers within larger multigrid codes. Max Win ’10 studied the robustness and efficiency of block smoothers to compute PageRank. Jim Dickson ’09 will extend these efforts in his research during the summer.
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