Department of Mathematics

West Chester University

**Mathematics Information**

Office: Room 101

25 University Avenue

West Chester, PA 19383

Phone (610) 436-2440

Fax (610) 738-0578

Email: Department Chair

Each Thursday there will be a mathematics seminar (usually in UNA 120 from 3:15-4:15), while colloquium talks will normally be on a Wednesday (usually in UNA 158 from 3:15-4:15).

These seminars/colloquium talks may be by visiting speakers, WCU faculty, or WCU students, and are open to all interested students and faculty.

Send an e-mail to jmclaughl@wcupa.edu, if you would like to be on the e-mail list to receive advance notice of upcoming talks.

Previous Semesters: Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Summer 2006, Spring 2006,

3:15 to 4:15PM UNA 162

**The Improved SNP Calling Algorithms for Illumina BeadArray Data**

**Abstract:** Genotype calling from high throughput platforms such as Illumina and Affymetrix is a critical step in data processing, so that accurate information on genetic variants can be obtained for phenotype-genotype association studies. A number of algorithms have been developed to infer genotypes from data generated through the Illumina BeadStation platform, including GenCall, GenoSNP, Illuminus, and CRLMM. Most of these algorithms are built on population-based statistical models to genotype every SNP in turn, such as GenCall with the GenTrain clustering algorithm, and require a large reference population to perform well. These approaches may not work well for rare variants where only a small proportion of the individuals carry the variant. A fundamentally different approach, implemented in GenoSNP, adopts a SNP-based model to infer genotypes of all the SNPs in one individual, making it an appealing alternative to call rare variants. However, compared to the population-based strategies, more SNPs in GenoSNP may fail the Hardy-Weinberg Equilibrium test. To take advantage of both strategies, we propose the two-stage SNP calling procedures, to improve call accuracy for both common and rare variants. The effectiveness of our approach is demonstrated through applications to genotype calling on a set of HapMap samples used for quality control purpose in a large case-control study of cocaine dependence. The increase in power with our proposed method is greater for rare variants than for common variants depending on the model.

*Gengxin Li is currently a Postdoctoral Associate in the Division of Biostatistics, Department of Epidemiology and Public Health at Yale University. She received a dual-major Ph.D. degree in Statistics and Quantitative Biology at Michigan State University. Her current research interests are high-dimensional data analysis, Bayesian method, Dirichlet process, longitudinal data analysis, Statistical genomics, Statistical genetics, Bioinformatics and Clinical Trials. *

3:15 to 4:15PM UNA 162

**Exact Relations for Fiber-Reinforced Elastic Composites**

Predicting the effective elastic properties of a composite material based on the elastic properties of its constituent materials is extremely difficult, even when the microstructure of the composite is known. However, there are special cases where certain properties in constituents always carry over to the composite, regardless of microstructure. We call such instances exact relations. The general theory of exact relations allows us to find all of these relations in a wide variety of contexts including elasticity, conductivity, and piezoelectricity. We combine this theory with certain results from representation theory to find all exact relations in the context of elasticity of fiber-reinforced polycrystalline composites and thereby generate new information about this widely-used class of materials.

*Meredith Hegg is currently a PhD student in the Department of Mathematics at Temple University. Her main area of research is currently Mechanics of Deformable Solids, and she expects to obtain her PhD in May 2012. Her thesis adviser is Dr. Yury Grabovsky. *

3:15 to 4:15PM Anderson 111

We all know one field that contains 0,1,2,..., but, logically, there is an earlier field that is defined as follows. We first fill in the addition table, subject to the condition that before we fill in the entry for *a+b*, we must have already filled in all entries *a'+b* and *a+b'* with *a'<a* and *b'<b*. Then, the entry at *a+b* is to be the least possible number that is consistent with the result's being a part of the addition table of a field.

We then tackle the multiplication table of a field with the given addition. Again, the entries are to be the least possible one's subject to this requirement; this construction produces a very strange field in which 8 is a fifth root of unity. Amazingly, this field actually has practical applications.

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu.

3:15 to 4:15PM UNA 162

**Refining Genetically-Inferred Relationships Using Treelet Smoothing **

**Abstract:** Heritability, or fraction of the total trait variance attributable to additive genetic effects, is an important concept in quantitative genetics. Originally, heritability was only measurable by examining groups of very closely related individuals, such as twin studies. More recently, methods have been proposed to analyze population samples containing only distantly related individuals using a random effects model. To do so they estimate the relatedness of all pairs of individuals in the population sample using a dense set of common genetic variants, such as SNPs, and evaluate their relationships to subject trait values. We build on their approach, focusing on improved estimates of pairwise familial relationships. We propose a new method for denoising genetically inferred relationship matrices, and refer to this general regularization approach of positive semi-definite matrices as Treelet Covariance Smoothing. On both simulated and real data, we show that better estimates of the relatedness amongst individuals lead to better estimates of the heritability.

3:15 to 4:15PM UNA 162

**On the Factorization of Eigenforms**

** **

Modular forms fall within the realm of complex analysis and number theory, with notable applications in theoretical physics. Hecke operators act on spaces of modular forms, and spectral theory implies the existence of eigenforms. My recent research, which will be presented at this talk, has investigated the factorizations of these eigenforms. This type of investigation is relatively new, having started in 1999 when Eknath Ghate and William Duke independently discovered that the product of two eigenforms is again an eigenform only when it is trivially so.

*Jeffrey Beyerl is a graduate student in the Department of Mathematics at Clemson University. His main area of research is presently in the field of modular forms, and he expects to obtain his PhD in May 2012. His thesis advisers are **Kevin James and Hui Xue.*

3:15 to 4:15PM UNA 120

**Borrowing Information across Genes and Experiments for Improved Error Variance Estimation in Microarray Data Analysis **

**Abstract: **Statistical inference for microarray experiments usually involves the estimation of error variance for each gene. Because the sample size available for each gene is often low, the usual unbiased estimator of the error variance can be unreliable. Shrinkage methods, including empirical Bayes approaches that borrow information across genes to produce more stable estimates, have been developed in recent years. Because the same microarray platform is often used for at least several experiments to study similar biological systems, there is an opportunity to improve variance estimation further by borrowing information not only across genes but also across experiments. We propose a lognormal model for error variances that involves random gene effects and random experiment effects. Based on the model, we develop an empirical Bayes estimator of the error variance for each combination of gene and experiment and call this estimator BAGE because information is Borrowed Across Genes and Experiments. A permutation strategy is used to make inference about the differential expression status of each gene. Simulation studies with data generated from different probability models and real microarray data show that our method outperforms existing approaches.

3:15 to 4:15PM UNA 162

**Twist Knots and the Uniform Thickness Property **

In 2007, Etnyre and Honda defined a new knot invariant called the Uniform Thickness Property in order to better understand Legendrian knots. The classification of Legendrian knots in R^3 with the standard contact structure has been a slow process in comparison to the topological classification in R^3. In this talk, we will discuss what makes Legendrian knots more delicate than topological knots, and how the Uniform Thickness Property can help in their classification. My current research investigates the Uniform Thickness Property with respect to positive twist knots which we will discuss towards in the second half of this talk.

*Whitney George is a graduate student in the Department of Mathematics at the University of Georgia. Her main area of research is presently in contact topology, and is focused towards knots and surfaces in R^3 with the standard contact structure, and she expects to obtain her PhD in May 2012. Her thesis adviser is Gordana Matic. *

3:15 to 4:15PM UNA 162

**Pointwise Convergence of Averages of L1 Functions on Sparse Sets.**

Joint work with P. LaVictoire (University of Wisconsin, Madison) and J. Rosenblatt (UIUC).

**Abstract:**The behavior of time averages when taken along subsets of the integers is a central

question in subsequence ergodic theory. The existence of transference principles enables us to talk

about the convergence of averaging operators in a universal sense; we say that a sequence {an} is

universally pointwise good for L1, for example, if the sequence of averages

1/NΣ_{n=0}^{N-1}f ◦ T^{-an}(x)

converges a.e. for any f ∈ L1 for every aperiodic measure preserving system (X; B; T; ). Only a

few methods of constructing a sparse sequence that is universally pointwise L1-good are known.

We will discuss how one can construct families of sets in Zd which are analogues of these sequences,

as well as some challenges and advantages presented by these higher-dimensional averages.

3:15 to 4:15PM UNA 155

Pi Mu Epsilon Presents

ALISSA CRANS (Loyola Marymount University)

In celebration of your mathematical achievements on this special day we will investigate fun facts

related to Leap Days! We'll discuss mathematicians associated to this day and various calendar

systems. In addition, we will explore the numerous interesting properties of the number 29. Of

course it's prime, but in fact, it's a twin prime, Sophie Germain prime, Lucas prime, Pell prime, and

Eisenstein prime. It's also a Markov number, Perrin number, tetranacci number and Stormer

number! We'll see all of this, and more, as we congratulate the newest members of Pi Mu Epsilon for

their wonderful accomplishments.

*Alissa S. Crans earned her B.S. in mathematics from the University of Redlands in 1999 and her Ph.D. in mathematics from the University of California at Riverside in 2004, under the guidance of John Baez. She is currently an Associate Professor of mathematics at Loyola Marymount University and has held positions at Pomona College, The Ohio State University, and the University of Chicago.
Alissa's research is in the field of higher-dimensional algebra and her current work, funded by an NSA Young Investigator Grant, involves categorifying algebraic structures called quandles with the goal of defining new knot and knotted surface invariants. She is also interested in the connections between mathematics and music, and enjoys playing the clarinet with the Santa Monica College wind ensemble.
Alissa is extremely active in helping students increase their appreciation and enthusiasm for mathematics through coorganizing the Pacific Coast Undergraduate Mathematics Conference together with Naiomi Cameron and Kendra Killpatrick, and her mentoring of young women in the Summer Mathematics Program (SMP) at Carleton College, the EDGE program, the Summer Program for Women in Mathematics at George Washington University, the Southern California Women in Mathematics Symposium, and the Career Mentoring Workshop. In addition, Alissa was an invited speaker at the MAA Spring Sectional Meeting of the So Cal/Nevada Section and the keynote speaker at the University of Oklahoma Math Day and the UCSD Undergraduate Math Day. She is a recipient of the 2011 Merten M. Hasse Prize for expository writing and the Henry L. Alder Award for distinguished teaching.
*

For further information e-mail rsullivan@wcupa.edu

3:15 to 4:15PM UNA 162

STEFAAN DELCROIX

3:15 to 4:15PM UNA 162

STEFAAN DELCROIX

**Locally Finite Simple Groups **

**Abstract:** A group $G$ is locally finite if every finite subset of $G$ generates a finite subgroup. In this talk, we study infinite, locally finite, simple groups (=LFS-groups). We will introduce some standard definitions and properties, divide the LFS-groups into three categories and provide examples of each category. Next, we study a specific category (LFS-groups of $p$-type) into more detail. This allows us to show some local characterization of each category. Time permitting, we discuss a general construction of LFS-groups of $p$-type.

*Born and raised in Belgium, Stefaan finished his masters in mathematics at the University of Ghent (in Belgium). He spent the next three years working on his Ph.D. at Michigan State University under the guidance of Prof. Ulrich Meierfrankenfeld. The subject of his thesis was locally finite simple groups of p-type and alternating type. In June 2000, Stefaan finished his Ph.D. at the University of Ghent. For two years, he
worked as a Visiting Assistant Professor at the University of Wyoming in Laramie. Since 2002, Stefaan has worked at California State University, Fresno.*

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu.

3:20 to 4:15PM UNA 155

We shall discuss some aspects of Euler's proof of Fermat's Last Theorem for exponent three.

This talk will be suitable for students who have taken (or currently taking) a course on Theory of

Numbers (Mat 414/514).

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

3:15 to 4:15PM UNA 162

**"Hybrid Proofs of the q-Binomial Theorem and other q-series Identities. I**"

The proof of a *q*-series identity, whether a series-to-series identity such as the second iterate of Heine’s transformation, a basic hypergeometric summation formula such as the *q*-Binomial Theorem or one of the Rogers-Ramanujan identities, generally falls into one of two broad camps.

In the one camp, there are a variety of analytic methods.

In the other camp there are a variety of combinatorial or bijective proofs, the simplest of course being conjugation of the Ferrer’s diagram for a partition.

In this series of talks we use a “hybrid” method to prove a number of basic hypergeometric identities. The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version.

3:15 to 4:15PM UNA 162

**"Hybrid Proofs of the q-Binomial Theorem and other q-series Identities. II"**

3:20 to 4:15PM UNA 155

Abstract: Tropical mathematics emerged in 1960's as a linear encoding of some problems in discrete

optimization and scheduling. In a nutshell, it studies "spaces" over the max-plus algebra, which is the set of

real numbers where taking maximum plays the role of addition, and addition plays the role of multiplication.

In the tropical mathematics, negative infinity plays the role of zero, hence any real number is "positive"

in the tropical sense. Hence, there are connections with nonnegative linear algebra (in particular, Perron-

Frobenius theory), and convex geometry. To this end, tropical spaces can be viewed as an analogue of convex cones, and many results of convex analysis have their tropical analogues, which will be reviewed.

Tropical linear two-sided systems Ax = Bx, where matrix-vector multiplication is defined using the

tropical arithmetics, are the algebraic encoding of tropical convex cones. Geometrically, such systems represent the tropical convex cones as intersection of tropical halfspaces. Methods for finding a solution to such two-sided systems stem from combinatorial game theory, more specifically, from the theory of deterministic mean-payoff games. We will also touch upon some problems like the tropical linear programming that can be viewed as parametric extension of two-sided systems, and give rise to parametric extensions of mean-payoff games.

All are welcome to join for tea in Students Lounge after the talk.

3:15 to 4:15PM UNA 162

**"Hybrid Proofs of the q-Binomial Theorem and other q-series Identities. III"**

3:20 to 4:15PM UNA 155

Hal Switkay

**"The Sensible Communication of Abstract Information"**

We consider the engagement of the senses in the process of communicating and learning the abstractions of mathematics. Examples are provided from the history of mathematics continuing through current developments, including Markov processes, analytic geometry, statistics, decision theory, 24-dimensional geometry, and the musical representation of groups.

This talk should be easily accessible to undergraduates.

*Hal M. Switkay earned his Ph.D. in mathematics at Lehigh University in the study of set theory. After graduation, his interests shifted towards symmetry, lattices, groups, and higher-dimensional geometry. He has taught mathematics, from remedial to advanced, has done public speaking, is a musician and composer, and has earned certification as a teacher of Tai Chi Easy and as a practitioner of reiki and Thai massage. He is currently enrolled in West Chester Universitys graduate certificate program in applied statistics. His business card lists the following interests: mathematics; music; philosophy; health and wellness; and syncretic
panendeism.*

All are welcome to join for tea in Students Lounge after the talk.

3:15 to 4:15PM Anderson 103

The first personal computing revolution took place not in Silicon Valley in the 1980s but in Pisa in the 13th Century. The medieval counterpart to Steve Jobs was a young Italian called Leonardo, better known today by the nickname Fibonacci. Thanks to a recently discovered manuscript in a library in Florence, the story of how this little known genius came to launch the modern commercial world can now be told.

Based on Devlin’s latest book The Man of Numbers: Fibonacci’s Arithmetical Revolution (Walker & Co, July 2011) and his co-published companion e-book Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years.

*Keith Devlin is a mathematician at Stanford University in California. He is a co-founder and Executive Director of the university's H-STAR institute, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He has written 31 books and over 80 published research articles. His books have been awarded the Pythagoras Prize and the Peano Prize, and his writing has earned him the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his "innovative work and longtime service in the field of mathematics and its relation to logic and linguistics." He is "the Math Guy" on National Public Radio. *

*He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He writes a monthly column for the Mathematical Association of America, "Devlin's Angle”.*

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

3:15 to 4:15PM UNA 162

**"Some Partition Bijections in Igor Pak's "PARTITION BIJECTIONS, A SURVEY""**

**Friday, April 27th, 2012**

2:00 to 3:00PM UNA 158

**Spring 2012 Mathematics Colloquium
**ELWYN BERLEKAMP University of California, Berkeley

**“Combinatorial Games: Hackenbush and Go” **

This talk will review the rudiments of combinatorial game theory [1] as exemplified by a game called Hackenbush. Positions are seen to have values, which are sums of numbers and infinitesimals, such that the winner depends on how the total value compares with zero.

We then discuss how refinements of this theory have been applied to the classical Asian board game called Go. The most important tool is the "cooling operator" [2], which maps combinatorial games into other combinatorial games. In the first application, many late stage Go endgame positions [3] are shown to be combinatorial games which, when cooled by 1, often reduce to familiar numbers and infinitesimals. Combinatorial game theory then enables its practitioner to win the endgame by one point. In the second application, Nakamura[4] has shown that liberties can also be viewed as combinatorial games which become familiar numbers and infinitesimals when cooled by 2. In a large class of interesting positions, this approach identifies the move(s), if any, which win the capturing race.

Although not prerequisite to this talk, more details can be found in these references:

[1] Berlekamp, Conway, and Guy: Winning Ways, Chap 1

[2] Berlekamp, Conway, and Guy: Winning Ways, Chap 6

[3] Berlekamp and Wolfe: Mathematical Go

[4] Nakamura, in Games of No Chance, vol 3

*Elwyn Berlekamp was an undergraduate at MIT; while there, he was a **Putnam Fellow** (1961). Professor Berlekamp completed his bachelor's and master's degrees in **electrical engineering** in 1962. Continuing his studies at MIT, he finished his Ph.D. in electrical engineering in 1964; his advisors were **Claude Shannon**, **Robert G. Gallager**, **Peter Elias** and **John Wozencraft**. Berlekamp taught at the **University of California, Berkeley** from 1964 until 1966, when he became a researcher at **Bell Labs**. In 1971, Berlekamp returned to Berkeley where, as of 2010, he is a Professor of the Graduate School.*

*He is a member of the **National Academy of Engineering** (1977)* *and the **National Academy of Sciences** (1999).* * He was elected a Fellow of the **American Academy of Arts and Sciences** in 1996.* * He received in 1991 the **IEEE Richard W. Hamming Medal**,* *and in 1998 the Golden Jubilee Award for Technological Innovation from the **IEEE Information Theory Society**.*

*Berlekamp is one of the inventors of the Welch-Berlekamp and **Berlekamp-Massey algorithms**, which are used to implement **Reed-Solomon error correction**. In the mid-1980s, he was president of **Cyclotomics, Inc.**, a corporation that developed error-correcting code technology.** **With **John Horton Conway** and **Richard K. Guy**, he co-authored **Winning Ways for your Mathematical Plays**, leading to his recognition as one of the founders of **combinatorial game theory**. He has studied various games, including Fox and Geese and other **fox games**, **dots and boxes**, and, especially, **Go**. With **David Wolfe**, Berlekamp co-authored the book Mathematical Go, which describes methods for analyzing certain classes of Go endgames.*

Note: Talks will be added to the schedule throughout the semester. Check back for updates.