## Colloquium Abstracts

**Wednesday, Aug 25, 2004**

9:00am, 421 Neville

**Suzhong Tian**, *Dept. of Mathematics and Statistics, University of Maine
*Statistical Inference for the Risk Ratio in 2 X 2 Binomial Trials with Structural Zero

Thesis defense; Advisor: Prof. Ramesh Gupta

In some statistical analyses, researchers may encounter the problem of analyzing correlated 2×2 table with a structural zero in one of the off diagonal cells. Structural zeros arise in situation where it is theoretically impossible for a particular cell to be observed. For instance, Agresti (1990) provided an example involving a sample of 156 calves born in Okeechobee County, Florida. Calves are first classified according to whether they get a pneumonia infection within certain time. They are then classified again according to whether they get a secondary infection within a period after the first infection clears up. Because subjects cannot, by definition, have a secondary infection without first having a primary infection, a structural void in the cell of the summary table that corresponds with no primary infection and has secondary infection is introduced. For discussion of this phenomenon, see Tang and Tang (2002), and Liu (1998).

The rate ratio (RR) between the secondary infection, given the primary infection, and the primary infection may be a useful measure of change in the pneumonia infection rates of the primary infection and the secondary infection. In this thesis, we will first develop and evaluate the large sample confidence intervals of RR. Then we will investigate the tests about RR and the power of these tests. An example, from the literature, will be provided to illustrate these procedures. Simulation studies will be carried out to examine the performance of these procedures.

**Wednesday, Nov 3, 2004**

2:30pm, 100 Neville Hall

**Dr. Ram C. Tripathi**, *Dept. of Management Science and Statistics, University of Texas at San Antonio
*Weighted Least-squares and Mximum Likelihood Estimation for Longitudinal Data with Truncated Observations: An Application in Pharmacokinetic Studies of Environmental Contaminants

Pharmacokinetic studies of biomarkers for environmental contaminants in humans are generally restricted to a few measurements per subject taken after the initial exposure. Subjects are selected for inclusion in the study if their measured body burden is above a threshold determined by the distribution of the biomarker in a control population, such selection procedures introduce bias in the weighted least-squares estimate of the decay rate lambda caused by the truncation. We propose and compare two estimators of lambda: (1) a bias-corrected weighted least-squares estimate and (2) maximum likelihood estimate based on truncated observations. Iterative methods are presented for obtaining these estimates. The estimates and their efficiencies are discussed in the context of a pharmacokinetic study of 2, 3, 7, 8-tetrachlorodibenzo-p-dioxin.

Dr. Ram Tripathi received his Ph.D. in Statistics from the University of Wisconsin-Madison in 1975. He is currently Professor of Statistics at the University of Texas at San Antonio. His research interests are in the areas of Biostatistics, Modeling of count data, Statistical inference, and Reliability theory.

**Thursday, Nov 4, 2004**

2:30pm, 100 Neville Hall

**Dr. Arthur Benjamin**, *Dept. of Mathematics, Harvey Mudd College
*Proofs That Really Count

Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using various tools. In this talk, we demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. You will enjoy the magic of Fibonacci numbers, Lucas numbers, continued fractions, and more. You can count on it! (This talk is based on research with Professor Jennifer Quinn, Associate Professor and Chair of Mathematics, Occidental College)

Dr. Benjamin is a Professor of Mathematics at Harvey Mudd College in Claremont, California. He received his PhD in Mathematical Sciences from Johns Hopkins University. Dr. Benjamin is co-author of two books on teaching mathematics and the inventor of the Mathemagics course, which teaches children and adults the secrets of rapid mental calculation. He is also a professional magician and combines these two talents in a dynamic presentation called Mathemagics! He frequently performs at the Magic Castle in Hollywood, California, and has presented Mathemagics to schools and organizations over the world. Dr. Benjamin has been featured in numerous publications, including *Mathematical Association of America: Math Horizons*, *The Los Angeles Times*, *USA Today*, *Scientific American*,*Discover Magazine*, *Omni Magazine*, *Esquire Magazine*, and *People Magazine*. He has also appeared on many programs including: *The Today Show*, *National Public Radio*, and *Amazing Discoveries*!

**Thursday, Nov 4, 2004**

7:00pm, 100 Donald P. Corbett Hall

**Dr. Arthur Benjamin**, *Dept. of Mathematics, Harvey Mudd College
*What do you get when you cross a mathematician with a magician? Mathemagician!

Dr. Benjamin is a Professor of Mathematics at Harvey Mudd College. He is one of the world’s fastest “lightning calculators” and a frequent performer at the Magic Castle in Hollywood. He is the author of several books and has presented his mixture of math and magic to audiences all over the world. He has appeared on the *Today Show* and National Public Radio and has been featured in *The Los Angeles Times*, *Scientific American*, and*Discover Magazine*.

**Thursday, Nov 11, 2004**

3:10pm, 100 Neville Hall

**Dan Look**, *Dept. of Mathematics, Boston University
*An Escape Trichotomy for Singularly Perturbed Complex Polynomials

In this talk I will discuss a trichotomy that occurs in the topology of the Julia sets of rational maps of the form *zn + A/zn*. We show that, depending upon how the critical orbit escapes to infinity, the Julia set is either a Cantor set, a Sierpinski curve, or a Cantor set of circles.

Dan Look graduated from UMaine with an MA in mathematics.

**Thursday, Dec 9, 2004**

2:00pm, 421 Neville Hall

**Katherine Merrill**, *Dept. of Mathematics and Statistics, University of Maine
*Ramanujan’s Entry 20

Katherine Merrill will present a talk on Ramanujan’s life and work. The talk will include a discussion of Ramanujan’s Entry 20 from his Notebooks involving the Riemann Zeta function for odd values. A discussion of other entries from Ramanujan’s notebook will be presented, along with a 100-year history of the influence of Ramanujan’s Entry 20. Finally, a review of the proof by Bruce Berndt of Entry 20 using contour integration will be given.

**Wednesday, Feb 2, 2005**

3:00pm, 421 Neville Hall

**Amanda Criner**, *Dept. of Mathematics and Statistics, University of Maine
*Internet Epidemiology

This talk discusses research on the spread of Internet worms with varied spatial dispersal strategies. We inspect a two dimensional lattice representation of the Internet modeled by a simulation and the pair approximation method. The simulation and model both demonstrate logistic behavior but with different growth rates.

This talk describes work Amanda is doing as an undergraduate research assistant with Dr. David Hiebeler, and in preparation for a talk Amanda will soon give at the Nebraska Conference for Undergraduate Women in Mathematics.

**Monday, Feb 7, 2005**

3:00pm, 105 DPC

**Dr. Richard Jordan**, *Dynamics Technology, Inc.
*Mathematical Modeling and Analysis of Spatial Epidemics

Traditionally, mathematical epidemiology models have tended to ignore spatial spread and geographic heterogeneities, or else the models assume that diseases spread spatially through local (i.e. nearest neighbor) contacts. In reality, for many diseases of interest, the spread takes place over a wide range of spatial scales. For example, SARS spread locally through villages, but at the same time the disease jumped continents, owing to airline travel.

In this talk we will introduce two different modeling approaches for the spatial spread of infectious diseases. The first approach amounts to a deterministic, multi-patch metapopulation model, where space is partitioned in nonoverlapping patches, which are connected via transportation/migration networks, allowing for both local and long-range movement of individuals. The second approach involves a stochastic (microscopic) model, in which individuals disperse spatially according to a probability kernel. We will present analytical and numerical results for these two types of models, and discuss differences and connections between them.

**Thursday, Feb 10, 2005**

1:10pm, 131 Barrows

**Katherine Merrill**, *Dept. of Mathematics and Statistics, University of Maine
*Ramanujan’s Formula for the Riemann Zeta Function Extended to L-Functions

Thesis defense; Advisor: Prof. David Bradley

Ramanujans formula for the Riemann-zeta function is one of his most celebrated. Beginning with M. Lerch in 1900, there have been many mathematicians who have worked with this formula. Many proofs of this formula have been given over the last 100 years utilizing many techniques and extending the formula.

Katherine Merrill will provide a proof of this formula by the Mittag-Leffler partial fraction expansion technique. In comparison to the most recent proof by utilizing contour integration, the proof in this thesis is based on a more natural growth hypothesis. In addition to a less artificial approach, the partial fraction expansion technique used in this thesis yields a stronger covergence result.

In addition to providing a new proof of this formula, the work in this thesis extends this formula to a series acceleration formula for Dirichlet L-series with periodic coefficients. The result is a generalized character analog, which can be reduced to the original formula.

**Thursday, Feb 10, 2005**

3:00pm, 105 DPC

**Dr. Anita Layton**, *Mathematics Dept., Duke University
*Mathematical Modeling of Renal Physiology

In this talk I will present mathematical models of the mammalian urine concentrating mechanism (UCM). The UCM gives mammals the capability to produce hypertonic urine, and is localized in the renal medulla. In the outer medulla, the UCM is driven by active NaCl transport from thick ascending limbs, coupled with a countercurrent flow configuration of nephrons and vessels; in the inner medulla, the underlying principles of the UCM remain to be determined. I will present a highly-detailed model of the renal medulla of the rat kidney that aims to assist in the better understanding of the mammalian UCM. The model incorporates and evaluates experimental findings not previously included in models, including the implications of recent immunolabeling experiments. I will also present a model of a papillary collecting duct undergoing peristaltic contractions; that model is based on the immersed boundary method. The immersed boundary motion equations are solved by means of the multi-implicit Picard integral deferred correction (MIPIDC) methods developed by us. The key feature of MIPIDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time-steps for each process.

**Thursday, Feb 24, 2005**

2:10pm, 421 Neville Hall

**Prof. Fernando Gouvea**, *Dept. of Mathematics, Colby College
*On p-adic numbers and their history

When most mathematicians hear the p-adic numbers described as the result of completing the field of rational numbers with respect to metrics that are different from the usual one, they wonder what the point is. This talk will give an elementary introduction to the p-adic numbers, examine their history, and discuss why they were invented and how they were used.

**Thursday, March 17, 2005**

3:00pm, 105 DPC

**Dr. Neil Portnoy**, *Stony Brook University (SUNY)
*How do preservice teachers’ understandings of function develop while engaging in a curriculum module in knot theory?

Undergraduate mathematics students in a college geometry course engaged in a curriculum module in knot theory centered on knot invariants. Knot invariants, functions from the set of knots into sets of numbers and sets of polynomials, provided tools to engage in the study of advanced mathematics but also provided challenges for students as they tried to reconcile these invariants with their existing understandings of function.

Dr. Portnoy is Director of Mathematics Education and Assistant Professor of Mathematics (on leave academic year 2004-5) at Stony Brook University (SUNY)

**Thursday, March 24, 2005**

11:00am, 421 Neville Hall

**Hui Tao**, *Dept. of Mathematics and Statistics, University of Maine
*An Investigation of False Discovery Rates in Multiple Testing Under Dependence

A typical microarray experiment often involves comparisons of hundreds or thousands of genes. Since a large number of genes are compared, simple use of a significance test without adjustment for multiple comparison artifacts could lead to a large chance of false positive findings.

In this presentation, we will study various methods that allow us to measure the overall error rate when testing multiple hypotheses. More specifically, we will examine the two models presented by Tsai et al. (2003). Both these models involve the distribution of the sum of Bernoulli random variables. Model I is derived under the assumption of independence Bernoulli trials and Model II assumes non-independent Bernoulli trials. Because of the overdispersion problem, this model is derived using a beta-binomial structure.

**Monday, March 28, 2005**

3:00pm, 105 DPC

**Dr. Andre Khalil**, *Jackson Laboratory
*Wavelets and Fractals: From Astrophysics to Bio-Medical Image Analysis

Since the end of the 80’s, the wavelet transform has been recognized as a tool to study fractal objects, providing a unified multifractal formalism for both functions and measures. In the first part, we present the 2D WTMM (Wavelet Transform Modulus Maxima) methodology, a wavelet-based multifractal formalism, and demonstrate its capabilities on synthetic surfaces.

In the second part, we use the 2D WTMM method to study the distribution of neutral hydrogen in the Galactic Plane of the Milky Way. The characterization of both the fractal properties and the anisotropic signatures (directional preferences) found in Galactic spiral arms and in the inter-arm regions will be discussed. In the context of digitized mammograms, we further illustrate the usefulness of the methodology in the study of texture segmentation of rough surfaces and the geometric characterization of clusters of microcalcifications, which can be early signs of breast cancer.

Finally, we end with a presentation of some current projects, including the study of bone structure and the analysis of cell membrane morphology.

**Thursday, March 31, 2005**

3:00pm, 421 Neville

**Dr. Charles Li**, *Dept of Mathematics, UCLA
*A new proof of the Petersson Trace Formula

The Selberg trace formula is an important tool for computing the trace of a linear operator in the regular representation of a group G. It has many applications in number theory. An early prototype of the trace formula was developed by Petersson in 1932. It relates the Fourier coefficient of cusp forms to Kloosterman sums and Bessel funnctions. It remains as a powerful tool to estimate Fourier coefficents. In this talk, we will give a new proof of the formula using the modern adelic setting. Some applications and generalizations will be discussed.

**Thursday, April 14, 2005**

11:00am, 421 Neville Hall

**Hui Tao**, *Dept. of Mathematics and Statistics, University of Maine
*An Investigation of False Discovery Rates in Multiple Testing Under Dependence

A typical microarray experiment often involves comparisons of hundreds or thousands of genes. Since a large number of genes are compared, simple use of a significance test without adjustment for multiple comparison artifacts could lead to a large chance of false positive findings.

In this presentation, we will review the five FDRs presented earlier. These five FDRs are computed under two models. Model I is derived under the assumption of independence and Model II assumes non-independent Bernoulli trials. The distribution of the non-independent Bernoulli trials is approximated by a beta-binomial model.

Instead of using a beta-binomial model, we shall obtain the exact distribution of the sum of non-independent and non-identically distributed Bernoulli random variables. This distribution is used to obtain the five FDRs. The results are then compared with those obtained by beta-binomial model.

**Monday, April 18, 2005**

3:10pm, 107 DPC

**Richard Haynes**, *Dept. of Mathematics, University of Chicago
*From the Exotic to the Mundane in Topology

(No abstract available.)

Richard did his undergraduate work at Williams College, and is currently a Ph.D. candidate in mathematics at the University of Chicago

**Friday, April 22, 2005**

3:10pm, 202 Shibles

**Nancy Austin**, *University of Southern Maine
*The Implications of Middle School Teacher Research Data for Teacher Certification

A recently published report, “Professional Development Needs of Middle Level Mathematics and Science Teachers in Maine”, details teacher data, instructional practices, teacher beliefs and professional development needs of Maine teachers. These findings have strong implications for school and university practices that support content and pedagogical knowledge and consequently impact certification. Please join me for a discussion of the findings and our roles as university faculty in the mathematical education of middle level teachers in Maine.

**Thursday, May 12, 2005**

2:00pm, 210 Neville

**Tao Hui**, *Dept. of Mathematics and Statistics, University of Maine
*An Investigation of False Discovery Rates in Multiple Testing Under Dependence

Thesis defense; Advisor: Prof. Ramesh C. Gupta

A typical microarray experiment often involves comparisons of hundreds or thousands of genes. Since a large number of genes are compared, simple use of a significance test without adjustment for multiple comparison artifacts could lead to a large chance of false positive findings.

In this presentation, we will study various methods that allow us to measure the overall error rate when testing multiple hypotheses. More specifically, we will examine the two models presented by Tsai et al. (2003). Both these models involve the distribution of the sum of Bernoulli random variables. Model I is derived under the assumption of independence Bernoulli trials and Model II assumes non-independent Bernoulli trials. Because of the over-dispersion problem, this model is derived using a beta-binomial structure.

Instead of using a beta-binomial model, we shall obtain the exact distribution of the sum of non-independent and non-identically distributed Bernoulli random variables. This distribution is used to obtain the five FDRs. The results are then compared with those obtained by beta-binomial model.

**Friday, July 15, 2005**

2:00pm, 419 Neville

**Brooke Feigon**, *Dept. of Mathematics, UCLA
*From Classical Automorphic Forms to Base Change and a Relative Trace Formula in the Local Field Setting

Classical automorphic forms are holomorphic functions on the upper half plane that transform in a particular way under linear fractional transformations. These classical automorphic forms can be embedded into a certain space of automorphic functions on groups over the adeles (a restricted direct product of local fields). “Base change” is a way of studying automorphic forms by relating the automorphic representations on one to group to automorphic representations on another group. I will explain this picture and then discuss recent work of mine on a relative trace formula over local fields that will yield additional information about base change.