Rose-Hulman Undergraduate Mathematics Conference
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Dr. Michael Rosen Dr. Allen Holder

Dr. Michael Rosen
Department of Mathematics, Brown University
Biography: Dr. Rosen received his Ph.D. from Princeton University in 1963 and began at Brown as an Instructor in 1962. He was promoted to Asst Professor one year later, to Associate Professor with tenure in 1967, and finally promoted to Full Professor in 1972. He retired from Brown in June 2005.

Dr. Rosen has published round 50 research papers and has written two books - "A Classical Introduction to Modern Number Theory" (co-authored with Kenneth Ireland) and "Number Theory in Function Fields", both in Springer's GTM Series. He has also co-edited with David Goss and David Hayes "The Arithmetic of Function Fields" published by Walter DeGruyter.

He has supervised 20 PhD theses, including one by a certain Josh Holden.

In 1999, Dr. Rosen won the Chauvenet Prize of the Mathematical Asociation of America for his article "Neils Henric Abel and Equations of the Fifth Degree". In 2001 he was awarded the Phillip Bray Award of Brown University for excellence in teaching within the physical sciences.

Title: The Zeta Function for Beginners
Abstract: The zeta function was introduced by Euler in the Eighteenth Century and has continued to fascinate mathematicians ever since. We will give the definition and discuss its basic properties. We will then explain the contributions of Bernard Riemann who wrote an exceedingly wonderful but short paper on the subject. His Riemann Hypothesis remainsone of the great unsolved problems of mathematics. We will conclude by discussing various generalizations and variants of Euler's zeta function and explain why they are interesting.
Title: Irreducible Polynomials Modulo p
Abstract: If you start with an irreducible polynomial with integral coeffcients, you can reduce the coeffcients modulo p (p a prime) and come up with a polynomial over the finite field Z/pZ. Is the reduced polynomial also irreducible? Not always. Are there infinitely many primes for which the reduced polynomal is irreducible? Not always. What can be said? It turns out that this is a deep and important question which leads into various parts of number theory, group theory, and Galois theory. We shall explain some of the connections and give some answers to these questions.


Dr. Allen Holder
Trinity University (TX)
Biography: Dr. Holder began a career in mathematics in 1998 after completing Ph.D. studies at the University of Colorado at Denver under the tutelage of Dr. Harvey Greenberg. His primary interest was in applied mathematics and specifically in optimization, a field that intersects numerous scientific disciplines. His current interests lie in applying optimization techniques to problems in medicine, biology, and economics. He holds a joint position in the department of Radiation Oncology at the University of Texas Health Science Center, and in close collaboration with his medical colleagues, he is investigating how to improve patient care by optimizing radiotherapy treatments. His initial work won the 2000 William Pierskalla award as the best annual paper in Operations Research and Health Care.

His work in biology addresses a problem in population genetics called the Pure Parsimony Problem. The basic question is what is the minimum amount of diversity needed in the previous generation to observe the current generation's genetic composition. Computational efforts on this problem have not been successful, and the mathematical attack of Dr. Holder's work provides alternative methods in graph theory.

In economics, Dr. Holder extended a result of Paul Samuelson, the 1970 Nobel Laureate in Economics. The original results is called the Nonsubstitution theorem and has been credited with transitioning economics from the neoclassic paradigm into modern economic theory. Dr. Holder extended this famous result to a dynamic problem and showed that the basic premise of the original result is maintained asymptotically.

Dr. Holder has a strong background in undergraduate research, and the work above has been accomplished largely through undergraduate efforts. He has research collaborations with 15 undergraduates spread over 7 articles and has directed numerous senior projects. He also has participated in the Trinity NSF-REU project for 5 summers.

Outside of mathematics, his hobbies include cycling, hiking, and auto mechanics.

Title: Optimal Oncology
Abstract: The interaction between the mathematical field of optimization and the medical field of oncology has provided some fertile and fascinating research over the last decade. The fundamental question is how do we best treat patients. Such questions address how to take advantage of current technology and how to best focus research efforts to develop innovative new techniques. We will discuss problems highlighting both.

Cancer is treated with three basic modalities: surgical, which physically removes a cancerous growth; drug therapy, which targets all fast proliferating cells; and radiotherapy, which locally irradiates tissues. Radiotherapy treatments are delivered by focusing beams of ionizing radiation on a patient so that the aggregate energy is highest over a targeted region. The goal is to deliver enough radiation to kill cancerous cells but not enough to harm surrounding structures. The design process has many facets, all of which have been aided by recent mathematical and computational results. We will discuss several of these recent advances.

A new treatment paradigm called photodynamic therapy is loosely a combination of drug therapy and radiotherapy. The idea is to develop a drug, called a photosensitizer, that when excited by an infrared light destroys tissue. There are two obstacles. First, it is difficult to chemically design a photosensitizer with a heightened affinity for cancerous cells, and hence, the drug is equally collected in all tissues. Second, the optical properties of infrared light are not favorable to targeting deep tissues. Combined, these two concerns mean that it is challenging to treat growths within the anatomy. We will show that even if the infrared light was focused optimally, the current drugs do not allow the treatment of tumors more than a few millimeters from the surface. This means that improved chemical properties are needed to make this a viable treatment. Moreover, we estimate the amount of drug improvement needed to successfully treat tumors several centimeters into the anatomy.