Renewal Systems, Sharp Eyed Snakes, and Shifts of Finite Type Aimee Johnson, Swarthmore College
One of the basic mathematical questions is when two seemingly different objects are really the "same". In this talk we will consider sets of bi-infinite strings of symbols which arise from different types of graphs, and explore what it means for these sets to be the "same". Once we agree what it means for these objects to be the same, we will try to determine whether they actually are the same. We will finish with a partial answer to this question that leaves us with a tantalizing open problem that is the subject of active mathematicalresearch. Aimee Johnson was an undergraduate student at University of California, Berkeley, and attended graduate school at the University of Maryland, College Park. There she studied ergodic theory under the direction of Daniel J. Rudolph. After a 3 year position at Tufts University, she joined the faculty at Swarthmore College, where she is an assistant professor. In her research she has considered measurable, topological, and symbolic dynamics systems.
2. Program Verification
Robert Kurshan, Bell Laboratories I will discuss a method for checking the correctness of certain types of computer programs. The method is based on a mathematical analysis of the program. This is in contrast with customary testing based upon program execution. The new form of analysis has been found to be more reliable than conventional testing, and is being used commercially in the development of programs implemented as integrated circuits. The same methodology is applicable to the development of "control-intensive" software programs as well. An interesting part of the story is the technology transfer process: how did an idea get from research to a commercially successful product. I will spend a few minutes on this as well. Robert Kurshan is a Distinguished Member of Technical Staff at Bell Laboratories, Murray Hill, NJ. He has worked there since receiving his Ph.D in mathematics in 1968, from the University of Washington. He spent two years as Visiting Professor at the Technion (Haifa, Israel) in the departments of Mathematics and Electrical Engineering. In addition, he has taught courses at U.C. Berkeley and N.Y.U. At Bell Labs, he did research in periodic sequences, digital filtering and approximation theory, before he began work in formal verification in 1983. He is an author of over 60 technical publications, holds eight patents in communications, digital filtering and verification, and is the author of the book Computer-Aided Verification of Coordinating Processes (Princeton Univ. Press, 1994), which is based upon courses he gave at U. C. Berkeley and the Technion. He designed and built the COSPAN verification system together with Zvi Har'El, Ronald H. Hardin, and a number of others, based upon the theory which is developed in this book. COSPAN has been in use (and continuous development) since 1986, having been applied to a number of commercial projects, as well as having been licensed to numerous universities for educational use. Currently, COSPAN is marketed for commercial use by Cadence Design Systems, Inc., under the trademark FormalCheck.
3. Modeling Radiation Transfer in the Human Body: A Hot Topic for Student Research Ann K. Stehney, Cedar Crest College It has been some years since Chernobyl and Three Mile Island were news, but exposure to radiation is still a "hot" topic. To assess the health risks associated with accidental, occupational, or environmental exposure, the International Commission on Radiological Protection has proposed increasingly sophisticated mathematical models for the transfer and deposition of radioactive substances in the human body. We suggest a variety of related problems that can be introduced as early as a first course in linear algebra, yet are suitable for a senior seminar in which students devise and test their own models. The questions lend themselves to both continuous and discrete methods, drawn from throughout the undergraduate math curriculum. The talk includes a preliminary report on joint work with chemist A. F. Stehney, based on his study of thorium workers. Ann Stehney obtained her Ph.D. in differential geometry from SUNY Stony Brook and joined the faculty of Wellesley College, where she became department chair and served as director of Educational Research and Development. She then spent ten years in applied mathematics at the Center for Communications Research, a government contractor in Princeton. Recently Associate Dean of Douglass College at Rutgers, she is now Vice President for Academic Affairs and Dean of Faculty at Cedar Crest College in Allentown, PA.
4. The Erdos Magic Joel Spencer, Courant Institute of Mathematical Sciences The Probabilistic Method is a lasting legacy of the late Paul Erdos. Here we examine two problems that Erdos started working on in the 1960s, both with recent improvements. (i) Erdos showed that given any 2n-1 n-element sets there exists a two coloring of the underlying points so that no set is monochromatic. Srinivasan and Radhukrishnan (building on a 1978 result of Beck) have recently done better. (ii) Fix 2 <=3D k < t (e.g.: k=3D3,t=3D5) In a universe of n elements how many t-sets can you have with no k-set contained in two (or more) of them. C n,k/C t,k is a natural counting bound. Erdos and Hanani conjectured that this was asymptotically attainable. This was shown by Rodl in1985. Here we examine an argument of the speaker using a Random Greedy Algorithm and an analogy to Birth Processes. Joel Spencer is a Professor of Mathematics and Computer Science at the Courant Institute, for Mathematical Sciences, New York University. His area of research is in Discrete Mathematics and Theoretical Computer Science. He obtained his Ph.D. from Harvard in 1970 under the direction of A. Gleason. He has been a visitor at M.I.T. and the Institute for Mathematics and Its Applications and has authored over 140 publications. A Sloane Foundation Fellowship, editorship of The Annals of Applied Probability and an associate editorship of the American Mathematical Monthly are among his many professional accomplishments. He was an invited speaker at the International Congress of Mathematicians at Zurich in 1994 and has lectured at an NSF-CBMS conference in Durango. The second edition of his book, The Probabilistic Method (with Noga Alon), published by Wiley, has recently appeared.
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Document last modified on September 20, 2000.