Math Department Talks
January 23rd, 3 pm, JMH 406
Math Club Talk
On the History of Number Theory
Dr. Michael Volpato (to be followed by a couple more talks)
January 30th, 3pm JMH 406
Math Club and FCRH Science Education Initiative talk on "Optimal Pentagonal Tilings"
Dr. Frank Morgan, Atwell Professor of Mathematics, Williams College
Abstract: Hales proved that the least perimeter way to tile the plane with unit areas is by regular hexagons. What is the least perimeter way to tile the plane with unit area pentagons? We will discuss some new results, examples, and open questions, including work by undergraduates.
The talk will be followed by a discussion on Undergraduate research experience at Williams College.
February 6, 3 pm, JMH 406
Math Club Talk
On the History of Number Theory II
Dr. Michael Volpato
Abstract: In this continuation talk, we delve into the history of elliptic curves. Taking our cue from Pierre de Fermat, who observed that integers which are the areas of rational right-triangles (i.e. "congruent numbers") correspond exactly to points with rational coordinates on a certain elliptic curve first considered by Diophantus of Alexandria. We will see how Sir Isaac Newton, with his fledgling calculus, was able to create new rational points on an elliptic curve from old ones --- in a way very reminicent of how Diophantus found rational points on a circle! And how Norweigian mathematician Niels Henrik Abel, while attempting to solve an integral with irrational denominator, showed how to take any two rational points on an elliptic curve and develop a third. Leading British mathematician Louis Mordell to prove (using ermat's method of infinite descent) that the rational points on an elliptic curve can all be generated (via the geometric methods of Newton and Abel) from a finite set of points. Understanding this finite set of generators of rational points on an elliptic curve still occupies much modern number-theoretic research!
February 13, 3 pm, JMH 406
Math Club Talk
On the History of Number Theory III
Dr. Michael Volpato
February 27th
Math Dept/Math Club talk on " Hyperbolic Geometry, Toll Booths and Mathematics Education Reform"
Dr. Jane Gilman, Rutgers University
Abstract: We discuss some methods used in Mathematics Education Reform in undergraduate and graduate level courses. This includes a discussion of team teaching and using hands on constructions. We will demonstrate this with so called Hyperbolic Paper and explain the connection between Non-Euclidean Geometry and the world of borders between countries and tariffs. The speaker will chair a question and answer session about NSF programs for faculty and students at all levels engaged in either research or teaching.
1) Report on Geometry and the Imagination, CBMS Issues in Mathematics Education, 3,AMS, Providence, Rhode Island, (1993), 131-135. http://andromeda.rutgers.edu/~gilman/publications/46_Report.pdf.
2) Report on MSRI 1994 Summer Workshop on Hyperbolic Geometry and Dynamical Systems, (with D.B.A. Epstein and W.P. Thurston), Notices of the AMS 42 (12), (1995) 1520-1527 http://andromeda.rutgers.edu/~gilman/publications/48_MSRI.pdf
April 3, 3 pm, JMH 406
Math Club Talk
Title: Using representation theory to win a Nobel prize
Speaker: Dr. Jeff Breeding II
Abstract: Murray Gell-Mann noticed that when baryons and mesons were organized into octets they possessed a symmetry that looked similar to the representation theory of the Lie algebra su(3). If the representation theory of su(3) really did describe this symmetry, there should be another particle, which was previously unobserved, having certain properties predicted by this su(3) model. Gell-Man predicted the existence of this particle in 1962. In 1964, a particle that closely matched his predictions was discovered. This led to the theory of quarks and Gell-Man won the Nobel Prize in Physics in 1969 for his work in particle physics.
April 18, 1 pm, JMH 406
Math Club Talk
Title: Prime factorization, complex analysis, and applications to
arithmetic geometry
Speaker: Dr. Jim Brown, Clemson University
Abstract: Given a positive integer n, one knows from grade school that there is a unique prime factorization of n. However, in more general rings determining when one has unique factorization is a very difficult problem. As there are cases when one cannot factor elements uniquely into primes (try 6 in Z[\sqrt{-5}]), one would also like some way to measure how far away from having unique factorization a ring is. The class group of a ring is a group that measures the failure of unique factorization. We will discuss some results about the class group that relate back to Fermat's last theorem, before moving on to more delicate results on divisibility of class groups. We will use the case of class groups of number fields to motivate more general results dealing with modular forms and Galois representations.
