U Mass Logo

T.W.I.G.S.
The "What Is ...?" Graduate Seminar

Room 1634, Lederle Graduate Research Tower
University of Massachusetts, Amherst 		ComplexCubicUnitAction
The Seminar meets Wed 3:00-4:00 in 1634 LGRT.    
Driving Directions   and   Campus Maps
Overview of the Seminar as well as Unofficial mottos and sponsors of TWIGS
Maintained by Farshid Hajir.

NOTE Here are the schedules for Fall 2002, Spring 2003, Fall 2003, Spring 2004, Fall 2004, Spring 2005, Fall 2005, Spring 2006, and Fall 2006.
Wednesday February7 Farshid Hajir, UMass Amherst. "What is the ABC Conjecture?"
Abstract: The ABC conjecture dates from 1985, so it is a "toddler" in the world of famous open problems in number theory, but it's already playing a major role in several parts of the subject. I will give its very simple statement (all you need to know is that integers factor into primes), describe some numerical data, and try to explain why it is an important conjecture. More than most TWIGS talks, this talk should be easily accessible to undergraduates as well.
Wednesday February 14 Tom Weston, UMass Amherst. "What is K-theory?"
Abstract: I'll answer the question of the title, and explain what K-theory has to do with zeta(3), the sum of the reciprocals of cubes of positive integers. POSTPONED TO NEXT WEEK DUE TO SNOW CLOSING.
Wednesday February 21 Tom Weston, UMass Amherst. "What is K-theory?"
Abstract: See last week's abstract.
Wednesday March 14 Farshid Hajir, UMass Amherst. "What is Chebyshev's theorem on real polynomials?"
Abstract: Given a monic real polynomial of degree n > 0, the maximum value of |p(x)| on [-1,1] is at least 1/2^(n-1). The proof of this deceptively simple-sounding statement will give us a chance to talk about trigonometric polynomials, especially a theorem of Riesz about them.
**Thursday** March 29 Jessica Sidman, Mt Holyoke. "What is intersection theory? (Part I)"
Abstract: In this talk, I will define intersection numbers in the setting of smooth surfaces and set up some questions regarding the geometry of surfaces embedded in projective space which can be answered using intersection theory. Note Special Day!
**Thursday** April 5 Jessica Sidman, Mt Holyoke. "What is intersection theory? (Part II)"
Abstract: I will raise some questions about the geometry of surfaces embedded in projective space and then answer them using intersection theory. I will concentrate on specific examples, including the 27 lines on a cubic surface in P^3. I will address the nature of intersection theory in higher dimensions very briefly.Note Special Day!
Wednesday April 11 Ralf Schiffler, UMass Amherst. "What is a path algebra?"
Abstract: A quiver (=oriented graph) is a set of vertices together with a set of arrows i->j where i,j are vertices. For example 1 -> 2 -> 3 -> 4 is a quiver with 4 vertices. For every quiver one can define its path algebra in terms of paths in the quiver. We will see, for example, that the path algebra of the quiver above is isomorphic to the algebra of upper triangular 4x4-matrices. Multiplication in the path algebra can be seen as "running around" in the quiver. This harmless little model turns out to be a very powerful tool in the study of algebras and their representations and also has applications in several other areas of mathematics.
Wednesday April 18 Floyd Williams, UMass Amherst. "What is the Rademacher-Zuckerman formula?/From there to black hole entropy?"
Abstract: In 1918,G.Hardy and S.Ramanujan obtained the asymptotic behavior of the partition p(n) of a natural number into a sum of natural numbers. Nineteen years later, H.Rademacher produced, in fact, an exact formula for p(n). This amazing exact formula,on the other hand, is but a very special case of a more general,amazing formula of Rademacher and H.Zuckerman (R-Z), published in 1938, for the coefficients of a so-called modular form of negative weight. In contrast to forms of positive weight, which were discussed by Professor Paul Gunnells in a previous TWIGS lecture, negative weight forms have special applications to black hole physics. We shall discuss the R-Z formula and indicate how the exact formula for p(n) is derived from it. We also condsider R-Z asymptotics and derive what we call an "abstract Cardy formula" - a formula which (suitable specialized) reduces to the famous John Cardy entropy formula (1982) in conformal field theory. The discussion assumes a basic knowledge of complex variables, but no knowledge of quantum field theory, nor of black holes.
Wednesday May 9 Keith Conrad, UConn. "What is a Carlitz Module?"
Abstract: Field extensions of Q obtained by adjoining a root of unity, such as Q(i), are an important testing ground for certain concepts in algebra, especially in number theory. If we replace the field Q by Fp(T), the rational functions over a finite field of size p, there is a replacement for roots of unity inside something called "the Carlitz module" (named after Leonard Carlitz). We will explain what the Carlitz module is, and what it is good for, using analogies with roots of unity.



The picture above was created by Paul Gunnells. It visualizes the natural action of the group of units of a complex cubic field on 3-space. Consult Paul for more details.

Last modified: Feb 2006 by Farshid Hajir