Title: Deligne's conjectures on critical values of L-functions
Abstract: We will explain how to attach an L-function to a motive, what
the critical points of this L-function are, and Deligne's conjectures on
the values of the L-function at critical points.
Time:
2:30pm-4:00pm
Location:
Room No. 215
Description:
h-Cobordism Thorem
Time:
4:00pm-5:00pm
Location:
Ramanujan Hall
Description:
Speaker: Kathleen Shannon, Salisbury University.
Title: Pascal's Triangle, Cellular Automata and Serendipity: A Mathematical Tale
Abstract: The talk will outline the development of the PascGalois Project. Its origins are in an exercise using Pascal's Triangle and modular arithmetic. Colors are assigned to the numbers 0, 1, ..., n-1, and Pascal's Triangle modulo n is drawn. The patterns in the triangle are then related to the properties of the cyclic group Zn. The process of drawing the triangles is then generalized to non-cyclic and non-abelian groups and the new patterns are examined in light of the properties of these groups. The images can help develop visual and intuitive understanding of concepts such as subgroup closure and quotient groups. They can also be used to discuss the relationship between mathematical properties and visual aesthetics. Finally we view Pascal's Triangle as a one-dimensional cellular automata and generalize to more general initial conditions and two dimensional automata. Many of the investigations in this project have been undertaken with students in undergraduate research projects and one outgrowth of the project has been the development of a set of visualization exercises to supplement the standard undergraduate course in abstract algebra. The web site for the project is at www.pascgalois.org.
Time:
3:30pm-5:00pm
Location:
Room No. 215
Description:
Title: Values of binary quadratic forms on integer pairs
Time:
3:00pm-5:00pm
Location:
Room 215
Description:
Groups of homotopy spheres
In a land-mark paper in 1956, J. Milnor showed that there are non standard differential structures on the 7-dimensional sphere. Six years later along with Kervaire, he introduced an abelian group structure on the set of equivalence classes of smooth structures on spheres of all dimension and determined these groups in several cases. We shall present some of the salient features of this work.
This is the second talk on this topic.
Time:
3:30pm
Location:
Ramanujan Hall
Description:
Title: Counting Zeros of Multivariate Laurent Polynomials and Mixed Volumes of Polytopes
Abstract. A result of D.N. Bernstein proved in the late seventies gives an upper bound
on the number of common solutions of n multivariate Laurent polynomials in
n indeterminates in terms of the mixed volumes of their Newton polytopes.
This bound refines the classical Bezout's bound. Bernstein's Theorem has several
proofs using techniques from numerical analysis, intersection theory and tori varieties.
B. Teissier proved the theorem using intersection theory. A proof using theory of toric
varieties can be found in the book by W. Fulton on the same subject.
In this talk, I will outline an algebraic proof similar to the standard proof of Bezout's Theorem.
This proof, found in collaboration with N.V. Trung, uses basic results about Hilbert functions
of multigraded algebras first proved by van der Waerden.