SCU Mathematics Colloquium Series Schedule

Winter 2000

Talks will be at 4:00 Wednesdays in O'Connor 205. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.

**Jan. 12**William A. Stein, University of California at BerkeleyTitle: The Conjecture of Birch and Swinnerton-Dyer

This talk concerns elliptic curves, which play a key role in cryptography and in the recent solution of Fermat's Last Theorem. I will describe how to attach various invariants to an elliptic curve, then I will explain how a conjecture of Birch and Swinnerton-Dyer ties these invariants together. After illustrating the conjecture with some examples, I'll discuss how little we know about the truth of the conjecture and report on joint work with Ed Schaefer. Some of these ideas are explained in H. Darmon's survey paper, which can be found in the book ``Modular Forms and Fermat's Last Theorem'', Springer-Verlag, 1997.

**Jan. 25 (Tues.)**A.A.Melikyan, Georgia Institute of TechnologyTitle: Singular Characteristics of Nonlinear First Order PDEs

The classical method of characteristics (MC) is a powerful tool for solving nonlinear first order PDEs arising in Control Theory and Mathematical Physics. The nonsmoothness of the generalized (e.g.,viscosity) solution and/or of the Hamiltonian (left hand side function of the PDE) is often referred to as an obstacle for the implementation of the MC. In this presentation we will describe a new technique, the method of singular characteristics, which allows one to construct nonsmooth solutions to nonlinear first order PDEs, or classical solutions with nonsmooth Hamiltonians. According to this method two types of characteristics should be associated with a PDE - regular (classical) and singular ones. Singular characteristics have the same differential-geometric nature as regular ones but have a different analytical description, and still are described by means of an ODE system in closed form. They may intersect regular characteristics. Weak waves of a certain type propagate along singular characteristics. Some applications in Optimal Control, Differential Games and Mathematical Physics will be discussed.

The talk will use some results of the book: A.A.Melikyan, Generalized characteristics of first order PDEs: Applications in Optimal Control and Differential Games, Birkhauser, Boston, 1998.

**Feb. 2**Tom Roby, California State University, HaywardTitle: Ups and Downs in Posets and Algebras

The natural operators Up and Down defined on certain partially ordered sets (posets) can be used to obtain enumerative and structural results about the posets. These results have connections with algebraic structures such as group representations. A canonical example is Young's Lattice of integer partitions, whose chains are equivalent to Young tableaux. The latter give information about the representations of the symmetric group and general linear group. For Young's Lattice the down and up operators satisfy DU-UD=I (the identity), thereby giving a concrete representation of the Weyl algebra.

In this talk we will consider the class of posets satisfying such a relation, first defined by R. Stanley, and those satisfying a similar cubic relation, due to P. Terwilliger. Generalizations to the abstract setting of ``Down-up Algebras'' will also appear.

- As part of the Colloquium Series, job applicants will be
presenting talks during the remainder of the quarter. The dates
will be posted when they are determined. Some of these presentations
may not be on Wednesday, but will normally be at 4:00pm.
Anyone wishing further information about any of these talks should contact the coordinator of the Colloquium Series, Prof. Rick Scott, rscott@schubert.scu.edu

If you have a disability and require a reasonable accommodation, please call Rick Scott 1-408-554-4460 or 1-800-735-2929 (TTY - California Relay).

The list of talks from previous quarters are available via this archive link.

*Last Updated: 7 January 2000*
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