| April 4, 2002 |
Abstract: This talk concerns geometric relations among abelian varieties that may be associated with sets of prime numbers. For example, a pair of prime numbers defines two natural fuchsian subgroups of SL(2,R), one inside SL(2,Z) and the second inside an indefinite quaternion division algebra over the rational field. These groups give rise to algebraic curves by familiar constructions of Shimura. The Selberg trace formula and a general theorem of Faltings allows one to infer the existence of a geometric relation between the Jacobians of the two curves. Optimally, however, one would like to produce such a relation by a direct geometric construction. I will explain how the search for such a construction led to one of the ingredients used in the proof of Fermat's Last Theorem and will describe a new result of D. Helm of Berkeley that sheds light on the relationship between the two Jacobians. Contact person: Bisi Agboola |
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NOTE DIFFERENT DAY
Friday
April 5, 2002 3:30pm-4:30pm |
Abstract: We give a brief description of Sobolev gradients, both in theoretical and numerical aspects. We use such gradients to determine numerically critical points of Ginzburg-Landau functionals of interest in superconductivity studies. We show calculated magnetic fields and superconducting currents. To further illustrate the method of Sobolev gradients we show results for transonic flow, elasticity and oil-water interface problems. Some general comments on partial differential equations are given. Contact person: Mike Crandall |
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| April 11, 2002 |
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| April 18, 2002 |
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| April 25, 2002 |
Abstract: Since Harvey and Lawson introduced the concept of calibrations and their geometries in 1982, these special geometries have played an increasingly important role in differential geometry and mathematical physics, with perhaps the most famous example being the notion of special Lagrangian geometry which (conjecturally) lies at the heart of mirror symmetry. In that same foundational paper, they emphasized the role of systems of PDE that could be understood by the technique of exterior differential systems and it has turned out that this technique is not just a convenient formalism but provides an incisive way to understand the local and global geometry of calibrations and their corresponding extremal submanifolds. In this talk, I will describe the foundations of the two subjects, explain some of the more recent applications of their interactions, and describe some of the open problems and questions that the two subjects suggest. Contact person: Xianzhe Dai |
| May 2, 2002 |
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| May 9, 2002 |
Abstract: In this talk I will discuss the numerical computation of polycrystals using level set methods. I will describe the application of this method to the computation of diamond thin films and MgO films grown using ion beam assisted deposition (IBAD). (This is joint work with X. Li, G. Russo, and D. Srolovitz.) Contact person: Bjorn Birnir |
| May 16, 2002 |
Abstract: The now nearly 40 years since Cohen's discovery of the technique of forcing has seen the proliferation of independence results within set theory and beyond. A more recent trend though has been the discovery that some of these otherwise unsolvable problems do arguably have solutions. So perhaps the view is now evolving that independence is not an insurmountable problem in set theory; perhaps there is after all an unambiguous conception of the transfinite realm, the vision is simply obscured by independence but not destroyed. I would go further and claim that fundamental questions such as that of Cantor's Continuum Hypothesis are solvable, or at the very least, in the case of the Continuum Hypothesis, that the theorem asserting otherwise has yet to be proved. I will survey some of the mathematical results and developments which lead to these admittedly curious claims. Contact person: Bisi Agboola |
| May 23, 2002 |
Abstract: Koszul algebras were first introduced in Algebraic Topology and they have also played an important role in Commutative Algebra, Lie Theory, and more recently also in Representation Theory and in Noncommutative Agebraic Geometry. Roughly speaking, a Koszul algebra is a graded algebra for which the graded irreducible modules have linear resolutions (these are very pretty projective resolutions). The purpose of this talk is to get more familiar with Koszul algebras and their representations. In particular I plan to talk about Koszul duality, rationality of Hilbert and Poincare series, Castelnuovo-Mumford regularity, and, if time permits, some longstanding problems from finite dimensional algebras in the Koszul algebra context. Contact person: Birge Huisgen-Zimmermann |
| May 30, 2002 |
Abstract: Let A be a finite subset of SU(M) generating a dense subgroup. I will describe an efficient algorithm that allows to approximate a given element of SU(M) by products of elements of A. Another version of this problem arises when one wants to approximate an element of U(2) by a quantum circuit using ancillas. This kind of approximation (relative to a certain standard gate set) can be achieved by circuits of doubly logarithmic depth. Contact person: Doug Moore |
| June 6, 2002 |
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