| January 10, 2002 |
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| January 17, 2002 |
Abstract: From a hard-nosed analytic standpoint, most applications of probability theory are unnecessarily elaborate expressions of the minimum principle. Nonetheless, probability theory has something to contribute. Namely, probability theory provides an intuitively appealing picture of what are otherwise austere analytic facts, and this picture has value in two important directions. First, it can be applied to the analysis of the equation to reveal mathematical properties which are otherwise invisible. Secondly, the picture can often suggest connections between the equation and the world outside of mathematics. By discussing a few examples, I will try in this lecture to convince the uninitiated that these claims have some validity. Contact person: Doug Moore |
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| January 24, 2002 |
Abstract: Crystal growth can be described by the motion of step edges (or island boundaries), across which the height of the crystal changes by a single atomic layer. The classical theory of Burton, Cabrera and Frank (BCF) assumes that a step edge is in equilibrium with the adjoining terraces. For typical examples of molecular beam epitaxy, however, the growth is driven by kinetics and the step edges are not in equilibrium. This talk will present a new microscopic theory for step edge dynamics, as a kinetic alternative to BCF. Two applications of the theory are described: First, for the island dynamics/level set method for epitaxial growth, the microscopic step edge model is used to derive the rate for breakup of small islands. Second, for a step edge with edge diffusion and detachment, the line tension is derived from the microscopic step edge model. Partial validation of the microscopic model and the applications will be presented. Contact person: Doug Moore |
| January 31, 2002 |
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| February 7, 2002 |
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| February 14, 2002 |
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| February 21, 2002 |
Abstract: The Riemann zeta function counts the number of prime numbers (maximal ideals) in the ring of integers. More generally, the Hasse-Weil zeta function counts the number of closed points (maximal ideals) on an algebraic variety. Among the most fundamental properties are the possible meromorphic continuation and Riemann hypothesis for these zeta functions. To understand how the zeta function varies in an algebraic family, we introduce a higher moment zeta function, which reduces to the classical zeta function for the first moment. In this lecture, we shall present a simple and expository introduction to these higher moment zeta functions as well as their limiting behavior when the moment goes to infinity. Contact person: Bisi Agboola |
| February 28, 2002 |
Abstract: I will discuss results about rigidity and vanishing of elliptic genus, intersection numbers on moduli spaces of bundles and mirror formulas for Calabi-Yau manifolds. They are all conjectured or inspired by physics and mathematically proved by localization methods combining with various parts of mathematics. Contact person: Xianzhe Dai |
| March 7, 2002 |
Abstract: John F. Nash shared the Nobel prize for economics in 1994 with John C. Harsanyi and Reinhard Selten, for what he claims was his "most trivial work". We will prove a generalization of Nash's equilibrium theorem by way of a result of Felix Browder and Ky Fan. Time permitting, we will also discuss the Kakutani-Fan and the Schauder-Tychonov fixed-point theorems and generalizations thereof. Prerequisites: elementary point-set topology and Brouwer's fixed-point theorem. Contact person: |
| March 14, 2002 |
Abstract: This talk will describe some ongoing joint work with Jacob Murre. Given any elliptic curve E over a field F, and a pair of F-rational points P, Q on E, one can look at the cycle of dimension zero on E x E given by {P,Q} := (P,Q) -(P,0) - (0,Q) + (0,0). By the parallellogram law, this cycle defines the trivial element of the Albanase variety. But {P,Q} need not be trivial modulo linear equivalence on E x E, and this what we investigate modulo primes p, when F is a number field and local field. The method is to study the Galois symbol, which takes values in H^2(F,E[p] \otimes E[p]). Contact person: Bisi Agboola |