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| MAY 2010 |
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| Department Colloquium |
| Topic: |
Elliptic curves and Hilbert's 10th problem |
| Presenter: |
Karl Rubin, University of California, Irvine |
| Date: |
Wednesday, May 5, 2010, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
In this talk I will introduce elliptic curves and discuss recent work with Barry Mazur on ranks of elliptic curves in families of quadratic twists. One application of our results, using ideas of Poonen and Shlapentokh, is that if a standard conjecture about elliptic curves (the Shafarevich-Tate conjecture) holds then Hilbert's 10th problem for the ring of integers of any number field has a negative answer. (Hilbert's 10th problem for a ring R asks whether there is an algorithm to decide whether a multivariate polynomial with coefficients in R has solutions in R. Around 1970 Matijasevich, building on work of Davis, Putnam, and Robinson, proved that no such algorithm exists when R is the ring of rational integers. This answered Hilbert's original question, but for many other rings of interest the question is still open.) |
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| Joint Princeton University and Institute for Advanced Study Number Theory Seminar |
| Topic: |
Selmer ranks of twists of elliptic curves |
| Presenter: |
Karl Rubin, UC Irvine |
| Date: |
Thursday, May 6, 2010, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses there are many twists with Mordell-Weil rank zero, and (assuming the Shafarevich-Tate conjecture) many others with Mordell-Weil rank one. The talk will conclude with some speculation about the density of twists with a given Selmer rank. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Front propagation and phase transitions for fractional diffusion equations |
| Presenter: |
Xavier Cabré, ICREA and Universitat Politècnica de Catalunya |
| Date: |
Friday, May 7, 2010, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
Long-range or \anomalous" di ffusions, such as di ffusions given by the frac- tional powers (-\Delta)^s of the Laplacian, attract lately interest in Physics, Biol- ogy, and Finance. From the mathematical point of view, nonlinear analysis for fractional di ffusions is being developed actively in the last years. In this talk, I will describe recent results concerning front propagation for the nonlinear fractional KPP heat equation, \partial_t tu+(-\Delta)^su = u(1-u) in (0;1)Rn, 0 u 1, with s 2 (0; 1). In collaboration with J.-M. Roquejo re, we establish that fronts propagate at exponential speed |in contrast with the classical case s = 1 for which there is propagation at a constant KPP speed. I will also describe works in collaboration with Y. Sire and E. Cinti. They concern the fractional elliptic Allen-Cahn equation (-\Delta)^su = f(u) in Rn with s 2 (0; 1), a model being the bistable nonlinearity f(u) = u - u3. Our main results concern the existence and properties of \layer" or heteroclinic solutions, as well as of minimizers of the equation. |
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| Analysis Seminar |
| Topic: |
On the soliton dynamics under a slowly varying medium for generalized KdV equations |
| Presenter: |
Claudio Munoz, University of Versailles |
| Date: |
Monday, May 10, 2010, Time: 4:00 p.m., Location: Fine Hall 110 |
| Abstract: |
We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly varying media induce on the soliton solution large dispersive effects in large time. Moreover, unlike gKdV equations, we prove that there is no pure-soliton solution in this regime. |
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| Analysis Seminar |
| Topic: |
Symmetry and regularity of solutions for nonlinear systems of Wolff type |
| Presenter: |
Wenxiong Chen, Yoshiva, NY |
| Date: |
Friday, May 14, 2010, Time: 4:00 p.m., Location: Fine Hall 110 |
| Abstract: |
http://www.math.princeton.edu/~seminar/2009-10-sem/ChenAbstract14May2010.pdf |
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| PACM Colloquium |
| Topic: |
Hydraulic Fractures: multiscale phenomena, asymptotic and numerical solutions |
| Presenter: |
Anthony Peirce, University of British Columbia |
| Date: |
Monday, May 17, 2010, Time: 4:30 p.m., Location: Fine Hall 214 |
| Abstract: |
Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. In this talk I provide examples of natural HF and situations in which HF are used in industrial problems. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy. I describe the governing equations in 1-2D as well as 2-3D models of HF, which involve a coupled system of degenerate nonlinear integro-partial differential equations as well as a free boundary. I demonstrate, via re-scaling the 1-2D model, how the active physical processes manifest themselves in the HF model and show how a balance between the dominant physical processes leads to special solutions. I discuss the challenges for efficient and robust numerical modeling of the 2-3D HF problem and some techniques recently developed to resolve these problems: including robust iterative techniques to solve the extremely stiff coupled equations and a novel Implicit Level Set Algorithm (ILSA) to resolve the free boundary problem. The efficacy of these techniques is demonstrated with numerical results. Relevant papers can be found at: http://www.math.ubc.ca/~peirce |
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