NTS Fall 2011/Abstracts
September 8
| Alexander Fish (Madison) |
| Title: Solvability of Diophantine equations within dynamically defined subsets of N |
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Abstract: Given a dynamical system, i.e. a compact metric space X, a homeomorphism T (or just a continuous map) and a Borel probability measure on X which is preserved under the action of T, the dynamically defined subset associated to a point x in X and an open set U in X is {n | T^n(x) is in U} which we call the set of return times of x in U. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points x in X. Among examples of such sets are normal sets which correspond to the system X = [0,1], T(x) = 2x mod 1, Lebesgue measure, U = [0, 1/2]. We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems |
September 15
| Chung Pang Mok (McMaster) |
| Title: Galois representation associated to cusp forms on GL2 over CM fields |
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Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs the compatible system of 2-dimensional p-adic Galois representations associated to a cuspidal automorphic representation of cohomological type on GL2 over a CM field, whose central character satisfies an invariance condition. A local-global compatibility statement, up to semi-simplification, can also be proved in this setting. This work relies crucially on Arthur's results on lifting from the group GSp4 to GL4.
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September 22
| Yifeng Liu (Columbia) |
| Title: tba |
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Abstract: tba |
September 29
| Nigel Boston (Madison) |
| Title: tba |
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Abstract: tba |
October 6
| Zhiwei Yun (MIT) |
| Title: tba |
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Abstract: tba |
October 13
| Zhiwei Yun (MIT) |
| Title: The probability that a curve over a finite field is smooth |
| Abstract: Given a fixed variety over a finite field, we ask what
proportion of hypersurfaces (effective divisors) are smooth. Poonen's work on Bertini theorems over finite fields answers this question when one considers effective divisors linearly equivalent to a multiple of a fixed ample divisor, which corresponds to choosing an ample ray through the origin in the Picard group of the variety. In this case the probability of smoothness is predicted by a simple heuristic assuming smoothness is independent at different points in the ambient space. In joint work with Erman, we consider this question for effective divisors along nef rays in certain surfaces. Here the simple heuristic of independence fails, but the answer can still be determined and follows from a richer heuristic that predicts at which points smoothness is independent and at which points it is dependent.
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October 27
| Zev Klagsburn (Madison) |
| Title: tba |
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Abstract: tba |
November 17
| Robert Harron (Madison) |
| Title: tba |
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Abstract: tba |
December 8
| Xinwen Zhu (Harvard) |
| Title: tba |
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Abstract: tba |
Organizer contact information
Zev Klagsbrun
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