NTS Fall 2011/Abstracts

September 8

Alexander Fish (Madison)

Title: Solvability of Diophantine equations within dynamically defined subsets of N

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

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.

September 22

Yifeng Liu (Columbia)

Title: tba

Abstract: tba

September 29

Nigel Boston (Madison)

Title: tba

Abstract: tba

October 6

Zhiwei Yun (MIT)

Title: tba

Abstract: tba

October 13

Melanie Matchett Wood (Madison)

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.

October 27

Zev Klagsburn (Madison)

Title: tba

Abstract: tba

November 17

Robert Harron (Madison)

Title: tba

Abstract: tba

December 8

Xinwen Zhu (Harvard)

Title: tba

Abstract: tba

Organizer contact information

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood

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