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 GL_{2} over CM fields 
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs the compatible system of 2dimensional padic Galois representations associated to a cuspidal automorphic representation of cohomological type on GL_{2} over a CM field, whose central character satisfies an invariance condition. A localglobal compatibility statement, up to semisimplification, can also be proved in this setting. This work relies crucially on Arthur's results on lifting from the group GSp_{4} to GL_{4}.

September 22
Yifeng Liu (Columbia) 
Title: Arithmetic inner product formula 
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for higher rank, relates the canonical height of special cycles on certain Shimura varieties and the central derivatives of Lfunctions. 
September 29
Nigel Boston (Madison) 
Title: Nonabelian CohenLenstra heuristics. 
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian pgroup A (p odd) arises as the pclass group of an imaginary quadratic field K is apparently proportional to 1/Aut(A). The Galois group of the maximal unramified pextension of K has abelianization A and one might then ask how frequently a given pgroup G arises. We develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is joint work with Michael Bush and Farshid Hajir. 
October 6
Zhiwei Yun (MIT) 
Title: Exceptional Lie groups as motivic Galois groups 
Abstract: More than two decades ago, Serre asked the following question: can exceptional Lie groups be realized as the motivic Galois group of some motive over a number field? The question has been open for exceptional groups other than G_{2}. In this talk, I will show how to use geometric Langlands theory to give a uniform construction of motives with motivic Galois groups E_{7}, E_{8} and G_{2}, hence giving an affirmative answer to Serre's question in these cases. 
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 20
Jie Ling (Madison) 
Title: tba 
Abstract: tba 
November 3
Zev Klagsburn (Madison) 
Title: tba 
Abstract: tba 
November 10
Luanlei Zhao (Madison) 
Title: tba 
Abstract: tba 
November 17
Robert Harron (Madison) 
Title: tba 
Abstract: tba 
December 1
Andrei Calderaru (Madison) 
Title: tba 
Abstract: tba 
December 8
Xinwen Zhu (Harvard) 
Title: tba 
Abstract: tba 
December 15
Shamgar Gurevich (Madison) 
Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation 
Abstract: tba 
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