Difference between revisions of "NTS ABSTRACTSpring2021"

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Classical Diophantine approximation quantifies the denseness of the set of rational vectors in their ambient Euclidean space. A far-reaching extension of the classical theory calls for quantifying the denseness of rational points in general homogeneous algebraic varieties. This was raised as an open problem by Serge Lang already half a century ago, but progress towards it was achieved only in a limited number of special cases. A systematic approach to this problem for homogeneous varieties associated with semi-simple groups has been developed in recent years, in joint work with A. Ghosh and A. Gorodnik.  The methods employ dynamical arguments and effective ergodic theory, and employ spectral estimates in the automorphic representation of semi-simple groups.  In the case of homogeneous spaces with semi-simple stability group, this approach leads to the derivation of pointwise uniform and almost sure Diophantine exponents, as well as analogs of Khinchin's and W. Schmidt's theorems, with some of the results being best possible.  We will explain some of the main results and some of the ingredients in their proof, focusing on some easily accessible examples.
 
Classical Diophantine approximation quantifies the denseness of the set of rational vectors in their ambient Euclidean space. A far-reaching extension of the classical theory calls for quantifying the denseness of rational points in general homogeneous algebraic varieties. This was raised as an open problem by Serge Lang already half a century ago, but progress towards it was achieved only in a limited number of special cases. A systematic approach to this problem for homogeneous varieties associated with semi-simple groups has been developed in recent years, in joint work with A. Ghosh and A. Gorodnik.  The methods employ dynamical arguments and effective ergodic theory, and employ spectral estimates in the automorphic representation of semi-simple groups.  In the case of homogeneous spaces with semi-simple stability group, this approach leads to the derivation of pointwise uniform and almost sure Diophantine exponents, as well as analogs of Khinchin's and W. Schmidt's theorems, with some of the results being best possible.  We will explain some of the main results and some of the ingredients in their proof, focusing on some easily accessible examples.
  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Carlo Pagano'''
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| bgcolor="#BCD2EE"  align="center" | On the negative Pell conjecture
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| bgcolor="#BCD2EE"  | The negative Pell equation has been studied since many centuries. Euler already provided an interesting criterion in terms of continued fractions. In 1995 Peter Stevenhagen proposed a conjecture for the frequency of the solvability of this equation, when one varies the real quadratic field. I will discuss an upcoming joint work with Peter Koymans where we establish Stevenhagen's conjecture.
 
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Revision as of 09:14, 26 February 2021

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Jan 28

Monica Nevins
Interpreting the local character expansion of p-adic SL(2)

The Harish-Chandra—Howe local character expansion expresses the character of an admissible representation of a p-adic group G as a linear combination with complex coefficients of the (finitely many!) Fourier transforms of nilpotent orbital integrals \(\widehat{\mu}_{\mathcal{O}}\) --- near the identity. Approaching from another direction: we can restrict the representation to any compact open subgroup K of G, obtaining its branching rules, which also describe the representation near the identity, in a different sense. We show that for G=SL(2,k), k a nonarchimedean local field, where the branching rules to maximal compact open subgroups K are known, each of these terms \(\widehat{\mu}_{\mathcal{O}}\) can be interpreted as the character \(\tau_{\mathcal{O}}\) of a representation of K, up to an error term arising from the zero orbit. Moreover, the irreducible components of \(\tau_{\mathcal{O}}\) are explicitly constructed from the K -orbits in \(\mathcal{O}\). This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.


Feb 4

Ke Chen
On CM points away from the Torelli locus

Coleman conjectured in 1980's that when g is an integer sufficiently large, the open Torelli locus T_g in the Siegel modular variety A_g should contain at most finitely many CM points, namely Jacobians of general curves of high genus should not admit complex multiplication. We show that certain CM points do not lie in T_g if they parametrize abelian varieties isogeneous to products of simple CM abelian varieties of low dimension. The proof relies on known results on Faltings height and Sato-Tate equidistributions. This is a joint work with Kang Zuo and Xin Lv.



Feb 11

Dmitry Gourevitch
Relations between Fourier coefficients of automorphic forms, with applications to vanishing and to Eulerianity

In recent works with H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, we found a way to express any automorphic form through its Fourier coefficients, using adelic integrals, period integrals and discrete summation – generalizing the Piatetski-Shapiro – Shalika decomposition for GL(n). I will explain the general idea behind our formulas, and illustrate it on examples. I will also show applications to vanishing and Eulerianity of Fourier coefficients.



Feb 18

Eyal Kaplan
The generalized doubling method, multiplicity one and the application to global functoriality

One of the fundamental difficulties in the Langlands program is to handle the non-generic case. The doubling method, developed by Piatetski-Shapiro and Rallis in the 80s, pioneered the study of L-functions for cuspidal non-generic automorphic representations of classical groups. Recently, this method has been generalized in several aspects with interesting applications. In this talk I will survey the different components of the generalized doubling method, describe the fundamental multiplicity one result obtained recently in a joint work with Aizenbud and Gourevitch, and outline the application to global functoriality. Parts of the talk are also based on a collaboration with Cai, Friedberg and Ginzburg.




Feb 25

Roger Van Peski
Random matrices, random groups, singular values, and symmetric functions

Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.




Mar 4

Amos Nevo
Intrinsic Diophantine approximation on homogeneous algebraic varieties

Classical Diophantine approximation quantifies the denseness of the set of rational vectors in their ambient Euclidean space. A far-reaching extension of the classical theory calls for quantifying the denseness of rational points in general homogeneous algebraic varieties. This was raised as an open problem by Serge Lang already half a century ago, but progress towards it was achieved only in a limited number of special cases. A systematic approach to this problem for homogeneous varieties associated with semi-simple groups has been developed in recent years, in joint work with A. Ghosh and A. Gorodnik. The methods employ dynamical arguments and effective ergodic theory, and employ spectral estimates in the automorphic representation of semi-simple groups. In the case of homogeneous spaces with semi-simple stability group, this approach leads to the derivation of pointwise uniform and almost sure Diophantine exponents, as well as analogs of Khinchin's and W. Schmidt's theorems, with some of the results being best possible. We will explain some of the main results and some of the ingredients in their proof, focusing on some easily accessible examples.



Mar 11

Carlo Pagano
On the negative Pell conjecture
The negative Pell equation has been studied since many centuries. Euler already provided an interesting criterion in terms of continued fractions. In 1995 Peter Stevenhagen proposed a conjecture for the frequency of the solvability of this equation, when one varies the real quadratic field. I will discuss an upcoming joint work with Peter Koymans where we establish Stevenhagen's conjecture.