Difference between revisions of "NTS Spring 2014/Abstracts"

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== April 10 ==
== September 12 ==
 
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall''' (Northwestern)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kartik Prasanna''' (Michigan)
 
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| bgcolor="#BCD2EE"  align="center" | Title: Endoscopy and cohomology growth on U(3)
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| bgcolor="#BCD2EE"  align="center" | Title: Algebraic cycles and Rankin-Selberg L-functions
 
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Abstract: I will use the endoscopic classification of automorphic forms on U(3) to determine the asymptotic cohomology growth of families of complex-hyperbolic 2-manifolds.
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Abstract: I will give a survey of a circle of results relating L-functions and algebraic cycles, starting with the Gross-Zagier formula and its various generalizations. This will lead naturally to a certain case of the Bloch-Beilinson conjecture which is closely related to Gross-Zagier but where one does not have a construction of the expected cycles. Finally, I will hint at a plausible construction of cycles in this "missing" case, which is joint work with A. Ichino, and explain what one can likely prove about them.
 
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== September 19 ==
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== April 17 ==
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Valerio Toledano Laredo''' (Northeastern)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Davide Reduzzi''' (Chicago)
 
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| bgcolor="#BCD2EE"  align="center" | Title: From Yangians to quantum loop algebras via abelian difference equations
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| bgcolor="#BCD2EE"  align="center" | Title: Galois representations and torsion in the coherent cohomology of
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Hilbert modular varieties
 
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Abstract: For a semisimple Lie algebra ''g'', the quantum loop algebra
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Abstract: Let ''F'' be a totally real number field, ''p'' a prime number
and the Yangian are deformations of the loop algebra ''g''[''z,&nbsp;''z&nbsp;&minus;&nbsp;1]
+
(unramified in ''F''), and ''M'' the Hilbert modular variety for ''F'' of some level
and the current algebra ''g''[''u''], respectively. These infinite-dimensional
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prime to ''p'', and defined over a finite field of characteristic ''p''. I will
quantum groups share many common features, though a
+
explain how exploiting the geometry of ''M'', and in particular the
precise explanation of these similarities has been missing
+
stratification induced by the partial Hasse invariants, one can attach
so far.
+
Galois representations to Hecke eigen-classes occurring in the coherent
 +
cohomology of ''M''. This is a joint work with Matthew Emerton and Liang Xiao.
 +
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In this talk, I will explain how to construct a functor between
+
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the finite-dimensional representation categories of these
 
two Hopf algebras which accounts for all known similarities
 
between them.
 
  
The functor is transcendental in nature, and is obtained from
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== April 24 ==
the discrete monodromy of an abelian difference equation
 
canonically associated to the Yangian.
 
  
This talk is based on a joint work with Sachin Gautam.
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Arul Shankar''' (Harvard)
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| bgcolor="#BCD2EE"  align="center" | Title: The average 5-Selmer rank of elliptic curves
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Abstract:  We use geometry-of-numbers techniques to show that the average size of the 5-Selmer group of
 +
elliptic curves is equal to 6. From this, we deduce an upper bound on the average rank of elliptic curves.
 +
Then, by constructing families of elliptic curves with equidistributed root number we show that the average rank is
 +
less than 1. This is joint work with Manjul Bhargava.
 
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== May 8 ==
== September 26 ==
 
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Haluk Şengün''' (Warwick/ICERM)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (UW-Madison)
 
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| bgcolor="#BCD2EE"  align="center" | Title: Torsion homology of Bianchi groups and arithmetic
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| bgcolor="#BCD2EE"  align="center" | Title: Jacobians of Random Graphs and Cohen Lenstra heuristics
 
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Abstract: Bianchi groups are groups of the form ''SL''(2,&nbsp;''R'') where ''R'' is the ring of integers of an imaginary quadratic field. They form an important class of arithmetic Kleinian groups and moreover they hold a key role for the development of the Langlands program for ''GL''(2) beyond totally real fields.
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Abstract: We will consider the question of the distribution of the Jacobians of random curves over finite fields.  Over a finite field, given a curve, we can associate to it the (finite) group of
 +
degree 0 line bundles on the curve.  This is the function field analog of the class group of a number field.
 +
We will discuss the relationship to the Cohen Lenstra heuristics for the distribution of class groups.  If the curve is reducible, a natural quotient of the Jacobian is the group of components, and we will focus on this aspect.  We are thus led to study Jacobians of random graphs, which go by many names (including the sandpile group and the critical group) as they have arisen in a wide variety of contexts.  We discuss new work proving a conjecture of Payne that Jacobians of random graphs satisfy a modified Cohen-Lenstra type heuristic.
  
In this talk, I will discuss several interesting questions related to the torsion in the homology of Bianchi groups. I will especially focus on the recent results on the asymptotic behavior of the size of torsion, and the reciprocity and functoriality (in the sense of the Langlands program) aspects of the torsion. Joint work with N.&nbsp;Bergeron and A.&nbsp;Venkatesh on the cycle complexity of arithmetic manifolds will be discussed at the end.
 
 
The discussion will be illustrated with many numerical examples.
 
 
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== October 3 ==
 
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Latest revision as of 13:10, 30 April 2014

January 23

Majid Hadian-Jazi (UIC)
Title: On a motivic method in Diophantine geometry

Abstract: By studying the variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in Diophantine geometry can be translated to problems concerning K-groups. Then we use some vanishing theorems to obtain concrete results.


January 30

Alexander Fish (University of Sydney, Australia)
Title: Ergodic Plunnecke inequalities with applications to sumsets of infinite sets in countable abelian groups

Abstract: By use of recent ideas of Petridis, we extend Plunnecke inequalities for sumsets of finite sets in abelian groups to the setting of measure-preserving systems. The main difference in the new setting is that instead of a finite set of translates we have an analogous inequality for a countable set of translates. As an application, by use of Furstenberg correspondence principle, we obtain new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Joint work with Michael Bjorklund, Chalmers.


February 13

John Voight (Dartmouth)
Title: Numerical calculation of three-point branched covers of the projective line

Abstract: We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone.


February 20

Nir Avni (Northwestern)
Title: Representation zeta functions

Abstract: I will talk about connections between the following: 1) Asymptotic representation theory of an arithmetic lattice G(Z). More precisely, the question of how many n-dimensional representations does G(Z) have. 2) The distribution of a random commutator in the p-adic analytic group G(Zp). 3) The complex geometry of the moduli spaces of G-local systems on a Riemann surface, and, more precisely, the structure of its singularities.


February 27

Jennifer Park (MIT)
Title: Effective Chabauty for symmetric power of curves

Abstract: While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^d(X) that are not parametrized by a projective space or a coset of an abelian variety, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.


March 11

Yueke Hu (Madison)
Title: Local integrals of triple product L-function and subconvexity bound

Abstract: Venkatesh proposed a strategy to prove the subconvexity bound in the level aspect for triple product L-function. With the integral representation of triple product L-function, if one can get an upper bound for the global integral and a lower bound for the local integrals, then one can get an upper bound for the L-function, which turns out to be a subconvexity bound. Such a subconvexity bound was obtained essentially for representations of square free level. I will talk about how to generalize this result to the case with higher ramifications as well as joint ramifications.


April 10

Kartik Prasanna (Michigan)
Title: Algebraic cycles and Rankin-Selberg L-functions

Abstract: I will give a survey of a circle of results relating L-functions and algebraic cycles, starting with the Gross-Zagier formula and its various generalizations. This will lead naturally to a certain case of the Bloch-Beilinson conjecture which is closely related to Gross-Zagier but where one does not have a construction of the expected cycles. Finally, I will hint at a plausible construction of cycles in this "missing" case, which is joint work with A. Ichino, and explain what one can likely prove about them.


April 17

Davide Reduzzi (Chicago)
Title: Galois representations and torsion in the coherent cohomology of

Hilbert modular varieties

Abstract: Let F be a totally real number field, p a prime number (unramified in F), and M the Hilbert modular variety for F of some level prime to p, and defined over a finite field of characteristic p. I will explain how exploiting the geometry of M, and in particular the stratification induced by the partial Hasse invariants, one can attach Galois representations to Hecke eigen-classes occurring in the coherent cohomology of M. This is a joint work with Matthew Emerton and Liang Xiao.


April 24

Arul Shankar (Harvard)
Title: The average 5-Selmer rank of elliptic curves

Abstract: We use geometry-of-numbers techniques to show that the average size of the 5-Selmer group of elliptic curves is equal to 6. From this, we deduce an upper bound on the average rank of elliptic curves. Then, by constructing families of elliptic curves with equidistributed root number we show that the average rank is less than 1. This is joint work with Manjul Bhargava.


May 8

Melanie Matchett Wood (UW-Madison)
Title: Jacobians of Random Graphs and Cohen Lenstra heuristics

Abstract: We will consider the question of the distribution of the Jacobians of random curves over finite fields. Over a finite field, given a curve, we can associate to it the (finite) group of degree 0 line bundles on the curve. This is the function field analog of the class group of a number field. We will discuss the relationship to the Cohen Lenstra heuristics for the distribution of class groups. If the curve is reducible, a natural quotient of the Jacobian is the group of components, and we will focus on this aspect. We are thus led to study Jacobians of random graphs, which go by many names (including the sandpile group and the critical group) as they have arisen in a wide variety of contexts. We discuss new work proving a conjecture of Payne that Jacobians of random graphs satisfy a modified Cohen-Lenstra type heuristic.


Organizer contact information

Robert Harron

Sean Rostami


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