Difference between revisions of "NTS/Abstracts"

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== January 23 ==
  
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<center>
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Majid Hadian-Jazi''' (UIC)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: On a motivic method in Diophantine geometry
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== January 30 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (University of Sydney, Australia)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Ergodic Plunnecke inequalities with applications to sumsets of infinite sets in countable abelian groups
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== February 13 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''John Voight''' (Dartmouth)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Numerical calculation of three-point branched covers of the projective line
 +
|-
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| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== February 20 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nir Avni''' (Northwestern)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Representation zeta functions
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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''('''Z'''<sub>''p''</sub>).
 +
3) The complex geometry of the moduli spaces of ''G''-local systems on a Riemann surface, and, more precisely, the structure of its singularities.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== February 27 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jennifer Park''' (MIT)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Effective Chabauty for symmetric power of curves
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== March 11 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yueke Hu''' (Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Local integrals of triple product ''L''-function and subconvexity bound
 +
|-
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| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== April 10 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kartik Prasanna''' (Michigan)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Algebraic cycles and Rankin-Selberg L-functions
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== April 17 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Davide Reduzzi''' (Chicago)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Galois representations and torsion in the coherent cohomology of
 +
Hilbert modular varieties
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== April 24 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Arul Shankar''' (Harvard)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: The average 5-Selmer rank of elliptic curves
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
== May 8 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (UW-Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Jacobians of Random Graphs and Cohen Lenstra heuristics
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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.
 +
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
<!--
 +
== October 3 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy''' (Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: The Artin–Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from '''P'''<sup>1</sup> to '''P'''<sup>1</sup> is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on '''P'''<sup>1</sup> that are finite quotients of endomorphisms of elliptic curves, and they have many interesting dynamical properties related to the geometry and arithmetic of elliptic curves. I show that the zeta function of a separable Lattes map in positive characteristic is a transcendental function.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== October 10 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Bogdan Petrenko''' (Eastern Illinois University)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Generating an algebra from the probabilistic standpoint
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: Let ''A'' be a ring whose additive group is free Abelian of finite
 +
rank. The topic of this talk is the following question: what is the
 +
probability that several random elements of ''A'' generate it as a ring? After
 +
making this question precise, I will show that it has an interesting
 +
answer which can be interpreted as a local-global principle. Some
 +
applications will be discussed. This talk will be based on my joint work
 +
with Rostyslav Kravchenko (University of Chicago) and Marcin Mazur
 +
(Binghamton University).
 +
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== October 17 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Anthony Várilly-Alvarado''' (Rice)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title:  Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space.  They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface.  I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X -> P^1 such that A is "vertical" for this map.  This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X -> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction.  The construction also gives a fast, practical algorithm for computing the Brauer group of X.  Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces.  This is joint work with Bianca Viray.
 +
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== October 24 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Garrett''' (Minnesota)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Spectra of pseudo-Laplacians on spaces of automorphic forms
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: Faddeev–Pavlov and Lax–Phillips observed that certain
 +
restrictions of the Laplacian to parts of automorphic continuous
 +
spectrum have discrete spectrum. Colin de Verdiere used this to prove
 +
meromorphic continuation of Eisenstein series, and proposed ways to
 +
exploit this idea to construct self-adjoint operators with spectra
 +
related to zeros of ''L''-functions. We show that simple forms of this
 +
construction produce at most very sparse spectra, due to
 +
incompatibility with pair correlations for zeros. Ways around some of
 +
the obstacles are sketched. (Joint with E. Bombieri.)
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== October 31 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jerry Wang''' (Princeton)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Pencils of quadrics and the arithmetic of hyperelliptic curves
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over '''Q''' of genus ''g'' have no points over any odd degree extension of '''Q'''. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser–Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== November 7 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''who?''' (where?)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: tba
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== November 12 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Counting extensions of number fields of given degree, bounded (rho)-discriminant, and specified Galois closure
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: A very basic question in algebraic number theory is: how many number fields are there? A natural way to order the fields of a fixed degree n is by discriminant, and classical results of Minkowski then assure us that there are only finitely many fields with a given discriminant.  We are also often interested in counting number fields, or relative extensions, with other properties, such as having a particular Galois closure.  A folk conjecture sometimes attributed to Linnik states that the number of extensions of degree n and absolute discriminant less than X is on the order of X.  A great deal of recent and ongoing work has been focused towards achieving upper bounds on counts of this nature (quite successfully, in degree 5 and lower), but there is comparatively little known in higher degrees, for relative extensions, or for sufficiently complicated Galois closures: the primary results are those of Schmidt and Ellenberg-Venkatesh.  I will discuss these results and my thesis work, in which I generalize several of their results and introduce another counting metric, the "rho-discriminant". 
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== November 21 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Lipnowski''' (Duke)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  |
 +
Abstract: tba
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== November 26 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Kane''' (Stanford)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Diffuse decompositions of polynomials
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: We study some problems relating to polynomials evaluated
 +
either at random Gaussian or random Bernoulli inputs.  We present some
 +
new work on a structure theorem for degree-''d'' polynomials with Gaussian
 +
inputs.  In particular, if ''p'' is a given degree-''d'' polynomial, then ''p''
 +
can be written in terms of some bounded number of other polynomials
 +
''q''<sub>1</sub>, ..., ''q''<sub>''m''</sub> so that the joint probability density function of
 +
''q''<sub>1</sub>(''G''), ..., ''q''<sub>''m''</sub>(''G'') is close to being bounded.  This says essentially
 +
that any abnormalities in the distribution of ''p''(''G'') can be explained by
 +
the way in which ''p'' decomposes into the ''q''<sub>''i''</sub>.  We then present some
 +
applications of this result.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== December 5 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jennifer Park''' (MIT)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: tba
 +
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
== December 12 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vivek Shende''' (Berkeley)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Equidistribution on the space of rank two vector bundles over the projective line
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: I will discuss how the algebraic geometry of hyperelliptic curves gives an approach to a function field analogue of the 'mixing conjecture' of Michel and Venkatesh.  (For a rather longer abstract, see the [http://arxiv.org/abs/1307.8237 arxiv posting] of the same name as the talk). This talk presents joint work with Jacob Tsimerman.
 +
|}                                                                       
 +
</center>
 +
 +
<br>
 +
 +
-->
 +
 +
== Organizer contact information ==
 +
 +
[http://www.math.wisc.edu/~rharron/ Robert Harron]
 +
 +
Sean Rostami
 +
 +
----
 +
Return to the [[NTS|Number Theory Seminar Page]]
 +
 +
Return to the [[Algebra|Algebra Group Page]]

Revision as of 13:07, 12 August 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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