Difference between revisions of "NTS/Abstracts"

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== January 23 ==
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== Aug 28 ==
  
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Majid Hadian-Jazi''' (UIC)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Lemke Oliver'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: On a motivic method in Diophantine geometry
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| bgcolor="#BCD2EE"  align="center" | ''The distribution of 2-Selmer groups of elliptic curves with two-torsion''
 
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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.
+
Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6.  In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun.
 
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== January 30 ==
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== Sep 04 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (University of Sydney, Australia)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Ergodic Plunnecke inequalities with applications to sumsets of infinite sets in countable abelian groups
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| bgcolor="#BCD2EE"  align="center" | ''Unramified deformation rings''
 
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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.  
+
Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.
 
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== February 13 ==
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== Sep 11 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Numerical calculation of three-point branched covers of the projective line
+
| bgcolor="#BCD2EE"  align="center" | The distribution of sandpile groups of random graphs &#42;&#42;&#42;
 
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|-
 
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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.
+
The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian.  An Erd&#337;s–R&#233;nyi random graph then gives some distribution of random abelian groups.  We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups.  We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution.  In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture.
 +
 
 +
&#42;&#42;&#42; ''This is officially a '''probability seminar''', but will occur in the usual NTS room B105 at a slightly '''earlier time''', 2:25 PM.''
 
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== February 20 ==
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== Sep 18 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Takehiko Yasuda'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Representation zeta functions
+
| bgcolor="#BCD2EE"  align="center" | ''Distributions of rational points and number fields, and height zeta functions''
 
|-
 
|-
 
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Abstract: I will talk about connections between the following:
+
In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function.
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.
 
 
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== February 27 ==
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== Sep 25 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Ramin Takloo-Bigash'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Effective Chabauty for symmetric power of curves
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| bgcolor="#BCD2EE"  align="center" | ''Counting orders in number fields and p-adic integrals''
 
|-
 
|-
 
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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.  
+
In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded
 +
discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration.  This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).
 
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== March 11 ==
+
== Oct 02 ==
  
 
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<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Pham Huu Tiep'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Local integrals of triple product ''L''-function and subconvexity bound
+
| bgcolor="#BCD2EE"  align="center" | ''Nilpotent Hall and abelian Hall subgroups''
 
|-
 
|-
 
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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.
+
To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent
 +
joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.
 
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</center>
 
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== April 10 ==
+
== Oct 09 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Michael Woodbury'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Algebraic cycles and Rankin-Selberg L-functions
+
| bgcolor="#BCD2EE"  align="center" | ''An Adelic Kuznetsov Trace Formula for GL(4)''
 +
|-
 +
| bgcolor="#BCD2EE"  |
 +
An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula.  Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications.  In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4).  I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.
 +
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 +
</center>
 +
 
 +
<br>
 +
 
 +
== Oct 16 ==
 +
 
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Grizzard'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | ''Small points and free abelian groups''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
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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.
+
Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values.  We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer.
 
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</center>
 
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== April 17 ==
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== Oct 23 ==
 +
 
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<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | ''A conjecture of Colmez''
 +
|-
 +
| bgcolor="#BCD2EE"  |
 +
In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable)  height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula.  In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, &Phi;) in terms of the log derivative at s=1 of some `Artin' L-function  associated to the CM type &Phi;. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport.
 +
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 +
</center>
 +
 
 +
<br>
 +
 
 +
== Oct 30 ==
 +
 
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<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Laura DeMarco'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | ''Elliptic curves and complex dynamics''
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields.  On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P<sup>1</sup>.
 +
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</center>
 +
<br>
 +
 
 +
== Nov 06 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Michael Magee'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Galois representations and torsion in the coherent cohomology of
+
| bgcolor="#BCD2EE"  align="center" | ''Zero sets of Hecke polynomials on the sphere''
Hilbert modular varieties
 
 
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|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: Let ''F'' be a totally real number field, ''p'' a prime number
+
The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.
(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.
 
 
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</center>
 
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== April 24 ==
+
== Nov 13 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Yiwei She'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: The average 5-Selmer rank of elliptic curves
+
| bgcolor="#BCD2EE"  align="center" | ''The Shafarevich conjecture for K3 surfaces''
 
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|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: We use geometry-of-numbers techniques to show that the average size of the 5-Selmer group of
+
Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces.  Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture.
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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</center>
 
</center>
  
 
<br>
 
<br>
== May 8 ==
+
 
 +
== Nov 20 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| 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="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Jacobians of Random Graphs and Cohen Lenstra heuristics
+
| bgcolor="#BCD2EE"  align="center" | ''Endoscopy and cohomology of unitary groups''
 +
|-
 +
| bgcolor="#BCD2EE"  |
 +
We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others.  We will show how this can be used to prove asymptotics for the L<sup>2</sup> Betti numbers of families of locally symmetric spaces.
 +
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 +
</center>
 +
 
 +
<br>
 +
 
 +
== Dec 04 ==
 +
 
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Joel Specter'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | ''Commuting Endomorphisms of the p-adic Unit Disk''
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| 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
+
When can a pair of endomorphisms of <math>\mathbf{Z}_p[[X]]/\mathbf{Z}_p</math> commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over <math>\mathbf{Z}_p</math>. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin.
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>
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<br>
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 +
== Dec 11 ==
  
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ila Varma'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | ''The mean number of 3-torsion elements in ray class groups of quadratic fields''
 +
|-
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In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the
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class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over <math>\mathbb{Q}</math>. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.
 
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== October 3 ==
 
== October 3 ==
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== Organizer contact information ==
 
== Organizer contact information ==
  
[http://www.math.wisc.edu/~rharron/ Robert Harron]
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Sean Rostami (srostami@math.wisc.edu)
 
 
Sean Rostami
 
  
 
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Latest revision as of 13:53, 28 November 2014

Aug 28

Robert Lemke Oliver
The distribution of 2-Selmer groups of elliptic curves with two-torsion

Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6. In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun.


Sep 04

Patrick Allen
Unramified deformation rings

Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.



Sep 11

Melanie Matchett Wood
The distribution of sandpile groups of random graphs ***

The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture.

*** This is officially a probability seminar, but will occur in the usual NTS room B105 at a slightly earlier time, 2:25 PM.



Sep 18

Takehiko Yasuda
Distributions of rational points and number fields, and height zeta functions

In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function.


Sep 25

Ramin Takloo-Bigash
Counting orders in number fields and p-adic integrals

In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).


Oct 02

Pham Huu Tiep
Nilpotent Hall and abelian Hall subgroups

To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.


Oct 09

Michael Woodbury
An Adelic Kuznetsov Trace Formula for GL(4)

An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.


Oct 16

Robert Grizzard
Small points and free abelian groups

Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer.


Oct 23

Tonghai Yang
A conjecture of Colmez

In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable) height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula. In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, Φ) in terms of the log derivative at s=1 of some `Artin' L-function associated to the CM type Φ. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport.


Oct 30

Laura DeMarco
Elliptic curves and complex dynamics

I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields. On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P1.


Nov 06

Michael Magee
Zero sets of Hecke polynomials on the sphere

The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.


Nov 13

Yiwei She
The Shafarevich conjecture for K3 surfaces

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture.


Nov 20

Simon Marshall
Endoscopy and cohomology of unitary groups

We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L2 Betti numbers of families of locally symmetric spaces.


Dec 04

Joel Specter
Commuting Endomorphisms of the p-adic Unit Disk

When can a pair of endomorphisms of [math]\mathbf{Z}_p[[X]]/\mathbf{Z}_p[/math] commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over [math]\mathbf{Z}_p[/math]. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin.


Dec 11

Ila Varma
The mean number of 3-torsion elements in ray class groups of quadratic fields

In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over [math]\mathbb{Q}[/math]. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.



Organizer contact information

Sean Rostami (srostami@math.wisc.edu)


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