Difference between revisions of "NTS ABSTRACTFall2019"

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== Jan 23 ==
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== Sep 5 ==
  
 
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{| 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%" | '''Yunqing Tang '''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Reductions of abelian surfaces over global function fields
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| bgcolor="#BCD2EE"  align="center" | The sup-norm problem for automorphic forms over function fields and geometry
 
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| bgcolor="#BCD2EE"  | For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.
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The sup-norm problem is a purely analytic question about
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automorphic forms, which asks for bounds on their largest value (when
 +
viewed as a function on a modular curve or similar space). We describe
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a new approach to this problem in the function field setting, which we  
 +
carry through to provide new bounds for forms in GL_2 stronger than
 +
what can be proved for the analogous question about classical modular
 +
forms. This approach proceeds by viewing the automorphic form as a  
 +
geometric object, following Drinfeld. It should be possible to prove
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bounds in greater generality by this approach in the future.
  
 
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== Jan 24 ==
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== Sep 12 ==
  
 
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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%" | '''Hassan-Mao-Smith--Zhu'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of  $S^{d-2}\subset S^d$
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| bgcolor="#BCD2EE"  align="center" | CM values of modular functions and factorization
 
|-
 
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| bgcolor="#BCD2EE"  | Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj},  $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$  point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the  diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.  
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The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.
  
 
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== Jan 31 ==
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== Sep 19 ==
  
 
<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%" | '''Kyle Pratt'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions
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| bgcolor="#BCD2EE"  align="center" | Proportion of ordinary curves in some families
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: I will discuss recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet $L$-functions. By way of motivation I will spend some time discussing the Lindel\"of Hypothesis, and work of Bettin, Chandee, and Radziwi\l\l. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.  
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An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so.  
  
 
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== Feb 7 ==
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== Oct 3 ==
  
 
<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%" | '''Shamgar Gurevich'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Harmonic Analysis on $GL_n$ over finite fields
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| bgcolor="#BCD2EE"  align="center" | On the modularity of elliptic curves over imaginary quadratic fields
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters.
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| bgcolor="#BCD2EE"  |  
For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:
 
$$trace (\rho(g))/dim (\rho),$$
 
for an irreducible representation $\rho$ of G and an element g of G.  For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio.  We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (TAMU).
 
  
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Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.
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== Feb 14 ==
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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" | The Lambda invariant and its CM values
 
|-
 
| bgcolor="#BCD2EE"  | Abstract:  The Lambda invariant which parametrizes  elliptic curves with two torsions (X_0(2)) has some interesting properties, some similar to that of the j-invariants, and some  not. For example,  $\lambda(\frac{d+\sqrt d}2)$  is a unit sometime. In this talk, I will briefly describe some of the properties.  This is joint work with Hongbo Yin and Peng Yu.
 
  
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<br>
</center>
 
  
== Feb 28 ==
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== Oct 10 ==
  
 
<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%" | '''Brian Lawrence'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Diophantine problems and a p-adic period map.
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| bgcolor="#BCD2EE"  align="center" | Sectional monodromy groups of projective curves
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract:  I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theoryJoint with Akshay Venkatesh.
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| bgcolor="#BCD2EE"  | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.
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<br>
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== March 7==
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== Oct 17 ==
  
 
<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%" | '''Masoud Zargar'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Sections of quadrics over the affine line
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| bgcolor="#BCD2EE"  align="center" | Generalized special cycles and theta series
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Abstract: Suppose we have a quadratic form Q(x) in d\geq 4 variables over F_q[t] and f(t) is a polynomial over F_q. We consider the affine variety X given by the equation Q(x)=f(t) as a family of varieties over the affine line A^1_{F_q}. Given finitely many closed points in distinct fibers of this family, we ask when there exists a section passing through these points. We study this problem using the circle method over F_q((1/t)). Time permitting, I will mention connections to Lubotzky-Phillips-Sarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari
+
| bgcolor="#BCD2EE"  | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.
  
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 +
                                                                        
 
</center>
 
</center>
  
== March 14==
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<br>
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== Oct 24 ==
  
 
<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%" | '''Elena Mantovan'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | p-adic automorphic forms, differential operators and Galois representations
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| bgcolor="#BCD2EE"  align="center" | Counting cohomological automorphic forms on $GL_3$
 
|-
 
|-
| bgcolor="#BCD2EE"  | A strategy pioneered by Serre and Katz in the 1970s yields a construction of p-adic families of modular forms via the study of Serre's weight-raising differential operator Theta. This construction is a key ingredient in Deligne-Serre's theorem associating Galois representations to modular forms of weight 1, and in the study of the weight part of Serre's conjecture. In this talk I will discuss recent progress towards generalizing this theory to automorphic forms on unitary and symplectic Shimura varieites. In particular, I will introduce certain p-adic analogues of Maass-Shimura weight-raising differential operators,  and  discuss their action on p-adic automorphic forms, and on the associated mod p Galois representations. In contrast with Serre's classical approach where q-expansions play a prominent role, our approach is geometric in nature and is inspired by earlier work of Katz and Gross.
+
| bgcolor="#BCD2EE"  | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.
This talk is based joint work with Eishen,  and also with Fintzen--Varma, and with Flander--Ghitza--McAndrew.
 
  
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</center>
 
</center>
  
== March 28==
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<br>
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== Nov 7 ==
  
 
<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%" | '''Adebisi Agboola'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |Relative K-groups and rings of integers
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| bgcolor="#BCD2EE"  align="center" | A zero density estimate for Dedekind zeta functions
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Suppose that F is a number field and G is a finite group. I shall discuss a conjecture in relative algebraic K-theory (in essence, a conjectural Hasse principle applied to certain relative algebraic K-groups) that implies an affirmative answer to both the inverse Galois problem for F and G and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of F. It also implies the weak Malle conjecture on counting tame G-extensions of F according to discriminant. The K-theoretic conjecture can be proved in many cases (subject to mild technical conditions), e.g. when G is of odd order, giving a partial analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield substantial new results concerning both the Galois module structure of rings of integers and the weak Malle conjecture.
+
| bgcolor="#BCD2EE"  | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$  with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average.  
  
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+
This is a joint work with Jesse Thorner.
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 +
                                                                        
 
</center>
 
</center>
  
== April 4==
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<br>
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== Nov 14 ==
  
 
<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%" | '''Wei-Lun Tsai'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |Hecke L-functions and $\ell$ torsion in class groups
+
| bgcolor="#BCD2EE"  align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: The canonical Hecke characters in the sense of Rohrlich form a
+
| bgcolor="#BCD2EE"  | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.
set of algebraic Hecke characters with important arithmetic properties.
+
In this talk, we will explain how one can prove quantitative
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|}                                                        
nonvanishing results for the central values of their corresponding
+
                                                                        
L-functions using methods of an arithmetic statistical flavor. In
 
particular, the methods used rely crucially on recent work of Ellenberg,  
 
Pierce, and Wood concerning bounds for $\ell$-torsion in class groups of
 
number fields. This is joint work with Byoung Du Kim and Riad Masri.
 
|}                                                                         
 
 
</center>
 
</center>
  
== April 11==
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<br>
  
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== Nov 21 ==
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Taylor Mcadam'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" |Almost-prime times in horospherical flows
 
|-
 
| bgcolor="#BCD2EE"  | Abstract: Equidistribution results play an important role in dynamical systems and their applications in number theory.  Often in such applications it is desirable for equidistribution to be effective (i.e. the rate of convergence is known). In this talk I will discuss some of the history of effective equidistribution results in homogeneous dynamics and give an effective result for horospherical flows on the space of lattices. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.
 
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</center>
 
 
 
== April 18==
 
  
 
<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%" | '''Ila Varma'''
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |Malle's Conjecture for octic $D_4$-fields.
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| bgcolor="#BCD2EE"  align="center" | Steenrod operations and the Artin-Tate pairing
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.
+
| bgcolor="#BCD2EE"  | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.  
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</center>
 
  
== April 25==
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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%" | '''Michael Bush'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" |Interactions between group theory and number theory
 
|-
 
| bgcolor="#BCD2EE"  | Abstract: I'll survey some of the ways in which group theory has helped us understand extensions of number fields with restricted ramification and why one might care about such things. Some of Nigel's contributions will be highlighted. A good portion of the talk should be accessible to those other than number theorists.
 
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</center>
 
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== April 25==
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<br>
  
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== Nov 26 ==
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rafe Jones'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" |Eventually stable polynomials and arboreal Galois representations
 
|-
 
| bgcolor="#BCD2EE"  | Abstract: Call a polynomial defined over a field K eventually stable if its nth iterate has a uniformly bounded number of irreducible factors (over K) as n grows. I’ll discuss some far-reaching conjectures on eventual stability, and recent work on various special cases. I’ll also describe some natural connections between eventual stability and arboreal Galois representations, which Nigel Boston introduced in the early 2000s.
 
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</center>
 
  
==April 25 NTS==
 
 
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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%" | '''Jen Berg'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |Rational points on conic bundles over elliptic curves with positive rank
+
| bgcolor="#BCD2EE"  align="center" | Counting Towers of Number Fields
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses the Brauer group, Br X, to preclude the existence of rational points on a variety X. In this talk, we'll explore the arithmetic of conic bundles over elliptic curves of positive rank over a number field k. We'll discuss the insufficiency of the known obstructions to explain the failures of the Hasse principle for such varieties over a number field. We'll further consider questions on the distribution of the rational points of X with respect to the image of X(k) inside of the rational points of the elliptic curve E. In the process, we'll discuss results on a local-to-global principle for torsion points on elliptic curves over Q. This is joint work in progress with Masahiro Nakahara.
+
| bgcolor="#BCD2EE"  | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.
|}                                                                       
 
</center>
 
  
== April 25==
+
|}                                                       
 
+
                                                                        
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Judy Walker'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" |Derangements of Finite Groups
 
|-
 
| bgcolor="#BCD2EE"  | Abstract: In the early 1990’s, Nigel Boston taught an innovative graduate-level group theory course at the University of Illinois that focused on derangements (fixed-point-free elements) of transitive permutation groups.  The course culminated in the writing of a 7-authored paper that appeared in Communications in Algebra in 1993.  This paper contained a conjecture that was eventually proven by Fulman and Guralnick, with that result appearing in the Transactions of the American Mathematical Society just last year.
 
|}                                                                        
 
 
</center>
 
</center>
  
 +
<br>
  
== May 2==
+
== Dec 05 ==
  
 
<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'''
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Benjamin Breen'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" |Unramified extensions of random global fields
+
| bgcolor="#BCD2EE"  align="center" | On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: For any finite group Gamma, I will give a "non-abelian-Cohen-Martinet Conjecture," i.e. a conjectural distribution on the "good part" of the Galois group of the maximal unramified extension of a global field K, as K varies over all Galois Gamma extensions of the rationals or rational function field over a finite field. I will explain the motivation for this conjecture based on what we know about these maximal unramified extensions (very little), and how we prove, in the function field case, as the size of the finite field goes to infinity, that the moments of the Galois groups of these maximal unramified extensions match out conjecture. This talk covers work in progress with Yuan Liu and David Zureick-Brown
+
| bgcolor="#BCD2EE"  | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.
|}                                                                         
+
 
 +
|}                                                        
 +
                                                                        
 
</center>
 
</center>
  
== May 9==
+
<br>
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown'''
 
|-
 
| bgcolor="#BCD2EE"  align="center" |Arithmetic of stacks
 
|-
 
| bgcolor="#BCD2EE"  | Abstract: I'll discuss several diophantine problems that naturally lead one to study algebraic stacks, and discuss a few results.
 
|}                                                                       
 
</center>
 

Latest revision as of 16:09, 25 November 2019

Return to [1]


Sep 5

Will Sawin
The sup-norm problem for automorphic forms over function fields and geometry

The sup-norm problem is a purely analytic question about automorphic forms, which asks for bounds on their largest value (when viewed as a function on a modular curve or similar space). We describe a new approach to this problem in the function field setting, which we carry through to provide new bounds for forms in GL_2 stronger than what can be proved for the analogous question about classical modular forms. This approach proceeds by viewing the automorphic form as a geometric object, following Drinfeld. It should be possible to prove bounds in greater generality by this approach in the future.


Sep 12

Yingkun Li
CM values of modular functions and factorization

The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.


Sep 19

Soumya Sankar
Proportion of ordinary curves in some families

An abelian variety in characteristic [math]\displaystyle{ p }[/math] is said to be ordinary if its [math]\displaystyle{ p }[/math] torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the [math]\displaystyle{ p }[/math] -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so.


Oct 3

Patrick Allen
On the modularity of elliptic curves over imaginary quadratic fields

Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.


Oct 10

Borys Kadets
Sectional monodromy groups of projective curves
Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.


Oct 17

Yousheng Shi
Generalized special cycles and theta series
We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.


Oct 24

Simon Marshall
Counting cohomological automorphic forms on $GL_3$
I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.


Nov 7

Asif Zaman
A zero density estimate for Dedekind zeta functions
Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average.

This is a joint work with Jesse Thorner.


Nov 14

Liyang Yang
Holomorphic Continuation of Certain $L$-functions via Trace Formula
In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.


Nov 21

Tony Feng
Steenrod operations and the Artin-Tate pairing
In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.


Nov 26

Brandon Alberts
Counting Towers of Number Fields
Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.


Dec 05

Benjamin Breen
On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics
What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.