NTS ABSTRACTSpring2019: Difference between revisions

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== Feb 13==
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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%" | '''Frank Calegari'''
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| bgcolor="#BCD2EE"  align="center" | Recent Progress in Modularity
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| bgcolor="#BCD2EE"  | Abstract:  We survey some recent work in modularity lifting, and also describe some applications of these results. This will be based partly on joint work with Allen, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, and also on joint work with Boxer, Gee, and Pilloni.
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== Feb 15 ==
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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%" | '''Junho Peter Whang'''
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| bgcolor="#BCD2EE"  align="center" | Integral points and curves on moduli of local systems
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| bgcolor="#BCD2EE"  | Abstract: We consider the Diophantine geometry of moduli spaces for 
special linear rank two local systems on surfaces with fixed boundary 
traces. After motivating their Diophantine study, we establish a 
structure theorem for their integral points via mapping class group 
descent, generalizing classical work of Markoff (1880). We also obtain 
Diophantine results for algebraic curves in these moduli spaces, 
including effective finiteness of imaginary quadratic integral points 
for non-special curves.
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== Feb 22 ==
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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%" | '''Yifan Yang'''
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| bgcolor="#BCD2EE"  align="center" | Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus
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| bgcolor="#BCD2EE"  | Abstract: In this talk we consider the rational torsion
subgroup of the generalized Jacobian of the modular
curve X_0(N) with respect to a reduced divisor given
by the sum of all cusps. When N=p is a prime, we find
that the rational torsion subgroup is always cyclic
of order 2 (while that of the usual Jacobian of X_0(p)
grows linearly as p tends to infinity, according to a
well-known result of Mazur). Subject to some unproven
conjecture about the rational torsions of the Jacobian
of X_0(p^n), we also determine the structure of the
rational torsion subgroup of the generalized Jacobian
of X_0(p^n). This is a joint work with Takao Yamazaki.
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== March 22 ==
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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%" | '''Fang-Ting Tu'''
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| bgcolor="#BCD2EE"  align="center" | Title: Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds
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| bgcolor="#BCD2EE"  | Abstract:
This is a joint work with Ling Long, Noriko Yui, and Wadim Zudilin. We establish the supercongruences for the rigid hypergeometric Calabi-Yau threefolds over rational numbers. These supercongruences were conjectured by Rodriguez-Villeagas in 2003. In this work, we use two different approaches. The first method is based on Dwork's p-adic unit root theory, and the other is based on the theory of hypergeometric motives and hypergeometric functions over finite fields. In this talk, I will introduce the first method, which allows us to obtain the supercongruences for ordinary primes.
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== April 12 ==
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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%" | '''Junehyuk Jung'''
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| bgcolor="#BCD2EE"  align="center" | Title: Quantum Unique Ergodicity and the number of nodal domains of automorphic forms
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Abstract: It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having a bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I will explain how one can prove that this is indeed true for the surfaces where the Laplacian is quantum uniquely ergodic, under certain symmetry assumptions. As an application, we prove that the number of nodal domains of Maass-Hecke eigenforms on a compact arithmetic triangles tends to $+\infty$ as the eigenvalue grows. I am going to also discuss the nodal domains of automorphic forms on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$. Under a minor assumption, I will give a quick proof that the real part of weight $k\neq 0$ automorphic form has only two nodal domains. This result captures the fact that a 3-manifold with Sasaki metric never admits a chaotic geodesic flow. This talk is based on joint works with S. Zelditch and S. Jang.
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== April 19 ==
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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%" | '''Hang Xue (Arizona)'''
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| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture.
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Abstract: I will explain the arithmetic analogue of the Gan--Gross--Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.
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== May 3 ==
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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%" | '''Matilde Lalin (Université de Montréal)'''
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| bgcolor="#BCD2EE"  align="center" | Title: The mean value of cubic $L$-functions over function fields.
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Abstract: We will start by exploring the problem of finding moments for  Dirichlet $L$-functions, including the first main results and the standard conjectures. We will then discuss the problem for function fields. We will then present  a result about the first moment of $L$-functions associated to cubic characters over  $\F_q(t)$, when $q\equiv 1 \bmod{3}$. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. This is joint work with C. David and A. Florea.
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== May 10 ==
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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%" | '''Hector Pasten (Harvard University)'''
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| bgcolor="#BCD2EE"  align="center" | Title: Shimura curves and estimates for abc triples.
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Abstract: I will explain a new connection between modular forms and the abc conjecture. In this approach, one considers maps to a given elliptic curve coming from various Shimura curves, which gives a way to obtain unconditional results towards the abc conjecture starting from good estimates for the variation of the degree of these maps. The approach to control this variation of degrees involves a number of tools, such as Arakelov geometry, automorphic forms, and analytic number theory. The final result is an unconditional estimate that lies beyond the existing techniques in the context of the abc conjecture, such as linear forms in logarithms.
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Revision as of 12:58, 16 January 2019

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Jan 23

Yunqing Tang


Jan 24

Hassan-Mao-Smith--Zhu
The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$
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 $S^{d-2}\subset S^d$.

March 28

Shamgar Gurevitch
Harmonic Analysis on GLn over finite fields
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters.

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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