Feb 06, 2023
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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09:30 AM  10:30 AM


Moduli Spaces for Dynamical Systems on Projective Space
Joseph Silverman (Brown University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
The space Mor_d^N of degree d morphisms f from P^N to itself is parameterized by the space of coefficients of the polynomials that define f. This space has a natural structure as an affine variety. Elements R of Aut(P^N), or equivalently of PGL_{N+1}, act on Mor_d^N via conjugation, f > R^{1} o f o R. Conjugation commutes with iteration, so it is the natural action to consider when studying the dynamics of f, since conjugate maps determine the same dynamical system up to simultaneous change of variables of the domain and range. In this talk I will discuss how geometric invariant theory yields a nice quotient Mor_d^N//Aut(P^N), how this quotient can be embedded in various larger spaces with even nicer properties, what happens when one adds dynamical level structure, some structure theorems and open problems, and, as time permits, various related topics such as field of moduli versus field of definition problems for dynamical systems.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


A Finite Index Conjecture for Iterated Galois Groups
Thomas Tucker (University of Rochester)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
Let f be a rational function of degree greater than 1 defined over a number field k and let x in k. Iterating f, we obtain
a sequence of field extensions K_n generated by the inverse images of x under f^n. Passing to the inverse limit one obtains an iterated Galois group G_x. There is a natural "generic" Galois group G attached to f in which G_x sits. Conjecturally, G_x will have finite index in G unless x is "special" in some fashion. While in its earliest form, this conjecture (originally made in a special case by Boston and Jones) was made in analogy with the Serre finite index theorem, it seems to have a very different flavor. We will discuss various results towards this conjecture, the diophantine techniques used in these results, a more general higherdimensional conjecture, and a very simple conjecture on the irreducibility of iterates of polynomials that is completely unresolved.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Heights of Rational Points on Stacks
Jordan Ellenberg (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
I'll report on a project with Matt Satriano and David ZureickBrown (who will be a Research Member later this semester) which in some sense started out in the MSRI "Rational and Integral Points on HigherDimensional Varieties" semester in 2006. Some natural arithmetic objects (like numbers) are parametrized by an algebraic variety, but many others (like elliptic curves, or number fields) are parametrized by more general algebraic objects called "stacks." Rational points on stacks are just as interesting as rational points on varieties, but a very useful tool for the study of rational points, a theory of *heights*, was missing in this case. I'll talk about the definition of heights we developed for this situation, make the case that it's a good definition, and discuss some of the refinements that have come in the time since we proposed it.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


On the Toric Locus of LAdic Local Systems (Joint work with J. Stix)
Anna Cadoret (Sorbonne University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
Given a ladic local system on a variety over a number field, one can define its toric locus, whic is the analogue of the CM locus of a variation of Hodge structure. I will discuss a work in progress on the sparcity of the points of bounded degree in the toric locus of local systems arising from geometry.
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Feb 07, 2023
Tuesday

09:30 AM  10:30 AM


On the Distribution of the Hodge Locus
Emmanuel Ullmo (IHES)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
The lecture will discuss a joint work with Gregorio Baldi and Bruno Klingler. Given a polarized Zvariation of Hodge structures over a complex, smooth quasiprojective variety S, we describe some properties of the Hodge locus, a countable union of algebraic subvarieties of S where exceptional Hodge tensors appear, by a result of Cattani, Deligne and Kaplan. We prove the geometric part of the ZilberPink conjecture in this context: the maximal atypical part of the Hodge locus of postive period demension arise in a finite number of families. In level at least 3, we show that the typical Hodge is empty and therefore the positive dimensional part of the Hodge locus is algebraic. For instance the Hodge locus of positive period dimension of the universal family of degree d smooth hypersurfaces in the projective space of dimension n+1 is algebraic. We also prove that if the typical Hodge locus is not empty, then the Hodge locus is analytically dense in S.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Some Effective Methods in the Context of the MordellLang Conjecture
Evelina Viada (Universität Göttingen)

 Location
 SLMath: Online/Virtual
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I would like to present links between the MordellLang and the Torsion Anoumalous conjectures. Then I intend to explain some methods to prove special cases that are effective or explicit. For some special families of curves, we manage to get all rational points.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Sparsity of Rational and Algebraic Points
Ziyang Gao (Leibniz Universität Hannover)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


A Product Formula for Periods
Pierre Colmez (Institut de Mathématiques de Jussieu)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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04:30 PM  06:20 PM


Reception

 Location
 SLMath: Front Courtyard
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Feb 08, 2023
Wednesday

09:30 AM  10:30 AM


Heights in the Isogeny Class of an Abelian Variety
Mark Kisin (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous A, and of height less than some fixed constant c.
In this talk, I will sketch a proof of the conjecture when the MumfordTate conjecture  which is known in many cases  holds for A. This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field.
This is joint work with Lucia Mocz.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


TateSemisimplicity Over Finite Fields
Ananth Shankar (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
Tate proved (amongst other things) that the action of Frobenius on the Tatemodule of an abelian variety over a finite field is semisimple. I will discuss the analogue of this question in the context of exceptional Shimura varieties. This is joint work with Ben Bakker and Jacob Tsimerman.
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12:00 PM  02:00 PM


Lunch

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03:00 PM  03:30 PM


Afternoon Tea

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Feb 09, 2023
Thursday

09:30 AM  10:30 AM


The Filtration Method in Diophantine Geometry
Min Ru (University of Houston)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
My talk will focus on the development and applications of the “filtration method”, which was introduced by Faltings and Wustholz in 1994 and developed by Evertse and Ferretti, and independently by CorvajaZannier. Further works were done by Levin, Autissier, and by Vojta and myself. In particular, I will present the recent work about a general arithmetic inequality, as an extension of the Schmidt's subspace theorem, established by Vojta and myself, as well as its applications. If time permits, I'll also describe its relationship with the algebrogeometric notion of the stability concerning the existence of KahlerEinstein metric on Fano varieties.
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


Modularity and Height Bounds
Hector Pasten (Pontificia Universidad Católica de Chile)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
In this talk I will explain how the modularity of elliptic curves leads to height bounds of different sorts. These have applications in explicit computation of integral points in varieties, and proving unconditional results related to the abc conjecture.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Quadratic Chabauty for Modular Curves
Jennifer Balakrishnan (Boston University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
We describe how padic height pairings can be used to determine the set of rational points on curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


Path Integrals and LFunctions
Minhyong Kim (International Centre for Mathematical Sciences)

 Location
 SLMath: Online/Virtual
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In the 1960s, Barry Mazur noticed an analogy between knots and primes that led to a view of the main conjecture of Iwasawa theory in which the padic Lfunction plays the role of the Alexander polynomial. In the late 1980s, Edward Witten reconstructed the Jones polynomial as a path integralin the sense of quantum field theoryassociated to SU(2) ChernSimons theory. This talk will review some of this history and present a weak arithmetic analogue of Witten’s formula. (This is joint work with Magnus Carlson, Heejoong Chung, Dohyeong Kim, Jeehoon Park, and Hwajong Yoo.)
 Supplements




Feb 10, 2023
Friday

09:30 AM  10:30 AM


FormalAnalytic Surfaces: Nori’s Bound in Arakelov Geometry and Archimedean Overflow
JeanBenoit Bost (Faculté des Sciences d’Orsay)

 Location
 SLMath: Online/Virtual
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 Abstract
In this talk, I will present various results, established in collaboration with François Charles, concerning formalanalytic arithmetic surfaces. These play the role, in Arakelov geometry, of germs of analytic or formal surfaces in complex algebraic geometry.
Firstly I will discuss various motivations for the study of formalanalytic arithmetic surfaces, notably algebraization results in Diophantine geometry and their recent developments by CalegariDimitrovTang. I will also recall classical theorems in complex geometry, due to Andreotti and Nori, of which our results are arithmetic analogues. I will present in some detail the arithmetic counterpart, in Arakelov geometry, of Nori’s degree bound. This arithmetic bound notably involves a new archimedean invariant, the overflow, attached to an analytic map from a pointed compact Riemann surface to another Riemann surface. Explicit expressions for this invariant play a key role in various applications of our results in Diophantine geometry.
Further applications of our arithmetic Nori’s bound will be presented in the talk of François Charles
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10:30 AM  11:00 AM


Break

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11:00 AM  12:00 PM


FormalAnalytic Arithmetic Surfaces: Finiteness Theorems, and Applications to Arithmetic Fundamental Groups
François Charles (École Normale Supérieure)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
In this talk, I will discuss several results obtained in collaboration with JeanBenoît Bost on the geometry of formalanalytic surfaces, as introduced in Bost's talk, of which this one is a followup.
Using numerical invariants of those surfaces, I will discuss finiteness results under suitable positivity conditions, with applications to qualitative finiteness results for certain algebras of power series satisfying "twisted" integrality conditions, motivated by concrete situations in the arithmetic geometry of modular curves.
I will also explain how to use formalanalytic arithmetic surfaces to obtain finiteness results for fundamental groups of certain arithmetic surfaces, including some integral models of modular curves.
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12:00 PM  02:00 PM


Lunch

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02:00 PM  03:00 PM


Unlikely Intersections and Applications to Diophantine Problems
Laura Capuano (Università degli Studi Roma Tre)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
The ZilberPink conjectures on unlikely intersections deal with intersections of subvarieties of a (semi)abelian variety or, more in general, of a Shimura variety, with “special” subvarieties of the ambient space. These conjectures generalize many classical results such as Faltings’ Theorem (Mordell Conjecture), Raynaud’s Theorem (ManinMumford Conjecture) and AndréOort Conjecture and have been studied by several authors in the last two decades.
Most proofs of results in this area follow the wellestablished PilaZannier strategy, first introduced by the two authors in 2008 to give an alternative proof of Raynaud’s theorem as a combination of results coming from ominimality (PilaWilkie’s theorem) with other diophantine ingredients. The talk will focus on a general introduction to these problems, on some results for semiabelian varieties and families of abelian varieties, and on applications to other problems of diophantine nature.
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03:00 PM  03:30 PM


Afternoon Tea

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03:30 PM  04:30 PM


Polynomials with Squarefree Discriminant
Arul Shankar (University of Toronto)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
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 Abstract
A classical question in analytic number theory is: given a polynomial with integer coefficients, how often does it take squarefree values? In arithmetic statistics, we are particularly interested in the case of discriminant polynomials. In this talk, I will present several different cases of this question. First, we will consider a classical result of DavenportHeilbronn which considers the case of discriminants of binary cubic forms. Motivated by this case, I will discuss two works, joint with Manjul Bhargava and Xiaoheng Wang, in which we consider the spaces of degreen polynomials and degreen binary forms. We will prove that a positive proportion of these polynomials and forms have squarefree discriminant.
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