[Csdmsemo] [math-ias] Updated Location: May 8, Joint IAS/Princeton University Number Theory Seminar

Fri May 4 12:42:06 EDT 2018

INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540

Mathematics Seminars
Week of May 7, 2018

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Please note:
*         Peter Sarnak has organized 3 special seminars this week which will
occur in Simonyi Hall 101 on Monday, Thursday, and Friday.
*         Tuesday, Math 8th’s Locally Symmetric Spaces Seminar will only be
an hour long.
*         Tuesday, Math 8th’s Number Theory Seminar will be held in Fine
Hall 214 from 4:30 – 5:30pm. 

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Monday, May 7

Special Seminar
Topic:                     Benjamini-Schramm convergence and eigenfunctions
on Riemannian manifolds
Speaker:                 Miklos Abert, Alfréd Rényi Institute of Mathematics
Time/Room:           3:15pm - 4:15pm/Simonyi Hall 101
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=137058

Tuesday, May 8

Locally Symmetric Spaces Seminar
Topic:                     Eigenfunctions and random waves on locally
symmetric spaces in the Benjamini-Schramm limit
Speaker:                 Nicolas Bergeron, l’université Pierre et Marie
Curie
Time/Room:           1:45pm - 2:45pm/Simonyi Hall 101
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=137079

Joint IAS/Princeton University Number Theory Seminar
Topic:                     Towards counting rational points on genus $g$
curves
Speaker:                 Ziyang Gao, Princeton University; Member, School of
Mathematics
Time/Room:           4:30pm - 5:30pm/Fine Hall 214, Princeton University
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=136748

Thursday, May 10

Working Group on Algebraic Number Theory
Speaker:                 To Be Announced
Time/Room:           2:00pm - 4:00pm/1201 Fine Hall, Princeton University

Special Probability Seminar
Topic:                     Percolation of sign clusters for the Gaussian
free field I
Speaker:                 Pierre-Francois Rodriguez, University of California
Time/Room:           2:00pm - 3:00pm/Simonyi Hall 101
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=136876

Joint IAS/Princeton University Number Theory Seminar
Topic:                     Goldfeld's conjecture and congruences between
Heegner points
Speaker:                 Chao Li, Columbia University
Time/Room:           4:30pm - 5:30pm/Fine Hall 214, Princeton University
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=136757

Friday, May 11

Special Probability Seminar
Topic:                     Percolation of sign clusters for the Gaussian
free field II
Speaker:                 Pierre-Francois Rodriguez, University of California
Time/Room:           2:00pm - 3:00pm/Simonyi Hall 101
Abstract Link:
http://www.math.ias.edu/seminars/abstract?event=136879

1. Benjamini-Schramm convergence and eigenfunctions on Riemannian manifolds 
   Miklos Abert 

Let M be a compact manifold with negative curvature. The Quantum Unique
Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the
Laplacian on M get equidistributed as the eigenvalue tends to infinity. A
weaker version, called Quantum Ergodicity by Shnirelman, Colin de Verdière,
and Zelditch says that this is true for a density 1 subsequence. On the
other hand, a conjecture of Berry, that has not been formulated in a
mathematically precise way, says that high eigenvalue eigenfunctions behave
like Gaussian random Euclidean waves. Using the framework of
Benjamini-Schramm convergence, we give a mathematically exact formulation of
Berry’s conjecture. This allows us to establish a relation to Quantum Unique
Ergodicity and to prove a version of Berry’s conjecture in a quite specific
setting.

Our approach is related to the recent result of Backhausz and Szegedy on
almost eigenfunctions of random regular graphs. A major difference is that
instead of expander graphs, here one deals with a hyperfinite sequence of
manifolds. In particular, opposed to the Backhausz-Szegedy theorem, Berry's
conjecture will not hold for almost eigenfunctions.

http://www.math.ias.edu/seminars/abstract?event=137058

2. Eigenfunctions and random waves on locally symmetric spaces in the
Benjamini-Schramm limit 
   Nicolas Bergeron 

I will consider the asymptotic behavior of Laplacian eigenfunctions on a
locally symmetric compact manifold as the volume approaches infinity. I will
formulate a precise conjecture "à la Berry" and will describe some partial
results obtained with Miklos Abert and Etienne Le Masson.

http://www.math.ias.edu/seminars/abstract?event=137079

3. Towards counting rational points on genus $g$ curves 
   Ziyang Gao 

We start by showing that for any 1-parameter family of genus $g>2$ curves,
the number of rational points is bounded by $g$, degree of the field, and
the Mordell-Weil rank. Apart from the classical Faltings-Vojta-Bombieri
method, the new input is a height inequality recently proved (joint with
Philipp Habegger). Then I'll explain some generalization of this method to
an arbitrary family of curves. I'll focus on how the mixed Ax-Schanuel for
universal abelian varieties, extension of a recent result of
Mok-Pila-Tsimerman, applies to this problem. This is work in progress, joint
with Vesselin Dimitrov and Philipp Habegger.

http://www.math.ias.edu/seminars/abstract?event=136748

4. Percolation of sign clusters for the Gaussian free field I 
   Pierre-Francois Rodriguez 

We consider level sets of the Gaussian free field on the $d$-dimensional
lattice, for $d>2$, above a given real-valued height $h$. This defines a
percolation model with strong, algebraically decaying correlations. We prove
a conjecture of Lebowitz asserting that the sign clusters of this field,
i.e. the level sets above height $h=0$, contain a unique infinite connected
component. As a central ingredient, we exploit a certain algebraic
correspondence which relates the free field to a Poissonian soup of
bi-infinite random walk trajectories known as random interlacements,
originally introduced by Sznitman to study local limits of random walks on
large, asymptotically transient graphs. The interlacement trajectories can
be fruitfully used to build large clusters. Representations of similar
flavor can be traced back to celebrated works of Symanzik and
Brydges-Fröhlich-Spencer.

Talk I will set the stage and provide an overview of the argument. Talk II
will fill in certain technical details, including state-of-the-art methods
to deal with percolation models in the presence of strong correlations, and
the extension of these techniques to a so-called "cable system", obtained by
joining adjacent vertices on the lattice by "cables", which provides a handy
continuous geometric structure. Based on joint work with Alexander Drewitz
and Alexis Prévost.

http://www.math.ias.edu/seminars/abstract?event=136876

5. Goldfeld's conjecture and congruences between Heegner points 
   Chao Li 

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of
Goldfeld asserts that a positive proportion of its quadratic twists should
have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$
has a rational 3-isogeny. We also prove the analogous result for the sextic
twists of $j$-invariant 0 curves. For a more general elliptic curve $E$, we
show that the number of quadratic twists of $E$ up to twisting discriminant
$X$ of analytic rank 0 (resp. 1) is $\gg\frac{X}{\log^{5/6}X}$, improving
the current best general bound towards Goldfeld's conjecture due to
Ono-Skinner (resp. Perelli-Pomykala). We prove these results by establishing
a congruence formula between $p$-adic logarithms of Heegner points based on
Coleman's integration. This is joint work with Daniel Kriz.

http://www.math.ias.edu/seminars/abstract?event=136757

6. Percolation of sign clusters for the Gaussian free field II 
   Pierre-Francois Rodriguez 

We consider level sets of the Gaussian free field on the d-dimensional
lattice, for d>2, above a given real-valued height h. This defines a
percolation model with strong, algebraically decaying correlations. We prove
a conjecture of Lebowitz asserting that the sign clusters of this field,
i.e. the level sets above height h=0, contain a unique infinite connected
component. As a central ingredient, we exploit a certain algebraic
correspondence which relates the free field to a Poissonian soup of
bi-infinite random walk trajectories known as random interlacements,
originally introduced by Sznitman to study local limits of random walks on
large, asymptotically transient graphs. The interlacement trajectories can
be fruitfully used to build large clusters. Representations of similar
flavor can be traced back to celebrated works of Symanzik and
Brydges-Fröhlich-Spencer.

Talk I will set the stage and provide an overview of the argument. Talk II
will fill in certain technical details, including state-of-the-art methods
to deal with percolation models in the presence of strong correlations, and
the extension of these techniques to a so-called "cable system", obtained by
joining adjacent vertices on the lattice by "cables", which provides a handy
continuous geometric structure. Based on joint work with Alexander Drewitz
and Alexis Prévost.

http://www.math.ias.edu/seminars/abstract?event=136879

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IAS Math Seminars Home Page:
http://www.math.ias.edu/seminars

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