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</o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" vlink="#954F72"><div class=WordSection1><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>INSTITUTE FOR ADVANCED STUDY<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>School of Mathematics<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Princeton, NJ 08540<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Mathematics Seminars<o:p></o:p></span></b></pre><pre><b><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Week of May 7, 2018<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>--------------------------------------------------------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></pre><pre><b><span style='font-size:12.0pt;font-family:"Times New Roman",serif'>Please note:<o:p></o:p></span></b></pre><pre style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo1'><![if !supportLists]><span style='font-size:11.0pt;font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Peter Sarnak has organized 3 special seminars this week which will occur in Simonyi Hall 101 on Monday, Thursday, and Friday.<o:p></o:p></span></pre><pre style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo1'><![if !supportLists]><span style='font-size:11.0pt;font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Tuesday, Math 8<sup>th</sup>’s Locally Symmetric Spaces Seminar will only be an hour long.<o:p></o:p></span></pre><pre style='margin-left:.5in;text-indent:-.25in;mso-list:l0 level1 lfo1'><![if !supportLists]><span style='font-size:11.0pt;font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'> </span></span></span><![endif]><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Tuesday, Math 8<sup>th</sup>’s Number Theory Seminar will be held in <u>Fine Hall 314 from 4:30 – 5:30pm</u>. <o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>--------------------------------------------------------------------------------------------------------------------------------------------------------------------------<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:12.0pt;font-family:"Times New Roman",serif'>Monday, May 7<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Special Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Benjamini-Schramm convergence and eigenfunctions on Riemannian manifolds<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Miklos Abert, Alfréd Rényi Institute of Mathematics<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 3:15pm - 4:15pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=137058">http://www.math.ias.edu/seminars/abstract?event=137058</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:12.0pt;font-family:"Times New Roman",serif'>Tuesday, May 8<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Locally Symmetric Spaces Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Eigenfunctions and random waves on locally symmetric spaces in the Benjamini-Schramm limit<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Nicolas Bergeron, l’université Pierre et Marie Curie<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 1:45pm - 2:45pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=137079">http://www.math.ias.edu/seminars/abstract?event=137079</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Joint IAS/Princeton University Number Theory Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Towards counting rational points on genus $g$ curves<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Ziyang Gao, Princeton University; Member, School of Mathematics<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 4:30pm - 5:30pm/Fine Hall 314, Princeton University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136748">http://www.math.ias.edu/seminars/abstract?event=136748</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:12.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:12.0pt;font-family:"Times New Roman",serif'>Thursday, May 10<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Working Group on Algebraic Number Theory<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: To Be Announced<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 2:00pm - 4:00pm/1201 Fine Hall, Princeton University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Special Probability Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Percolation of sign clusters for the Gaussian free field I<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Pierre-Francois Rodriguez, University of California<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 2:00pm - 3:00pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136876">http://www.math.ias.edu/seminars/abstract?event=136876</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Joint IAS/Princeton University Number Theory Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Goldfeld's conjecture and congruences between Heegner points<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Chao Li, Columbia University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 4:30pm - 5:30pm/Fine Hall 214, Princeton University<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136757">http://www.math.ias.edu/seminars/abstract?event=136757</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><span style='font-size:12.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre><b><span style='font-size:12.0pt;font-family:"Times New Roman",serif'>Friday, May 11<o:p></o:p></span></b></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><pre style='margin-bottom:6.0pt'><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Special Probability Seminar<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Topic: Percolation of sign clusters for the Gaussian free field II<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Speaker: Pierre-Francois Rodriguez, University of California<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Time/Room: 2:00pm - 3:00pm/Simonyi Hall 101<o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=136879">http://www.math.ias.edu/seminars/abstract?event=136879</a><o:p></o:p></span></pre><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'><o:p> </o:p></span></pre><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>1. Benjamini-Schramm convergence and eigenfunctions on Riemannian manifolds </span></b><span style='font-family:"Times New Roman",serif'><br> Miklos Abert <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<br> <br>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.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=137058">http://www.math.ias.edu/seminars/abstract?event=137058</a><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><br><br><br><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>2. Eigenfunctions and random waves on locally symmetric spaces in the Benjamini-Schramm limit</span></b><span style='font-family:"Times New Roman",serif'> <br> Nicolas Bergeron <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=137079">http://www.math.ias.edu/seminars/abstract?event=137079</a><br><br><br><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>3. Towards counting rational points on genus $g$ curves</span></b><span style='font-family:"Times New Roman",serif'> <br> Ziyang Gao <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=136748">http://www.math.ias.edu/seminars/abstract?event=136748</a><br><br><br><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>4. Percolation of sign clusters for the Gaussian free field I</span></b><span style='font-family:"Times New Roman",serif'> <br> Pierre-Francois Rodriguez <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<br> <br>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.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=136876">http://www.math.ias.edu/seminars/abstract?event=136876</a><br><br><br><o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>5. Goldfeld's conjecture and congruences between Heegner points</span></b><span style='font-family:"Times New Roman",serif'> <br> Chao Li <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<o:p></o:p></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=136757">http://www.math.ias.edu/seminars/abstract?event=136757</a><br><br><br><b><o:p></o:p></b></span></p><p class=MsoNormal style='margin-bottom:12.0pt'><b><span style='font-family:"Times New Roman",serif'>6. Percolation of sign clusters for the Gaussian free field II</span></b><span style='font-family:"Times New Roman",serif'> <br> Pierre-Francois Rodriguez <o:p></o:p></span></p><p style='margin-bottom:3.0pt'><span style='font-size:11.0pt'>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.<br> <br>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.<o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><a href="http://www.math.ias.edu/seminars/abstract?event=136879">http://www.math.ias.edu/seminars/abstract?event=136879</a><o:p></o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><o:p> </o:p></span></p><pre><span style='font-size:11.0pt;font-family:"Times New Roman",serif'>-------------------------------------<o:p></o:p></span></pre><p class=MsoNormal><span style='font-family:"Times New Roman",serif'><br><br>IAS Math Seminars Home Page:<br><a href="http://www.math.ias.edu/seminars">http://www.math.ias.edu/seminars</a><o:p></o:p></span></p></div></body></html>