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<pre>INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540
Mathematics Seminars
Week of October 23, 2017
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To view mathematics in titles and abstracts, please click on the talk's link.
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Monday, October 23
Computer Science/Discrete Mathematics Seminar I
Topic: A nearly optimal lower bound on the approximate degree of AC$^0$
Speaker: Mark Bun, Princeton University
Time/Room: 11:00am - 12:15pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=128781">http://www.math.ias.edu/seminars/abstract?event=128781</a>
Members' Seminar
Topic: Geometry and arithmetic of sphere packings
Speaker: Alex Kontorovich, Rutgers University
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=129299">http://www.math.ias.edu/seminars/abstract?event=129299</a>
Princeton/IAS Symplectic Geometry Seminar
Topic: Wrapped Fukaya categories and functors
Speaker: Yuan Gao, Stonybrook University
Time/Room: 4:00pm - 5:00pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=132409">http://www.math.ias.edu/seminars/abstract?event=132409</a>
Tuesday, October 24
Locally Symmetric Spaces Seminar
Topic: Cohomology of arithmetic groups and Eisenstein series - an introduction
Speaker: Joachim Schwermer, Universität Wien; Member, School of Mathemtics
Time/Room: 10:00am - 11:45am/Physics Library, Bloomberg Hall 201
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=133236">http://www.math.ias.edu/seminars/abstract?event=133236</a>
Computer Science/Discrete Mathematics Seminar II
Topic: On the strength of comparison queries
Speaker: Shay Moran, University of California, San Diego; Member, School of Mathematics
Time/Room: 10:30am - 12:30pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=129013">http://www.math.ias.edu/seminars/abstract?event=129013</a>
Locally Symmetric Spaces Seminar
Topic: Motivic correlators and locally symmetric spaces II
Speaker: Alexander Goncharov, Yale University; Member, School of Mathematics and Natural Sciences
Time/Room: 1:45pm - 4:15pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=133457">http://www.math.ias.edu/seminars/abstract?event=133457</a>
Joint IAS/Princeton University Number Theory Seminar
Topic: Elliptic curves of rank two and generalised Kato classes
Speaker: Francesc Castella, Princeton University
Time/Room: 4:45pm - 5:45pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=133488">http://www.math.ias.edu/seminars/abstract?event=133488</a>
Wednesday, October 25
Analysis Seminar
Topic: Nematic liquid crystal phase in a system of interacting dimers
Speaker: Ian Jauslin, Member, School of Mathematics
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=132443">http://www.math.ias.edu/seminars/abstract?event=132443</a>
Mathematical Conversations
Topic: To Be Announced
Speaker: Toniann Pitassi, University of Toronto; Visiting Professor, School of Mathematics
Time/Room: 6:00pm - 7:00pm/Dilworth Room
Thursday, October 26
Working Group on Algebraic Number Theory
Speaker: To Be Announced
Time/Room: 2:00pm - 4:00pm/S-101
Joint IAS/Princeton University Number Theory Seminar
Topic: A converse theorem of Gross-Zagier and Kolyvagin: CM case
Speaker: Ye Tian, Chinese Academy of Sciences
Time/Room: 4:30pm - 5:30pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=131088">http://www.math.ias.edu/seminars/abstract?event=131088</a>
</pre>
1 A nearly optimal lower bound on the approximate degree of AC$^0$
<br>
Mark Bun
<br>
<br>
<p>The approximate degree of a Boolean function $f$ is the least
degree of a real polynomial that approximates $f$ pointwise to
error at most $1/3$. For any constant $\delta > 0$, we exhibit
an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$.
This improves over the best previous lower bound of
$\Omega(n^{2/3})$ due to Aaronson and Shi, and nearly matches the
trivial upper bound of $n$ that holds for any function.</p>
<p>Our lower bound follows from a new hardness amplification
theorem, which shows how to increase the approximate degree of a
given function while preserving its computability by
constant-depth circuits. I will also describe several applications
of our results in communication complexity and cryptography.</p>
<p>This is joint work with Justin Thaler and is available at <a>https://eccc.weizmann.ac.il/report/2017/051/</a>.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=128781">http://www.math.ias.edu/seminars/abstract?event=128781</a><br>
<br>
2 Geometry and arithmetic of sphere packings
<br>
Alex Kontorovich
<br>
<br>
<p>We introduce the notion of a "crystallographic sphere packing,"
which generalizes the classical Apollonian circle packing. Tools
from arithmetic groups, hyperbolic geometry, and dynamics are used
to show that, on one hand, there is an infinite zoo of such
objects, while on the other, there are essentially finitely many
of these, in all dimensions. No familiarity with any of these
topics will be assumed.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=129299">http://www.math.ias.edu/seminars/abstract?event=129299</a><br>
<br>
3 Wrapped Fukaya categories and functors
<br>
Yuan Gao
<br>
<br>
<p>Inspired by homological mirror symmetry for non-compact
manifolds, one wonders what functorial properties wrapped Fukaya
categories have as mirror to those for the derived categories of
the mirror varieties, and also whether homological mirror symmetry
is functorial. Comparing to the theory of Lagrangian
correspondences for compact manifolds, some subtleties are seen in
view of the fact that modules over non-proper categories are
complicated. In this talk, the story concerning the fundamental
construction of Fourier-Mukai type functors of wrapped Fukaya
categories is discussed, under slightly modified framework of
wrapped Floer theory. Applications of the relevant techniques to
be presented include the Kunneth formula and restriction maps.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=132409">http://www.math.ias.edu/seminars/abstract?event=132409</a><br>
<br>
4 Cohomology of arithmetic groups and Eisenstein series - an
introduction
<br>
Joachim Schwermer
<br>
<br>
<p>I intend to cover some basic questions and material regarding the
phenomena in the cohomology of an arithmetic group "at infinity"
when the corresponding locally symmetric space originating with an
algebraic $k$-group $G$ of positive $k$-rank is non-compact [$k$
an algebraic number field]. The theory of Eisenstein series plays
a fundamental role in this discussion.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=133236">http://www.math.ias.edu/seminars/abstract?event=133236</a><br>
<br>
5 On the strength of comparison queries
<br>
Shay Moran
<br>
<br>
<p>Joint work with Daniel Kane (UCSD) and Shachar Lovett (UCSD)</p>
<p>We construct near optimal linear decision trees for a variety of
decision problems in combinatorics and discrete geometry.</p>
<p>For example, for any constant $k$, we construct linear decision
trees that solve the $k$-SUM problem on $n$ elements using $O(n
\log^2 n)$ linear queries. This settles a problem studied by
[Meyer auf der Heide ’84, Meiser ‘93, Erickson ‘95, Ailon and
Chazelle ‘05, Gronlund and Pettie '14, Gold and Sharir ’15,
Cardinal et al '15, Ezra and Sharir ’16] and others.</p>
<p>The queries we use are comparison queries, which compare the sums
of two $k$-subsets. When viewed as linear queries, comparison
queries are $2k$-sparse and have only $\{-1,0,1\}$ coefficients.
We give similar constructions for sorting sumsets $A+B$ and for
deciding the SUBSET-SUM problem, both with optimal number of
queries, up to poly-logarithmic terms.</p>
<p>Our constructions are based on the notion of ``inference
dimension", recently introduced by the authors in the context of
active classification with comparison queries. This can be viewed
as another contribution to the fruitful link between machine
learning and discrete geometry, which goes back to the discovery
of the VC dimension.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=129013">http://www.math.ias.edu/seminars/abstract?event=129013</a><br>
<br>
6 Motivic correlators and locally symmetric spaces II
<br>
Alexander Goncharov
<br>
<br>
<p>According to Langlands, pure motives are related to a certain
class of automorphic representations.</p>
<p>Can one see mixed motives in the automorphic set-up? For
examples, can one see periods of mixed motives in entirely
automorphic terms? The goal of this and the next lecture is to
supply some examples.</p>
<p>We define motivic correlators describing the structure of the
motivic fundamental group $\pi_1^{\mathcal M}(X)$ of a curve.
Their relevance to the questions raised above is explained by the
following examples.</p>
<p>1. Motivic correlators have an explicit Hodge realization given
by the Hodge correlator integrals, providing a new description of
the real mixed Hodge structure of the pro-nilpotent completion of
$\pi_1(X)$. When $X$ is a modular curve, the simplest of them
coincide with the Rankin-Selberg integrals, and the rest provide
an "automorphic" description of a class of periods of mixed
motives related to (products of) modular forms.</p>
<p>2. We use motivic correlators to relate the structure of
$\pi_1^{\mathcal M}(\mathbb G_m − \mu N )$ to the geometry of the
locally symmetric spaces for the congruence subgroup $\Gamma_1 (m;
N ) \subset \mathrm{GL}_m(\mathbb Z)$. Then we use the geometry of
the latter, for $m \leq 4$, to understand the structure of the
former.</p>
<p>3. This mysterious relation admits an "explanation" for $m = 2$:
we define a canonical map \[ \mu : \text{modular complex} \to
\text{the weight two motivic complex of the modular curve.} \]</p>
<p>Here the complex on the left calculates the singular homology of
the modular curve via modular symbols. The map $\mu$ generalizes
the Belinson-Kato Euler system in $K_2$ of the modular curves.</p>
<p>Composing the map μ with the specialization to a cusp, we recover
the correspondence above at $m = 2$.</p>
<p>4. Yet specializing to CM points on modular curves, we get a new
instance of the above correspondence, now between $\pi_1^{\mathcal
M}(E − E[\mathcal N])$ and geometry of arithmetic hyperbolic
threefolds. Here $E$ is a CM elliptic curve, and $\mathcal N
\subset \mathrm{Aut}(E)$ is an ideal.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=133457">http://www.math.ias.edu/seminars/abstract?event=133457</a><br>
<br>
7 Elliptic curves of rank two and generalised Kato classes
<br>
Francesc Castella
<br>
<br>
<p>The generalised Kato classes of Darmon-Rotger arise as $p$-adic
limits of diagonal cycles on triple products of modular curves,
and in some cases, they are predicted to have a bearing on the
arithmetic of elliptic curves over $Q$ of rank two. In this talk,
we will report on a joint work in progress with Ming-Lun Hsieh
concerning a special case of the conjectures of Darmon-Rotger.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=133488">http://www.math.ias.edu/seminars/abstract?event=133488</a><br>
<br>
8 Nematic liquid crystal phase in a system of interacting dimers
<br>
Ian Jauslin
<br>
<br>
<p>In 1979, O. Heilmann and E.H. Lieb introduced an interacting
dimer model with the goal of proving the emergence of a nematic
liquid crystal phase in it. In such a phase, dimers spontaneously
align, but there is no long range translational order. Heilmann
and Lieb proved that dimers do, indeed, align, and conjectured
that there is no translational order. I will discuss a recent
proof of this conjecture. This is joint work with Elliott H. Lieb.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=132443">http://www.math.ias.edu/seminars/abstract?event=132443</a><br>
<br>
9 A converse theorem of Gross-Zagier and Kolyvagin: CM case
<br>
Ye Tian
<br>
<br>
<p>Let $E$ be a CM elliptic curves over rationals and $p$ an odd
prime ordinary for $E$. If the $\mathbb Z_p$ corank of $p^\infty$
Selmer group for $E$ equals one, then we show that the analytic
rank of $E$ also equals one. This is joint work with Ashay
Burungale.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=131088">http://www.math.ias.edu/seminars/abstract?event=131088</a><br>
<br>
IAS Math Seminars Home Page:<br>
<a href="http://www.math.ias.edu/seminars">http://www.math.ias.edu/seminars</a>
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