[iasmath-seminars] Mathematics Seminars -- Week of October 23, 2017
Anthony Pulido
apulido at ias.edu
Fri Oct 20 18:00:39 EDT 2017
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: http://www.math.ias.edu/seminars/abstract?event=128781
Members' Seminar
Topic: Geometry and arithmetic of sphere packings
Speaker: Alex Kontorovich, Rutgers University
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=129299
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: http://www.math.ias.edu/seminars/abstract?event=132409
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: http://www.math.ias.edu/seminars/abstract?event=133236
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: http://www.math.ias.edu/seminars/abstract?event=129013
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: http://www.math.ias.edu/seminars/abstract?event=133457
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: http://www.math.ias.edu/seminars/abstract?event=133488
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: http://www.math.ias.edu/seminars/abstract?event=132443
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: http://www.math.ias.edu/seminars/abstract?event=131088
1 A nearly optimal lower bound on the approximate degree of AC$^0$
Mark Bun
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.
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.
This is joint work with Justin Thaler and is available at
https://eccc.weizmann.ac.il/report/2017/051/.
http://www.math.ias.edu/seminars/abstract?event=128781
2 Geometry and arithmetic of sphere packings
Alex Kontorovich
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.
http://www.math.ias.edu/seminars/abstract?event=129299
3 Wrapped Fukaya categories and functors
Yuan Gao
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.
http://www.math.ias.edu/seminars/abstract?event=132409
4 Cohomology of arithmetic groups and Eisenstein series - an introduction
Joachim Schwermer
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.
http://www.math.ias.edu/seminars/abstract?event=133236
5 On the strength of comparison queries
Shay Moran
Joint work with Daniel Kane (UCSD) and Shachar Lovett (UCSD)
We construct near optimal linear decision trees for a variety of
decision problems in combinatorics and discrete geometry.
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.
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.
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.
http://www.math.ias.edu/seminars/abstract?event=129013
6 Motivic correlators and locally symmetric spaces II
Alexander Goncharov
According to Langlands, pure motives are related to a certain class of
automorphic representations.
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.
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.
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.
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.
3. This mysterious relation admits an "explanation" for $m = 2$: we
define a canonical map \[ \mu : \text{modular complex} \to \text{the
weight two motivic complex of the modular curve.} \]
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.
Composing the map μ with the specialization to a cusp, we recover the
correspondence above at $m = 2$.
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.
http://www.math.ias.edu/seminars/abstract?event=133457
7 Elliptic curves of rank two and generalised Kato classes
Francesc Castella
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.
http://www.math.ias.edu/seminars/abstract?event=133488
8 Nematic liquid crystal phase in a system of interacting dimers
Ian Jauslin
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.
http://www.math.ias.edu/seminars/abstract?event=132443
9 A converse theorem of Gross-Zagier and Kolyvagin: CM case
Ye Tian
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.
http://www.math.ias.edu/seminars/abstract?event=131088
IAS Math Seminars Home Page:
http://www.math.ias.edu/seminars
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