[iasmath-seminars] Mathematics Seminars -- Week of October 30, 2017
Anthony Pulido
apulido at ias.edu
Fri Oct 27 19:04:47 EDT 2017
INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540
Mathematics Seminars
Week of October 30, 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 30
Computer Science/Discrete Mathematics Seminar I
Topic: Fooling intersections of low-weight halfspaces
Speaker: Rocco Servedio, Columbia University
Time/Room: 11:00am - 12:15pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=128784
Members' Seminar
Topic: High density phases of hard-core lattice particle systems
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=129314
Princeton/IAS Symplectic Geometry Seminar
Topic: Weinstein manifolds through skeletal topology
Speaker: Laura Starkston, Stanford University
Time/Room: 4:00pm - 5:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=132422
Tuesday, October 31
Locally Symmetric Spaces Seminar
Topic: Cohomology of arithmetic groups and Eisenstein series - an introduction (continued)
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=133736
Computer Science/Discrete Mathematics Seminar II
Topic: Cap-sets in $(F_q)^n$ and related problems
Speaker: Zeev Dvir, Princeton University; von Neumann Fellow, School of Mathematics
Time/Room: 10:30am - 12:30pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=129016
Locally Symmetric Spaces Seminar
Topic: Motivic correlators and locally symmetric spaces III
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=133460
Joint IAS/Princeton University Number Theory Seminar
Topic: Nonlinear descent on moduli of local systems
Speaker: Junho Peter Whang, Princeton University
Time/Room: 4:45pm - 5:45pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=133506
Wednesday, November 1
Analysis Seminar
Topic: Structure theorems for intertwining wave operators
Speaker: Wilhelm Schlag, University of Chicago; Visiting Professor, School of Mathematics
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=132446
Mathematical Conversations
Topic: To Be Announced
Speaker: Nadav Cohen, Member, School of Mathematics
Time/Room: 6:00pm - 7:00pm/Dilworth Room
Thursday, November 2
Analysis Seminar
Topic: Two-bubble dynamics for the equivariant wave maps equation
Speaker: Jacek Jendrej, University of Chicago
Time/Room: 11:00am - 12:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=133767
Working Group on Algebraic Number Theory
No seminar - Fall break at Princeton University
Speaker: No seminar - Fall break at Princeton University
Time/Room: -
Joint IAS/Princeton University Number Theory Seminar
Topic: On the notion of genus for division algebras and algebraic groups
Speaker: Andrei Rapinchuk, University of Virginia
Time/Room: 4:30pm - 5:30pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=131091
Friday, November 3
Special Seminar
Topic: Introduction to works of Takuro Mochizuki
Speaker: Pierre Deligne, Professor Emeritus, School of Mathematics
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=133715
1 Fooling intersections of low-weight halfspaces
Rocco Servedio
A weight-$t$ halfspace is a Boolean function $f(x)=\mathrm{sign}(w_1 x_1
+ \cdots + w_n x_n - \theta)$ where each $w_i$ is an integer in
$\{-t,\dots,t\}.$ We give an explicit pseudorandom generator that
$\delta$-fools any intersection of $k$ weight-$t$ halfspaces with seed
length poly$(\log n, \log k,t,1/\delta)$. In particular, our result
gives an explicit PRG that fools any intersection of any quasipoly$(n)$
number of halfspaces of any polylog$(n)$ weight to any $1/$polylog$(n)$
accuracy using seed length polylog$(n).$
Prior to this work no explicit PRG with seed length $o(n)$ was known
even for fooling intersections of $n$ weight-1 halfspaces to constant
accuracy.
Our analysis introduces new coupling-based ingredients into the standard
Lindeberg method for establishing quantitative central limit theorems
and associated pseudorandomness results.
Joint work with Li-Yang Tan.
http://www.math.ias.edu/seminars/abstract?event=128784
2 High density phases of hard-core lattice particle systems
Ian Jauslin
In this talk, I will discuss the behavior of hard-core lattice particle
systems at high fugacities. I will first present a collection of models
in which the high fugacity phase can be understood by expanding in
powers of the inverse of the fugacity. I will then discuss a model in
which this expansion diverges, but which can still be solved by
expanding in other high fugacity variables. This model is an interacting
dimer model, introduced by O.Heilmann and E.H.Lieb in 1979 as an example
of a nematic liquid crystal. Heilmann and Lieb proved that the dimers
spontaneously align, and conjectured that they do not exhibit long range
positional order. I will present a recent proof of this conjecture.
http://www.math.ias.edu/seminars/abstract?event=129314
3 Weinstein manifolds through skeletal topology
Laura Starkston
We will discuss how to study the symplectic geometry of $2n$-dimensional
Weinstein manifolds via the topology of a core $n$-dimensional complex
called the skeleton. We show that the Weinstein structure can be
homotoped to admit a skeleton with a unique symplectic neighborhood.
Then we further work to reduce the remaining singularities to a simple
combinatorial list coinciding with Nadler's arboreal singularities. We
will discuss how arboreal singularities occur naturally in a Weinstein
skeleton, and what information about the symplectic manifold one might
hope to extract out of an arboreal complex.
http://www.math.ias.edu/seminars/abstract?event=132422
4 Cohomology of arithmetic groups and Eisenstein series - an
introduction (continued)
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=133736
5 Cap-sets in $(F_q)^n$ and related problems
Zeev Dvir
A cap set in $(F_q)^n$ is a set not containing a three term arithmetic
progression. Last year, in a surprising breakthrough, Croot-Lev-Pach and
a follow up paper of Ellenberg-Gijswijt showed that such sets have to be
of size at most $c^n$ with $c < q$ (as $n$ goes to infinity). The simple
algebraic proof of this result has since led to new progress and
insights on several other related problems in combinatorics and
theoretical computer science. In this survey I will describe these
results including some new work (joint with B. Edelman) relating to
matrix rigidity.
http://www.math.ias.edu/seminars/abstract?event=129016
6 Motivic correlators and locally symmetric spaces III
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=133460
7 Nonlinear descent on moduli of local systems
Junho Peter Whang
In 1880, Markoff studied a cubic Diophantine equation in three variables
now known as the Markoff equation, and observed that its integral
solutions satisfy a form of nonlinear descent. Generalizing this, we
consider families of log Calabi-Yau varieties arising as moduli spaces
for local systems on topological surfaces, and prove a structure theorem
for their integral points using mapping class group dynamics. The result
is reminiscent of the finiteness of class numbers for linear arithmetic
group actions on homogeneous varieties, and this Diophantine perspective
guides us to obtain new extensions of classical results on hyperbolic
surfaces along the way.
http://www.math.ias.edu/seminars/abstract?event=133506
8 Structure theorems for intertwining wave operators
Wilhelm Schlag
We will describe an implementation of the Wiener theorem in $L^1$ type
convolution algebras in the setting of spectral theory. In joint work
with Marius Beceanu we obtained a structure theorem for the wave
operators by this method.
http://www.math.ias.edu/seminars/abstract?event=132446
9 Two-bubble dynamics for the equivariant wave maps equation
Jacek Jendrej
I will consider the energy-critical wave maps equation with values in
the sphere in the equivariant case, that is for symmetric initial data.
It is known that if the initial data has small energy, then the
corresponding solution scatters. Moreover, the initial data of any
scattering solution has topological degree 0. I try to answer the
following question: what are the non-scattering solutions of topological
degree 0 and the least possible energy? I will show how to construct
such threshold solutions. Then I will describe the dynamical behavior of
any threshold solution: in one time direction it is close to a
superposition of two stationary states, in the other time direction it
scatters. This is a joint work with Andrew Lawrie (MIT).
http://www.math.ias.edu/seminars/abstract?event=133767
10 On the notion of genus for division algebras and algebraic groups
Andrei Rapinchuk
Let $D$ be a central division algebra of degree $n$ over a field $K$.
One defines the genus gen$(D)$ of $D$ as the set of classes $[D']$ in
the Brauer group Br$(K)$ where $D'$ is a central division $K$-algebra of
degree $n$ having the same isomorphism classes of maximal subfields as
$D$. I will review the results on gen$(D)$ obtained in the last several
years, in particular the finiteness theorem for gen$(D)$ when $K$ is
finitely generated of characteristic not dividing $n$. I will then
discuss how the notion of genus can be extended to arbitrary absolutely
almost simple algebraic $K$-groups using maximal $K$-tori in place of
maximal subfields, and report on some recent progress in this direction.
(Joint work with V. Chernousov and I. Rapinchuk.)
http://www.math.ias.edu/seminars/abstract?event=131091
11 Introduction to works of Takuro Mochizuki
Pierre Deligne
I will give examples and motivations, about the local systems/Higgs
bundles correspondence, the case of variations of Hodge structures and
the case of irregular singularities. I hope this will help to enjoy the
forthcoming lectures of T. Mochizuki the week of November 13.
http://www.math.ias.edu/seminars/abstract?event=133715
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