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INSTITUTE FOR ADVANCED STUDY
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<pre>School of Mathematics
Princeton, NJ 08540
Mathematics Seminars
Week of October 9, 2017
****Please note that this week's CSDM seminars will take place in the West Building Lecture Hall.
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To view mathematics in titles and abstracts, please click on the talk's link.
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Monday, October 9
Emerging Topics Working Group
Topic: An introduction to quantum chaos
Speaker: Stéphane Nonnemacher, Université Paris-Sud
Time/Room: 11:00am - 12:00pm/S-101
Computer Science/Discrete Mathematics Seminar I
Topic: Barriers for rank methods in arithmetic complexity
Speaker: Rafael Oliveira, University of Toronto
Time/Room: 11:00am - 12:15pm/West Building Lecture Hall
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=128775" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=128775</a>
Members' Seminar
Topic: Analysis and topology on locally symmetric spaces
Speaker: Akshay Venkatesh, Stanford University; Distinguished Visiting Professor, School of Mathematics
Time/Room: 2:00pm - 3:00pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=129293" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=129293</a>
Emerging Topics Working Group
Topic: Fractal uncertainty principle and its applications
Speaker: Semyon Dyatlov, Massachusetts Institute of Technology
Time/Room: 4:30pm - 5:30pm/S-101
Princeton/IAS Symplectic Geometry Seminar
Topic: Wrapped Floer theory and Homological mirror symmetry for toric Calabi-Yau manifolds
Speaker: Yoel Groman, Columbia University
Time/Room: 4:45pm - 5:45pm/West Building Lecture Hall
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=132506" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=132506</a>
Tuesday, October 10
Locally Symmetric Spaces Seminar
Topic: Cohomology of arithmetic groups and automorphic forms: an introduction
Speaker: Laurent Clozel, Université Paris-Sud 11; Member, School of Mathematics
Time/Room: 10:00am - 11:45am/Physics Library, Bloomberg Hall 201
Emerging Topics Working Group
Topic: Semiclassical analysis, chaotic dynamics, and fractal uncertainty principle
Speaker: Semyon Dyatlov, Massachusetts Institute of Technology
Time/Room: 10:30am - 11:30am/S-101
Computer Science/Discrete Mathematics Seminar II
Topic: Structural aspects of the null-cone problem in invariant theory
Speaker: Ankit Garg, Microsoft Research
Time/Room: 10:30am - 12:30pm/West Building Lecture Hall
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=129007" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=129007</a>
Emerging Topics Working Group
Topic: Proof of fractal uncertainty principle
Speaker: Ruixiang Zhang, Member, School of Mathematics
Time/Room: 11:30am - 12:30pm/S-101
Locally Symmetric Spaces Seminar
Topic: Transfer operators for (relative) functoriality "beyond endoscopy" I
Speaker: Yiannis Sakellaridis, Rutgers University; von Neumann Fellow, School of Mathematics
Time/Room: 1:45pm - 4:15pm/West Building Lecture Hall
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=132632" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=132632</a>
Emerging Topics Working Group
Topic: Control of eigenfunctions on hyperbolic surfaces
Speaker: Long Jin, Purdue University
Time/Room: 4:30pm - 5:30pm/S-101
Wednesday, October 11
Emerging Topics Working Group
Topic: An introduction to Dolgopyat's method
Speaker: Frédéric Naud, Université Avignon
Time/Room: 11:00am - 12:00pm/S-101
Emerging Topics Working Group
Topic: Fractal uncertainty principle: improving over the volume bound
Speaker: Semyon Dyatlov, Massachusetts Institute of Technology
Time/Room: 2:00pm - 3:00pm/S-101
Emerging Topics Working Group
Topic: Limit sets in higher dimensions
Speaker: Michael Magee, Durham University
Time/Room: 3:15pm - 4:15pm/S-101
Thursday, October 12
Emerging Topics Working Group
Topic: Long time propagation of waves and the hyperbolic parametrix
Speaker: Stéphane Nonnemacher, Universite Paris-Sud
Time/Room: 11:00am - 12:00pm/S-101
Working Group on Algebraic Number Theory
Speaker: To Be Announced
Time/Room: 2:00pm - 4:00pm/Jadwin 111, Princeton University
Joint IAS/Princeton University Number Theory Seminar
Topic: On residues of Eisenstein series - through a cohomological lens
Speaker: Joachim Schwermer, University of Vienna
Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=131082" moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=131082</a>
Friday, October 13
Emerging Topics Working Group
Topic: TBA
Speaker: TBA
Time/Room: 11:00am - 12:00pm/S-101
</pre>
1 Barriers for rank methods in arithmetic complexity <br>
Rafael Oliveira <br>
<br>
<p>Arithmetic complexity is considered (for many good reasons)
simpler to understand than Boolean complexity. And indeed, we
seem to have significantly more lower bound techniques and
results in arithmetic complexity than in Boolean complexity.
Despite many successes and rapid progress, however, foundational
challenges, like proving super-polynomial lower bounds on
circuit or formula size for explicit polynomials, or
super-linear lower bounds on explicit 3-dimensional tensors,
remain elusive.</p>
<p>At the same time (and possibly for similar reasons), we have
plenty more excuses, in the form of “barrier results” for
failing to prove basic lower bounds in Boolean complexity than
in arithmetic complexity. Efforts to find barriers to arithmetic
lower bound techniques seem harder, and despite some attempts we
have no excuses of similar quality for these failures in
arithmetic complexity.</p>
<p>In this talk we will give the first unconditional barriers for
rank methods, which were long recognized as encompassing and
abstracting almost all known arithmetic lower bounds to-date,
including the most recent impressive successes. In this talk, we
will show that</p>
<ul>
<li>Rank methods cannot prove better than $\Omega_d(n^{\lfloor
d/2 \rfloor})$ lower bound on the tensor rank of any
$d$-dimensional tensor of side $n$. (In particular, they
cannot prove super-linear, indeed even $> 8n$ tensor rank
lower bounds for any 3-dimensional tensors.)</li>
<li>Rank methods cannot prove better than $\Omega_d(n^{\lfloor
d/2 \rfloor})$ lower bound on the Waring rank of any
$n$-variate polynomial of degree $d$. (In particular, they
cannot prove such lower bounds on stronger models, including
depth-3 circuits.)</li>
</ul>
<p>The bounds above nearly match the best explicit bounds we know
for these models, and hence offer an explanation why the rank
methods got stuck there. Time permitting, we will discuss how
these techniques can be extended to barriers for other
arithmetic models.</p>
<p>Joint work with Klim Efremenko, Ankit Garg and Avi Wigderson</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=128775"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=128775</a><br>
<br>
2 Analysis and topology on locally symmetric spaces <br>
Akshay Venkatesh <br>
<br>
<p>Locally symmetric spaces are a class of Riemannian manifolds
which play a special role in number theory. In this talk, I will
introduce these spaces through example, and show some of their
unusual properties from the point of view of both analysis and
topology. I will conclude by discussing their (still very
mysterious) relationship with algebraic geometry. </p>
<a href="http://www.math.ias.edu/seminars/abstract?event=129293"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=129293</a><br>
<br>
3 Wrapped Floer theory and Homological mirror symmetry for toric
Calabi-Yau manifolds <br>
Yoel Groman <br>
<br>
<p>Consider a Lagrangian torus fibration a la SYZ over a non
compact base. Using techniques from arXiv:1510.04265, I will
discuss the construction of wrapped Floer theory in this
setting. Note that this setting is generally not exact even near
infinity. The construction allows the formulation of a version
of the homological mirror symmetry conjecture for open manifolds
which are not exact near infinity. According to time
constraints, I will apply this to prove homological mirror
symmetry in the case where the A-model is the complement of an
anti-canonical divisor in a toric Calabi Yau manifold. </p>
<a href="http://www.math.ias.edu/seminars/abstract?event=132506"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=132506</a><br>
<br>
4 Structural aspects of the null-cone problem in invariant theory
<br>
Ankit Garg <br>
<br>
<p>Invariant theory studies the actions of groups on vector spaces
and what polynomial functions remain invariant under these
actions. An important object related to a group action is the
null cone, which is the set of common zeroes of all homogeneous
invariant polynomials. I will talk about the structural aspects
of the null cone from a computational and optimization
perspective. These will include the Hilbert-Mumford and
Kempf-Ness theorems which imply that null cone membership is in
NP intersect coNP (ignoring bit-size issues). I will explain how
this should be thought of as a noncommutative generalization of
linear programming duality, which arises when the group is
commutative (group of invertible diagonal matrices aka algebraic
tori). </p>
<a href="http://www.math.ias.edu/seminars/abstract?event=129007"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=129007</a><br>
<br>
5 Transfer operators for (relative) functoriality "beyond
endoscopy" I <br>
Yiannis Sakellaridis <br>
<br>
<p>"Beyond endoscopy", broadly interpreted, is the idea that
functoriality should be realized as a comparison between stable
trace formulas. The nature of this comparison, however, remains
completely unclear.</p>
<p>Broadening our scope to include the relative Langlands program
(replacing groups by spherical varieties), in this series of
talks we will revisit examples of relative trace formula
comparisons that have appeared in the literature, and study the
local "transfer operators" that realize these comparisons. Some
structure will begin to emerge, that will be discussed further
in subsequent talks, later in the semester.</p>
<p>The examples that will be discussed include:</p>
<p>(1) comparison between the Kuznetsov formula and the stable
trace formula of $SL(2)$ (which first appeared in the thesis of
Rudnick);</p>
<p>(2) comparison between the Kuznetsov formula and the relative
trace formula for the variety $T\backslash PGL(2)$, where $T$ is
a torus;</p>
<p>(3) comparison between the Kuznetsov formula for $GL(2)$ and
the "trace formula" for a torus (Venkatesh's thesis).</p>
<p>Paradoxically (because functoriality was supposed to solve the
problem of analytic continuation of L-functions), "beyond
endoscopy" calls for the insertion of L-functions into trace
formulas, and some treatment of their meromorphic continuation.
This treatment is most successful when their functional equation
can be expressed as a Poisson summation formula for certain
"Hankel transforms" between spaces of orbital integrals. Thus,
alongside the aforementioned transfer operators, we will also
discuss two examples of Hankel transforms, namely:</p>
<p>(4) the Hankel transform for the standard L-function of $GL(n)$
on the Kuznetsov formula (contained in a paper of Jacquet);</p>
<p>(5) the Hankel transform for the symmetric square L-function of
$GL(2)$ on the Kuznetsov formula (extracted from the
Rankin–Selberg method).</p>
<p>Again, some structure will be visible, but we will also stumble
on a precise local version of Sarnak's objection to the
extension of these methods to higher symmetric powers.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=132632"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=132632</a><br>
<br>
6 On residues of Eisenstein series - through a cohomological lens
<br>
Joachim Schwermer <br>
<br>
<p>The cohomology of an arithmetic subgroup of a reductive
algebraic group defined over a number field is closely related
to the theory of automorphic forms. We discuss in which way
residues of Eisenstein series contribute non-trivially to the
subspace of square-integrable classes in these cohomology
groups.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=131082"
moz-do-not-send="true">http://www.math.ias.edu/seminars/abstract?event=131082</a><br>
<br>
IAS Math Seminars Home Page:<br>
<a href="http://www.math.ias.edu/seminars" moz-do-not-send="true">http://www.math.ias.edu/seminars</a>
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