[iasmath-seminars] Mathematics Seminars -- Week of October 9, 2017 - CSDM seminars in West Building Lecture Hall

[Anthony V. Pulido]

INSTITUTE FOR ADVANCED STUDY

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:	http://www.math.ias.edu/seminars/abstract?event=128775

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:	http://www.math.ias.edu/seminars/abstract?event=129293

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:	http://www.math.ias.edu/seminars/abstract?event=132506



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:	http://www.math.ias.edu/seminars/abstract?event=129007

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:	http://www.math.ias.edu/seminars/abstract?event=132632

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:	http://www.math.ias.edu/seminars/abstract?event=131082



Friday, October 13

Emerging Topics Working Group
Topic: 		TBA
Speaker: 	TBA
Time/Room: 	11:00am - 12:00pm/S-101

1 Barriers for rank methods in arithmetic complexity
    Rafael Oliveira

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.

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.

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

  * 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.)
  * 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.)

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.

Joint work with Klim Efremenko, Ankit Garg and Avi Wigderson

http://www.math.ias.edu/seminars/abstract?event=128775

2 Analysis and topology on locally symmetric spaces
    Akshay Venkatesh

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.

http://www.math.ias.edu/seminars/abstract?event=129293

3 Wrapped Floer theory and Homological mirror symmetry for toric 
Calabi-Yau manifolds
    Yoel Groman

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.

http://www.math.ias.edu/seminars/abstract?event=132506

4 Structural aspects of the null-cone problem in invariant theory
    Ankit Garg

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).

http://www.math.ias.edu/seminars/abstract?event=129007

5 Transfer operators for (relative) functoriality "beyond endoscopy" I
    Yiannis Sakellaridis

"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.

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.

The examples that will be discussed include:

(1) comparison between the Kuznetsov formula and the stable trace 
formula of $SL(2)$ (which first appeared in the thesis of Rudnick);

(2) comparison between the Kuznetsov formula and the relative trace 
formula for the variety $T\backslash PGL(2)$, where $T$ is a torus;

(3) comparison between the Kuznetsov formula for $GL(2)$ and the "trace 
formula" for a torus (Venkatesh's thesis).

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:

(4) the Hankel transform for the standard L-function of $GL(n)$ on the 
Kuznetsov formula (contained in a paper of Jacquet);

(5) the Hankel transform for the symmetric square L-function of $GL(2)$ 
on the Kuznetsov formula (extracted from the Rankin–Selberg method).

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.

http://www.math.ias.edu/seminars/abstract?event=132632

6 On residues of Eisenstein series - through a cohomological lens
    Joachim Schwermer

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.

http://www.math.ias.edu/seminars/abstract?event=131082

IAS Math Seminars Home Page:
http://www.math.ias.edu/seminars
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