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<pre>INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540
Mathematics Seminars
Week of October 2, 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 2
Computer Science/Discrete Mathematics Seminar I
Topic: Crossing the logarithmic barrier for dynamic boolean data structure lower bounds
Speaker: Omri Weinstein, Columbia University
Time/Room: 11:00am - 12:15pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=128772">http://www.math.ias.edu/seminars/abstract?event=128772</a>
Seminar on Theoretical Machine Learning
Topic: Hyperparameter optimization: a spectral approach
Speaker: Elad Hazan, Princeton University
Time/Room: 12:30pm - 1:45pm/White-Levy Room
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=133015">http://www.math.ias.edu/seminars/abstract?event=133015</a>
Members' Seminar
no seminar: postdoctoral talks
Speaker: no seminar: postdoctoral talks
Time/Room: -
Short talks by postdoctoral members
Topic: Weights of mod $p$ automorphic forms
Speaker: Bao V. Le Hung, Member, School of Mathematics
Time/Room: 2:00pm - 2:15pm/S-101
Topic: Multivariate trace inequalities
Speaker: Marius Lemm, Member, School of Mathematics
Time/Room: 2:15pm - 2:30pm/S-101
Topic: $A_\infty$ structures as a language for open Gromov-Witten theory
Speaker: Sara Tukachinsky, Member, School of Mathematics
Time/Room: 2:30pm - 2:45pm/S-101
Topic: $p$-adic L-functions and Iwasawa main conjectures
Speaker: Zheng Liu, Member, School of Mathematics
Time/Room: 4:00pm - 4:15pm/S-101
Topic: Zeroes of Laplace eigenfunctions and propagation of smallness
Speaker: Aleksandr Logunov, Member, School of Mathematics
Time/Room: 4:15pm - 4:30pm/S-101
Topic: The distribution of primes and zeros of Riemann's Zeta function
Speaker: James Maynard, Member, School of Mathematics
Time/Room: 4:30pm - 4:45pm/S-101
Tuesday, October 3
Computer Science/Discrete Mathematics Seminar II
Topic: Elementary open problems in Algebra (with consequences in computational complexity)
Speaker: Avi Wigderson, Herbert H. Maass Professor, School of Mathematics
Time/Room: 10:30am - 12:30pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=129001">http://www.math.ias.edu/seminars/abstract?event=129001</a>
Locally Symmetric Spaces Seminar
Topic: Motivic correlators and locally symmetric spaces
Speaker: Alexander Goncharov, Yale University; Member, School of Mathematics and Natural Sciences
Time/Room: 1:30pm - 4:00pm/S-101
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=132702">http://www.math.ias.edu/seminars/abstract?event=132702</a>
Wednesday, October 4
Short talks by postdoctoral members
Topic: On weak epsilon nets and the Radon number
Speaker: Shay Moran, Member, School of Mathematics
Time/Room: 2:00pm - 2:15pm/S-101
Topic: From representations of the symmetric group to branched covers of the disk
Speaker: Amitai Netser Zernik, Member, School of Mathematics
Time/Room: 2:15pm - 2:30pm/S-101
Topic: To Be Announced
Speaker: Behnam Neyshabur, Member, School of Mathematics
Time/Room: 2:30pm - 2:45pm/S-101
Topic: Twisted integral orbit parametrizations
Speaker: Aaron Pollack, Member, School of Mathematics
Time/Room: 4:00pm - 4:15pm/S-101
Topic: Algebraic groups in positive characteristic
Speaker: Srimathy Srinivasan, Member, School of Mathematics
Time/Room: 4:15pm - 4:30pm/S-101
Topic: Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds
Speaker: Michael Lipnowski, Member, School of Mathematics
Time/Room: 4:30pm - 4:45pm/S-101
Thursday, October 5
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: Unlikely intersections for algebraic curves in positive characteristic
Speaker: David Masser, University of Basel
Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University
Abstract Link: <a href="http://www.math.ias.edu/seminars/abstract?event=131079">http://www.math.ias.edu/seminars/abstract?event=131079</a>
Friday, October 6
Short talks by postdoctoral members
Topic: The local multiplicity problem for the Ginzburg-Rallis model and the generalized Shalika model
Speaker: Chen Wan, Member, School of Mathematics
Time/Room: 11:00am - 11:15am/S-101
Topic: Formulas related to Zastava spaces
Speaker: Jonathan Wang, Member, School of Mathematics
Time/Room: 11:15am - 11:30am/S-101
Topic: Finer virtual structure for moduli spaces of holomorphic curves
Speaker: Dingyu Yang, Member, School of Mathematics
Time/Room: 11:30am - 11:45am/S-101
Topic: Geometry of Shimura varieties
Speaker: Rong Zhou, Member, School of Mathematics
Time/Room: 2:00pm - 2:15pm/S-101
Topic: The relative trace formula approach to the global Gan-Gross-Prasad conjecture for unitary groups
Speaker: Michal Zydor, Member, School of Mathematics
Time/Room: 2:15pm - 2:30pm/S-101
Topic: The geometry and topology of minimal surfaces in $\mathbb R^3$ of finite total curvature
Speaker: Otis Chodosh, Member, School of Mathematics
Time/Room: 2:30pm - 2:45pm/S-101
Mathematical Conversations
Topic: To Be Announced
Speaker: Akshay Venkatesh, Stanford University; Distinguished Visiting Professor, School of Mathematics
Time/Room: 6:00pm - 7:00pm/Dilworth Room
Saturday, October 7
Workshop on Topology: Identifying Order in Complex Systems
Topic: Discrete conformal geometry of polyhedral surfaces and its applications
Speaker: Feng Luo, Rutgers University
Time/Room: 9:00am - 10:00am/David Rittenhouse Laboratory A4, University of Pennsylvania
Workshop on Topology: Identifying Order in Complex Systems
Topic: On Morse index estimates for minimal surfaces
Speaker: Davi Maximo, University of Pennsylvania
Time/Room: 10:30am - 11:30am/David Rittenhouse Laboratory A4, University of Pennsylvania
Workshop on Topology: Identifying Order in Complex Systems
Topic: Fast predictive models for image registration
Speaker: Marc Niethammer, University of North Carolina, Chapel Hill
Time/Room: 1:30pm - 2:30pm/David Rittenhouse Laboratory A4, University of Pennsylvania
Workshop on Topology: Identifying Order in Complex Systems
Topic: Personalized cancer therapy using molecular landscape topology and thermodynamics
Speaker: Edward Rietman, University of Massachusetts, Amherst
Time/Room: 3:00pm - 4:00pm/David Rittenhouse Laboratory A4, University of Pennsylvania
Workshop on Topology: Identifying Order in Complex Systems
Topic: Consistent manifold representation for topological data analysis
Speaker: Tim Sauer, George Mason University
Time/Room: 4:30pm - 5:30pm/David Rittenhouse Laboratory A4, University of Pennsylvania
</pre>
1 Crossing the logarithmic barrier for dynamic boolean data
structure lower bounds
<br>
Omri Weinstein
<br>
<br>
<p>This paper proves the first super-logarithmic lower bounds on the
cell-probe complexity of dynamic boolean (a.k.a. decision) data
structure problems, a long-standing milestone in data structure
lower bounds.</p>
<p>We introduce a new method for proving dynamic cell probe lower
bounds and use it to prove a $\tilde{\Omega}(\log^{1.5} n)$ lower
bound on the operational time of a wide range of boolean data
structure problems, most notably, on the query time of dynamic
range counting over $\mathbb{F}_2$ ([Pat07]). Proving an
$\omega(\lg n)$ lower bound for this problem was explicitly posed
as one of five important open problems in the late Mihai
Patrascu's obituary. This result also implies the first
$\omega(\lg n)$ lower bound for the classical 2D range counting
problem, one of the most fundamental data structure problems in
computational geometry and spatial databases. We derive similar
lower bounds for boolean versions of dynamic polynomial evaluation
and 2D rectangle stabbing, and for the (non-boolean) problems of
range selection and range median.</p>
<p>Our technical centerpiece is a new way of ``weakly" simulating
dynamic data structures using efficient one-way communication
protocols with small advantage over random guessing. This
simulation involves a surprising excursion to low-degree
(Chebyshev) polynomials which may be of independent interest, and
offers an entirely new algorithmic angle on the ``cell sampling"
method.</p>
<p>Joint work with Kasper Green Larsen and Huacheng Yu.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=128772">http://www.math.ias.edu/seminars/abstract?event=128772</a><br>
<br>
2 Hyperparameter optimization: a spectral approach
<br>
Elad Hazan
<br>
<br>
<p>Modern machine learning algorithms often involve complicated
models with tunable parameters, called hyperparameters, that are
set during training. Hyperparameter tuning/optimization is
considered an art. (See e.g. <a>http://www.argmin.net/2016/06/20/hypertuning/</a>
)</p>
<p>We give a simple, fast algorithm for hyperparameter optimization
inspired by techniques from the analysis of Boolean functions. We
focus on the high-dimensional regime where the canonical example
is training a neural network with a large number of
hyperparameters. The algorithm - an iterative application of
compressed sensing techniques for orthogonal polynomials -
requires only uniform sampling of the hyperparameters and is thus
easily parallelizable. Experiments for training deep nets on
Cifar-10 show that compared to state-of-the-art tools (e.g.,
Hyperband and Spearmint), our algorithm finds significantly
improved solutions. Additionally, our method comes with provable
guarantees and yields a quasi-polynomial time algorithm for
learning decision trees under the uniform distribution with the
best known sample complexity.</p>
<p>The talk covers this paper paper: <a>https://arxiv.org/abs/1706.00764</a>
as well as work in progress.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=133015">http://www.math.ias.edu/seminars/abstract?event=133015</a><br>
<br>
3 Elementary open problems in Algebra (with consequences in
computational complexity)
<br>
Avi Wigderson
<br>
<br>
<p>I will survey some elementary (to state!) problems on groups,
matrices, and tensors, and discuss their motivations arising from
several major problems in computational complexity theory. On each
problem there was some exciting recent progress which may raise
hope it can be resolved. No special background will be assumed.
</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=129001">http://www.math.ias.edu/seminars/abstract?event=129001</a><br>
<br>
4 Motivic correlators and locally symmetric spaces
<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=132702">http://www.math.ias.edu/seminars/abstract?event=132702</a><br>
<br>
5 Unlikely intersections for algebraic curves in positive
characteristic
<br>
David Masser
<br>
<br>
<p>In the last two decades there has been much study of what happens
when an algebraic curve in \(n\)-space is intersected with two
multiplicative relations<br>
\[x_1^{a_1} \cdots x_n^{a_n} = x_1^{b_1} \cdots x_n^{b_n} = 1
\tag{\(\times\)}\]<br>
for \((a_1, \ldots ,a_n),(b_1,\ldots, b_n)\) linearly independent
in \({\bf Z}^n\). Usually the intersection with the union of all
\((\times)\) is at most finite, at least in zero characteristic.
This often becomes false in positive characteristic, and I gave in
2014 a substitute conjecture and proved it for \(n = 3\). I will
describe all this together with more recent work with Dale
Brownawell where we do the same for additive relations \((+)\);
now an extra Frobenius structure has to be added, and there are no
longer any direct analogues in zero characteristic.</p>
<a href="http://www.math.ias.edu/seminars/abstract?event=131079">http://www.math.ias.edu/seminars/abstract?event=131079</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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