[math-ias] Mathematics Seminars -- Week of December 12, 2016
Anthony Pulido
apulido at ias.edu
Fri Dec 9 13:30:25 EST 2016
INSTITUTE FOR ADVANCED STUDY
School of Mathematics
Princeton, NJ 08540
Mathematics Seminars
Week of December 12, 2016
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To view mathematics in titles and abstracts, please click on the talk's link.
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Monday, December 12
Computer Science/Discrete Mathematics Seminar I
Topic: On gradient complexity of measures on the discrete cube
Speaker: Ronen Eldan, Weizmann Institute of Science
Time/Room: 11:15am - 12:15pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=104244
Members' Seminar
Topic: 2-associahedra and functoriality for the Fukaya category
Speaker: Nathaniel Bottman, Member, School of Mathematics
Time/Room: 1:15pm - 2:15pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=113975
Tuesday, December 13
Computer Science/Discrete Mathematics Seminar II
Topic: Sum of squares lower bounds for refuting any CSP
Speaker: Pravesh Kothari, Member, School of Mathematics
Time/Room: 10:30am - 12:30pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=104484
Homological Mirror Symmetry Reading Group
Topic: Gamma-integral structures reading group
Speaker: To Be Announced
Time/Room: 10:30am - 12:00pm/Dilworth Room
Princeton/IAS Symplectic Geometry Seminar
Topic: Positive loops of loose Legendrians and applications
Speaker: Guogang Liu, Université de Nantes
Time/Room: 3:00pm - 4:00pm/Fine 224, Princeton University
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=113215
Princeton/IAS Symplectic Geometry Seminar
Topic: Log geometric techniques for open invariants in mirror symmetry
Joint with Princeton University Algebraic Geometry Seminar
Speaker: Hülya Argüz, Member, School of Mathematics
Time/Room: 4:30pm - 5:30pm/Fine 322, Princeton University
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=121464
Auroux Watching Seminar
Topic: To Be Announced
Speaker: To Be Announced
Time/Room: 5:00pm - 7:00pm/S-101
Wednesday, December 14
Homological Mirror Symmetry (minicourse)
Topic: Numerical invariants from bounding chains
Speaker: Jake Solomon, Hebrew University of Jerusalem; Visitor, School of Mathematics
Time/Room: 10:45am - 12:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=113024
Reading Group on Homological Mirror Symmetry and K3 Surfaces
Topic: To Be Announced
Speaker: To Be Announced
Time/Room: 1:00pm - 2:30pm/Dilworth Room
Mathematical Conversations
Topic: How to measure a Lagrangian cobordism
Speaker: Joshua Sabloff, Haverford College; Member, School of Mathematics
Time/Room: 6:00pm - 7:00pm/Dilworth Room
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=103474
Thursday, December 15
Working Group on Algebraic Number Theory
Speaker: To Be Announced
Time/Room: 2:00pm - 4:00pm/Fine 401, Princeton University
Joint IAS/Princeton University Number Theory Seminar
Topic: On the spectrum of Faltings' height
Speaker: Juan Rivera-Letelier, University of Rochester
Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=114975
Friday, December 16
Term I ends - SOM
Time/Room: 8:00am - 9:00am
Homological Mirror Symmetry (minicourse)
Topic: Numerical invariants from bounding chains
Speaker: Jake Solomon, Hebrew University of Jerusalem; Visitor, School of Mathematics
Time/Room: 10:45am - 12:00pm/S-101
Abstract Link: http://www.math.ias.edu/seminars/abstract?event=113034
1 On gradient complexity of measures on the discrete cube
Ronen Eldan
The motivating question for this talk is: What does a sparse Erdős–Rényi
random graph, conditioned to have twice the number of triangles than the
expected number, typically look like? Motivated by this question, In
2014, Chatterjee and Dembo introduced a framework for obtaining Large
Deviation Principles (LDP) for nonlinear functions of Bernoulli random
variables (this followed an earlier work of Chatterjee-Varadhan which
used limit graph theory to answer this question in the dense regime).
The aforementioned framework relies on a notion of "low complexity"
functions on the discrete cube, defined in terms of the covering numbers
of their gradient. The central lemma used in their proof provides a
method of estimating the log-normalizing constant $\log \sum_{x \in
\{-1,1\}^n} e^{f(x)}$, which applies for functions attaining low
complexity and some additional smoothness properties. In this talk, we
will introduce a new notion of complexity for measures on the discrete
cube, namely the mean-width of the gradient of the log-density. We prove
a general structure theorem for such measures which goes beyond the
discrete cube. In particular, we show that a measure $\nu$ attaining low
complexity (with no extra smoothness assumptions needed) are close to a
product measure in the following sense: there exists a measure $\tilde
\nu$ a small "tilt" of $\nu$ in the sense that their log-densities
differ by a linear function with small slope, such that $\tilde \nu$ is
close to a product measure in transportation distance. An easy corollary
of our result is a strengthening of the framework of Chatterjee-Dembo,
which in particular simplifies the derivation of LDPs for subgraph
counts, and improves the attained bounds.
http://www.math.ias.edu/seminars/abstract?event=104244
2 2-associahedra and functoriality for the Fukaya category
Nathaniel Bottman
The Fukaya category of a symplectic manifold is a robust intersection
theory of its Lagrangian submanifolds. Over the past decade, ideas
emerging from Wehrheim-Woodward's theory of quilts have suggested a way
to produce maps between the Fukaya categories of different symplectic
manifolds. I have proposed that these maps should be controlled by
compactified moduli spaces of marked parallel lines in the plane, called
"2-associahedra". In this talk I will describe the 2-associahedra, with
a focus on their topological and combinatorial aspects; in particular, I
will produce combinatorial data that are in bijection with the strata of
the 2-associahedra, and describe a generating function technique for
computing the number of dimension-$m$ strata in a particular
2-associahedron.
http://www.math.ias.edu/seminars/abstract?event=113975
3 Sum of squares lower bounds for refuting any CSP
Pravesh Kothari
Let $P:\{0,1\}^k \to \{0,1\}$ be a $k$-ary predicate. A random instance
of a constraint satisfaction problem (CSP(P)) where each of the $\Delta
n$ constraints is $P$ applied to $k$ literals on $n$ variables chosen at
random is unsatisfiable with high probability whenever the /density /of
constraints, $\Delta \gg 1.$ The /refutation /problem asks for an
efficient proof of unsatisfiability of such an instance that works
correctly with high probability. We show that whenever the predicate $P$
fails to support a $t$-/wise-uniform/ probability distribution over its
satisfying assignments, the Sum-of-Squares (SoS) algorithm of degree $d
= \Theta(\frac{n}{\Delta^{2/(t-2)} \log \Delta})$ (that runs in time
$n^{O(d)}$) /cannot /refute a random instance of CSP(P). In particular,
polynomial time SoS algorithm requires $\sim n^{t/2}$ constraints to
refute CSPs with predicates that support $t$-wise-uniform distribution
on their satisying assignments. This matches the bounds known for
special cases such as 3XOR and 3SAT [Grigoriev 2001, Schonebeck 08].
Together with the recent work [Lee, Raghavendra, Steurer 2015], it also
yields that /any /polynomial-size semidefinite programming relaxation
for refutation requires at least $\sim n^{t/2}$ constraints. More
generally, for every constraint predicate~$P$, we get a three-way
hardness tradeoff between the density of constraints, the SOS degree
(hence running time), and the strength of the refutation. By recent
known algorithmic results of [Allen, O'Donnell, Witmer 2015] and
[Raghavendra, Rao, Schramm 2016], our full three-way tradeoff is
/tight/, up to lower-order factors. Our results carry over to the more
general $\delta$-refutation problem: we show that if $P$ is
$\delta$-close to supporting a $t$-wise uniform distribution on
satisfying assignments, then the
degree-$\Theta(\frac{n}{\Delta^{2/(t-1)} \log \Delta})$ SoS algorithm
cannot $(\delta+o(1))$-refute a random instance of CSP$(P)$. Our results
also extend with no change to CSPs over larger alphabets and subsume all
previously known lower bounds for semialgebraic refutations of random
CSPs. They are also the first to show a distinction between the degree
SOS needs to weakly refute random CSPs, versus $\delta$-refute them.
http://www.math.ias.edu/seminars/abstract?event=104484
4 Positive loops of loose Legendrians and applications
Guogang Liu
In this talk, I will give a simple and geometrical proof of the
following theorem from my thesis: Every loose Legendrian is in a
positive loop amongst Legendrian embeddings. The idea is that we add
wrinkles to a loose Legendrian and rotate the wrinkles positively then
resolve the wrinkles without changing the isotopy class of the initial
Legendrian. Finally, I will give some application, especially we can
define a strong partial order on the universal cover of groups of
contactomorphisms.
http://www.math.ias.edu/seminars/abstract?event=113215
5 Log geometric techniques for open invariants in mirror symmetry
Hülya Argüz
We would like to discuss an algebraic-geometric approach to some open
invariants arising naturally on the A-model side of mirror symmetry.
http://www.math.ias.edu/seminars/abstract?event=121464
6 Numerical invariants from bounding chains
Jake Solomon
I'll begin with a leisurely introduction to Fukaya A-infinity algebras
and their bounding chains. Then I'll explain how to use bounding chains
to define open Gromov-Witten invariants. The bounding chain invariants
can be computed using an open analog of the WDVV equations. This leads
to an explicit understanding of the homotopy type of certain Fukaya
A-infinity algebras. Also, the bounding chain invariants generalize
Welschinger's real enumerative invariants. A nice example is real
projective space considered as a Lagrangian submanifold of complex
projective space. This is joint work with Sara Tukachinsky.
http://www.math.ias.edu/seminars/abstract?event=113024
7 How to measure a Lagrangian cobordism
Joshua Sabloff
After introducing Lagrangian cobordisms from the perspective of a
(Legendrian) knot theorist, we will explore notions of length and width.
We will end by musing about what these measurements might say about the
asymmetry of Lagrangian cobordisms.
http://www.math.ias.edu/seminars/abstract?event=103474
8 On the spectrum of Faltings' height
Juan Rivera-Letelier
The arithmetic complexity of an elliptic curve defined over a number
field is naturally quantified by the (stable) Faltings height. Faltings'
spectrum is the set of all possible Faltings' heights. The corresponding
spectrum for the Weil height on a projective space and the Neron-Tate
height of an Abelian variety is dense on a semi-infinite interval. We
show that, in contrast, Faltings' height has 2 isolated minima. We also
determine the essential minimum of Faltings' height up to 5 decimal
places. This is a joint work with Jose Burgos-Gil and Ricardo Menares.
http://www.math.ias.edu/seminars/abstract?event=114975
9 Numerical invariants from bounding chains
Jake Solomon
I'll begin with a leisurely introduction to Fukaya A-infinity algebras
and their bounding chains. Then I'll explain how to use bounding chains
to define open Gromov-Witten invariants. The bounding chain invariants
can be computed using an open analog of the WDVV equations. This leads
to an explicit understanding of the homotopy type of certain Fukaya
A-infinity algebras. Also, the bounding chain invariants generalize
Welschinger's real enumerative invariants. A nice example is real
projective space considered as a Lagrangian submanifold of complex
projective space. This is joint work with Sara Tukachinsky.
http://www.math.ias.edu/seminars/abstract?event=113034
IAS Math Seminars Home Page:
http://www.math.ias.edu/seminars
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