Mathematics Colloquia
Talks are at 3:30 p.m. in South Hall, Room 6635, on
Thursdays
Colloquium Committee: Xianzhe Dai
Feature Event, Spring 2015
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Thursday
May 28, 2015
3:30-4:30pm, SH 6635
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Yitang Zhang, University of New
Hampshire
Title: Small gaps between
primes
Abstract: The twin prime conjecture states
that there are infitely many pairs of distinct
primes which differ by 2. Until recently
this conjecture had seemed to be out of reach
with current techniques. However, in 2013, the
author proved that there are infitely many
pairs of distinct primes which differ by no
more than B with B= 7*10^7. The value of
B has been considerably improved by Polymath8
(a cooperative team) and Maynard. In
this talk we shall describe the basic ideas
which lead to the proofs of the above results.
In particular, a breakthrough on the
distribution of primes in arithmetic
progressions will be introduced.
Click here
Contact person: Dave
Morrison
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Winter and Spring 2015
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Thursday
March 5, 2015
3:30-4:30pm, SH 6635
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James Zhang, University of
Washington, Seattle
Title: The Tits Alternative
Abstract: In 1972, J. Tits proved the
following dichotomy: every subgroup
of the linear automorphism group of a finite
dimensional vector
space is either virtually solvable or contains
a free subgroup
of rank two. Automorphism groups of
deformations of a polynomial
ring have been studied extensively by many
mathematicians, for
example, J. Alev, J. Dixmier, M. Kontsevich,
T. Lenagan, M. Yakimov
and others. In this talk, we will explain why
the discriminant
controls some global structures of a family of
automorphism groups.
By using the discriminant, a new version of
the Tits alternative can
be proved. This talk will be suitable for a
general audience.
Contact person: Ken
Goodearl
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Thursday
April 14, 2015
3:30-4:30pm, SH 6635
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Yuri Tschinkel, Simons Foundation
Title:
Contact person: Dave
Morrison
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Thursday
April 16, 2015
3:30-4:30pm, SH 6635
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Simon Rubinstein-Salzedo, Stanford
Title: Dessins d'enfants
and origamis
Abstract: We begin by looking at
branched covers of the projective line with
three branch points. Such covers became part
of mainstream mathematics
thanks to Grothendieck, who was interested in
such covers as a way of
understanding the absolute Galois group of the
rational numbers and hence
learning more about the sorts of number fields
that exist. In this case, we
can explicitly construct some interesting
number fields. We then talk about
the analogous problem for covers of elliptic
curves with one branch point.
In this case, the computations are far more
difficult, but in theory, they
allow us similar opportunities and more. We
shall see how one can go about
writing down explicit examples of covers of
elliptic curves in special
cases. We shall also see that both covers of
the projective line and covers
of elliptic curves have pictorial
representations that capture a wealth of
combinatorial and topological information, and
that are fun to study.
Contact person: Birge
Huisgen-Zimmermann
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Thursday
May 7, 2015
3:30-4:30pm, SH 6635
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Skip Garibaldi, IPAM
Title: Some people have all
the luck
Abstract: Winning a prize of $600 or more in
the lottery is a remarkable
thing – probably none of your friends or
family have ever won one. But
when we investigated records of all claimed
lottery prizes in Florida, we
discovered that some people had won hundreds
of them! Such people seem to
be not just lucky, but suspiciously
lucky. I will explain what we thought
they might have been up to, what mathematics
says about it, what further
investigations revealed, and the law
enforcement actions and state policy
changes that occurred as a consequence of our
theorems.
This talk is about joint work with Richard
Arratia, Lawrence Mower, and
Philip B. Stark.
Contact person: Bill
Jacob
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Thursday
May 21, 2015
3:30-4:30pm, Sh 6635
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Chelsea Walton, MIT
Title: Quantum symmetry
Abstract: Symmetry has long been a crucial
notion in mathematics and
physics. Groups arose to axiomatize the notion
of symmetry; namely, groups
are comprised of a set of invertible
transformations of an object of
interest. But it is common practice to replace
the object of study X with an
algebra A of functions on X. Symmetries of X
are then realized as the set of
group actions on A. (Here, the group is a set
of automorphisms of A.)
So let's kick this up a notch- let's study
symmetries of quantum objects.
Indeed such objects are impossible to
visualize, yet there are natural
noncommutative algebras B that arise as
'quantum function algebras' on these
objects. We can certainly still consider group
actions on B in this setting.
But the aim of this talk is to convince you
that studying actions of "Hopf
algebras" (or of "quantum groups") on B
is more appropriate.
Classification results (including some from
the analytic point-of-view) and
lots of examples will be included.
Contact person: Ken
Goodearl
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Thursday
May 28, 2015
3:30-4:30pm, Sh 6635
|
Yitang Zhang, University of New
Hampshire
Title: Small gaps between
primes
Abstract: The twin prime conjecture states
that there are infitely many pairs of distinct
primes which differ by 2. Until recently
this conjecture had seemed to be out of reach
with current techniques. However, in 2013, the
author proved that there are infitely many
pairs of distinct primes which differ by no
more than B with B= 7*10^7. The value of
B has been considerably improved by Polymath8
(a cooperative team) and Maynard. In
this talk we shall describe the basic ideas
which lead to the proofs of the above results.
In particular, a breakthrough on the
distribution of primes in arithmetic
progressions will be introduced.
Click here
Contact person: Dave
Morrison
|
Fall 2014
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Thursday
October 16, 2014
3:30-4:30pm, SH 6635
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Eitan Tadmor, University of
Maryland
Title: Images, PDEs and critical
regularity spaces:
Hierarchical construction of their
nonlinear solutions
Abstract: Edges are noticeable features in
images which can be extracted from noisy data
using different variational models. The
analysis of such variational models leads to
the question of representing general images as
the of divergence of uniformly bounded vector
fields.
We construct uniformly bounded solutions of
div(U)=f for general f’s in the critical
regularity space L^d(R^d). The study of this
equation and related problems was motivated by
recent results of Bourgain & Brezis. The
intriguing aspect here is that although the
problems are linear, the construction of their
solution is not. These constructions are
special cases of a rather general framework
for solving linear equations in critical
regularity spaces. The solutions are realized
in terms of nonlinear hierarchical
representations U= \sum_j u_j which we
introduced earlier in the context of image
processing. The u_j's are constructed
recursively as proper minimizers, yielding a
multi-scale decomposition of the solutions U.
Contact person: Xu
Yang
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Thursday
Nov 20, 2014
3:30-4:30pm, SH 6635
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Anna Wienhard, Caltech
Title:
Abstract:
Contact person: Daryl
Cooper
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Tuesday (Note the unusual date)
Dec 2, 2014
3:30-4:30pm, Room TBA
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Zhen-Su She, Peking
University
Title:
Abstract:
Contact person:
Bjorn Birnir
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Spring 2014
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Thursday
April 17, 2014
3:30-4:30pm, SH 6635
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Svitlana Mayboroda, University of
Minnesota
Title: Localization of
eigenfunctions and associated free
boundary problems
Abstract: The phenomenon of wave localization
permeates acoustics, quantum physics,
elasticity, energy engineering. It was used in
construction of the noise abatement walls,
LEDs, optical devices. Anderson localization
of quantum states of electrons has become one
of the prominent subjects in quantum physics,
as well as harmonic analysis and probability.
Yet, no methods predict specific spatial
location of the localized waves.
In this talk I will present recent results
revealing a universal mechanism of spatial
localization of the eigenfunctions of an
elliptic operator and emerging operator
theory/analysis/geometric measure theory
approaches and techniques. We prove that for
any operator on any domain there exists a
``landscape" which splits the domain into
disjoint subregions and indicates location,
shapes, and frequencies of the localized
eigenmodes. In particular, the landscape
connects localization to a certain multi-phase
free boundary problem, regularity of
minimizers, and geometry of free boundaries.
This is joint work with D. Arnold, G. David,
M. Filoche, and D. Jerison.
Contact person: Gustavo
Ponce
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Thursday
May 12-15, 2014
3:30-4:30pm, SH 6635
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Distinguished Lectures
Alice Chang and Paul Yang, Princeton
University
Monday, May 12, Dinstinguished
Lectures/Distinguished Women in Math
Alice Chang
Title: On a class of conformal
covariant operators and conformal
invariants
Abstract: In 2005, Graham-Zworski introduced
a continuous family of conformal covari- ant
operators of high orders via scattering theory
on conformal compact Einstein manifolds. This
class of operators P° and their associated
curvature Q° has played important roles in
problems in conformal geometry and in the
study of some geometric invariants in the
Ads/CFT setting. In the talk, I will survey
some of the recent progress in this field, and
also report some recent joint work with
Jeffrey Case about positivity property of this
class of operators under curvature
assumptions.
Wednesday, May 14, Dinstinguished Lectures
Paul Yang
Title: CR geometry in 3-D
Abstract: In this talk, I report on the
Embedding problem for 3-D CR structures. As a
consequence of the embedding criteria, we
obtain a positive mass theorem. Further
application is the analysis of the new
operator on the pluriharmonic functions
and the associated Q'curvature. The work were
joint with Case, Chanillo, Cheng, Chiu, and
Malchiodi.
Thursday, May 15, Dinstinguished Lectures
Paul Yang
Title: A fourth order operator in
conformal geometry
Abstract: In this talk, I report on the
Paneitz operator and the Q-curvature equation.
We obtain criteria for the sign of the Green's
function for this operator, and hence the
solvability of the Q-curvature equation in all
dimensions. This is a joint work with Fengbo
Hang.
Contact person: Xianzhe
Dai
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Thursday
June 5, 2014
3:30-4:30pm, SH 6635
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Sergey Fomel, University of
Texas-Austin
Title: Wave Equations and Wave
Extrapolations in Seismic Imaging
Abstract: Seismic imaging is a
multibillion-dollar enterprise aimed at
extracting information about the Earth
interior from reflected seismic waves. The
main task of reflection imaging is to take
seismic measurements from multiple experiments
with sources and receivers on the surface of
the Earth and to extrapolate seismic waves
numerically back in time and space to the
moment of their reflection. This problem leads
to novel partial-differential and
pseudo-differential equations describing
different parts of the imaging process. A
constructive way for numerical wave
extrapolation follows from low-rank
approximations of Fourier symbols.
Contact person: Xu
Yang
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Winter 2014
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Thursday
March 6, 2014
3:30-4:30pm, KITP
(note the unusual place)
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David Gross, KITP
Title: Yang-Mills Theory ( Video Link )
Abstract: (from the CMI website)The laws of
quantum physics stand to the world of
elementary particles in the way that Newton's
laws of classical mechanics stand to the
macroscopic world. Almost half a century ago,
Yang and Mills introduced a remarkable new
framework to describe elementary particles
using structures that also occur in geometry.
Quantum Yang-Mills theory is now the
foundation of most of elementary particle
theory, and its predictions have been tested
at many experimental laboratories, but its
mathematical foundation is still unclear. The
successful use of Yang-Mills theory to
describe the strong interactions of elementary
particles depends on a subtle quantum
mechanical property called the "mass gap": the
quantum particles have positive masses, even
though the classical waves travel at the speed
of light. This property has been discovered by
physicists from experiment and confirmed by
computer simulations, but it still has not
been understood from a theoretical point of
view. Progress in establishing the existence
of the Yang-Mills theory and a mass gap and
will require the introduction of fundamental
new ideas both in physics and in mathematics.
Contact person: Xianzhe
Dai
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Fall 2013
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Thursday
Oct. 17, 2013
3:30-4:30pm, SH 6635
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Greg Galloway, University of Miami
& MSRI
Title: On the topology of black
holes and beyond
Abstract:
In recent years there has been an explosion of
interest in black holes in higher dimensional
gravity. This, in particular, has led to questions
about the topology of black holes in higher
dimensions. In this talk we review Hawking's
classical theorem on the topology of black holes
in 3+1 dimensions (and its connection to black
hole uniqueness) and present a generalization of
it to higher dimensions. The latter is a
geometric result which places restrictions on the
topology of black holes in higher
dimensions. We shall also discuss recent
work on the topology of space exterior to a black
hole. This is closely connected to the
Principle of Topological Censorship, which roughly
asserts that the topology of the region outside of
all black holes (and white holes) should be
simple. All of the results to be
discussed rely on the recently developed theory of
marginally outer trapped surfaces, which are
natural spacetime analogues of minimal surfaces in
Riemannian geometry. This talk is based
primarily on joint work with Rick Schoen and with
Michael Eichmair and Dan Pollack.
Contact person: Guofang
Wei
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Thursday
Oct. 24, 2013
3:30-4:30pm, SH 6635
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Mike Freedman, Microsoft & UCSB
Title: P vs NP (
Video Link )
Abstract: The title refers to the iconic
problem of separating the class of (decision)
problems for which we can find solutions quickly
from those where it is merely possible to check
solutions quickly. For mathematicians this
problem asks "Is there really anything to your
subject?". For if P=NP it would not be much
harder to find proof than to read them. Maybe P
vs.NP is undecidable. I'll discuss why some
people say the problem is irrelevant. I'll
say a little about the little that is known
from: diagonalization, oracle reduction,
and "natural proofs" and mention a program that
exists to solve a related problem ( in algebraic
complexity). Then I will tell you my own
thoughts, if I have any.
Contact person: Xianzhe
Dai
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Thursday
Oct. 31, 2013
3:30-4:30pm, SH 6635
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Bisi Agboola, UCSB
Title: The Conjecture of
Birch and Swinnerton-Dyer
Abstract: The problem of finding integer
solutions to Diophantine equations is one
that has fascinated mathematicians for thousands
of years. Although we now
know (thanks to the work of Davis, Matiyasevich,
Putnam and Robinson
resolving Hilbert's 10th problem in the
negative) that it is impossible to
do this in general, it ought to be possible to
say a great deal in special
cases. For example, when the equation in
question defines an elliptic
curve, a remarkable conjecture due to Birch and
Swinnerton-Dyer implies
that the behaviour of the solutions is governed
by the properties of an
analytic object (whose very existence is a deep
problem in and of itself),
namely the L-function attached to the elliptic
curve. In this talk, I
shall explain some of the ideas that go into the
formulation of the Birch
and Swinnerton-Dyer conjecture, and I shall
discuss some aspects of what
is currently known about the conjecture.
Contact person: Xianzhe
Dai
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Thursday
Nov. 7, 2013
3:30-4:30pm, SH 6635
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Lenny Ng, Duke University
Title: Knot invariants
via the cotangent bundle
Abstract:In recent years, symplectic geometry
has emerged as a key tool in the study of
low-dimensional topology. One approach,
championed by Arnol'd, is to study the topology
of a smooth manifold through the symplectic
geometry of its cotangent bundle, building on
the familiar concept of phase space from
classical mechanics. I'll describe how one can
use this approach to construct an invariant of
knots called "knot contact homology". This
invariant is still pretty mysterious 10 years
on, but I'll outline some surprising relations
to representations of the knot group and to
mirror symmetry.
Contact person: Zhenghan
Wang
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Spring 2013
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April 18, 2013
3:30-4:30pm, SH 6635
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Jeff Stopple
Title: Music of the
Primes: from Pythagoras to Riemann
Abstract: The Riemann Hypothesis is considered
to be one of the most difficult unsolved
problems in mathematics. One interpretation, due
to physicist Sir Michael Berry, is that “the
primes have music in them.” This talk requires
only calculus to see what is meant by
this. We will hear the (conjectural)
music.
Contact person: Xianzhe
Dai
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Thursday
May 2, 2013
3:30-4:30pm, SH 6635
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Bjorn Birnir
Title: The Navier-‐Stokes
Millennium Problem:
Laminar versus Turbulent Flow
Abstract: One of the Millennium Problems is to
prove the existence and uniqueness of strong
solutions to the initial value problem for the
Navier-‐Stokes equation. This problem
is still open. In this talk we will discuss this
problem andits relation to the turbulence
problem. This is theproblem to find and prove
the statistical properties of turbulent flow.
We explain how these problems were identified,
more than 150 years ago, by O. Reynolds
and which problems one has to solve depends on
the dimensionless Reynolds number that
he defined. We will discuss the recent solution
of the turbulence problem and how the techniques
developed in its resolution may eventually lead
to the solution of the millennium
problem.
Contact person: Xianzhe
Dai
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Thursday
May 9, 2013
3:30-4:30pm, SH 6635
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Jesse Peterson,
Vanderbilt
Title: Characters and
invariant random subgroups for lattices in Lie
groups
Abstract: A character on a group $G$ is a
conjugation invariant function $\tau$ with
$\tau(e) = 1$ and such that for $g_1, \ldots,
g_n \in G$, the matrix $[\tau(g_j^{-1}g_i)]$ is
always non-negative definite. For finite groups,
the set of extreme points in the space of
characters are in one to one correspondence with
the set of irreducible representations, and have
been extensively studied. The study of
characters on infinite groups was initiated in
1964 by Thoma who classified all characters for
the group of finite permutations of $\mathbb N$.
In my talk I will discuss the classification of
characters on certain lattices in Lie groups,
generalizing results of Margulis, and present
several applications related to "random
subgroups", and rigidity for representations. In
contrast to the combinatorial nature of Thoma's
result, the techniques involved in studying
characters on lattices come from ergodic theory,
representation theory, and von Neumann algebras.
Contact person: Chuck
Akemann
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Thursday
May 23, 2013
3:30-4:30pm, SH 6635
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Dave Morrison
Title: Soap bubbles, the
Hodge conjecture, and all that:
finding good geometric representatives
for topological classes
Abstract: Soap bubbles minimize area, and many
of the beautiful aspects of minimal surface
theory can be illustrated by dipping wire frames
into soap solutions and observing the surface
which forms. I will introduce the Hodge
conjecture -- one of the Clay Foundation
Millenium Prize Problems -- from a somewhat
unconventional viewpoint, relating it to a
generalization of the minimal surface problem to
higher dimensional ambient spaces equipped with
interesting geometric structures.
Contact person: Xianzhe
Dai
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