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October 3, 2006 |
Nils Dencker
Lund University |
Solvability and the Nirenberg-Treves Conjecture |
In the 50's, the consensus was that all linear partial differential equations
were solvable. Therefore, it came as a surprise 1957 when Hans Lewy found a
non-solvable complex vector field. The vector field is a natural one, it is the
Cauchy-Riemann operator on the boundary of a strictly pseudo-convex domain. The
fact is that almost all linear PDE's are unsolvable, because of the Hörmander
bracket condition.
A rapid development in the 60's lead to the conjecture by Nirenberg and Treves
in 1969: condition (Ψ) is necessary and
sufficient for the solvability of partial differential equations of principal
type. This is a condition which involves only the sign changes of the imaginary
part of the highest order term of the operator along the bicharacteristics of
the real part.
The Nirenberg-Treves conjecture has recently been proved, see Annals of
Mathematics, 163:2, 2006. We shall present the background and the ideas of the
proof.
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October 17, 2006 |
Joseph Zaks
University of Haifa |
The Rational Cases of the Beckman-Quarles Theorem |
The Beckman-Quarles Theorem states that every mapping of Ed to
Ed which preserves distances one is an isometry, for all d
>= 2.
It was known that for d = 2, 3 and 4, there exist unit-preserving
mappings from the rational space Qd to Qd
which are not isometries.
A. Tyszka proved that every unit-preserving mapping from Q8 to
Q8 is an isometry. I extended it to many more values of d.
Recently, my Ph.D. student W. Hibi completed a proof that for all d >= 5,
every unit-preserving mapping from Qd to Qd
is an isometry.
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October 24, 2006 |
Gregory Lawler
University of Chicago
PIMS 10th Anniversary Distinguished Lecturer |
Conformal Invariance and Two-dimensional Statistical Physics |
A number of lattice models in two-dimensional statistical physics are
conjectured to exhibit conformal invariance in the scaling limit at criticality.
In this talk, I will try to explain what the previous sentence means, focusing
on three elementary examples: simple random walk, self-avoiding walk,
loop-erased random walk. I will describe the limit objects, Schramm-Loewner
Evolution (SLE), the Brownian loop soup, and the normalized partition functions, and show how conformal invariance can be
used to calculate quantities ("critical exponents") for the model. I will also
describe why (in some sense) there is only a one-parameter family of conformally
invariant limits. In conformal field theory, this family is parametrized by
central charge.
This talk is for a general mathematical audience. No knowledge of statistical
physics will be assumed.
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October 31, 2006 |
Lun-Yi Tsai
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Mathematics & Art |
In this talk the artist Lun-Yi Tsai will present his art work that is inspired
by mathematical ideas. He will talk about how he came to study math and how his
mathematical education has become one of the main wellsprings of his artistic
creativity. One of the main questions he would like to address is what areas of
research mathematics are amenable to visual presentation. Coming from the
viewpoint that mathematics is an art, he would like to engage mathematicians in
a collaborative effort to bring the abstract beauty of their own research to the
general public. Essentially, this is a "Call for Mathematicians (and their
theorems)."
He will also talk about the process by which he creates his mathematical
artworks, which have titles such as Hopf Fibration, Brouwer's Fixed Point, Chain
Complex, Quadric Parametrization, Purple Vector Bundle, Three Views of Tangent
Space.
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November 14, 2006 |
Hendrik Lenstra
Universiteit Leiden and UC Berkeley
VIGRE Distinguished Lecturer |
Searching for abc Triples |
Now that Fermat's Last Theorem has been proved, many number theorists view the
abc conjecture as their new Holy Grail. Its elementary formulation, to be
given in the lecture, contrasts sharply with our complete lack of insight into
the subject matter of the conjecture. Recently a project was started that aims
to improve our understanding by means of numerical experimentation. The lecture
is devoted to the mathematically interesting aspects of the project.
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November 21, 2006 |
Richard Kenyon
University of British Columbia |
Dimers and Crystal Surfaces |
This is joint work with Andrei Okounkov. We study a simple model of crystalline
surfaces. Microscopically, these are random discrete surfaces, arising in the
so-called dimer model, or domino tiling model. The law of large numbers implies
that at large scales the surfaces take on definite shapes, which are smooth
surfaces satisfying a certain PDE, similar in certain respects to the minimal
surface equation. We show how this equation can be solved via complex analytic
functions, and investigate the behavior of solutions, in particular the
formation of facets. This is the first model of facet formation which can be
analytically solved.
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January 11, 2007 |
Klaus Schmidt
University of Vienna
PIMS 10th Anniversary Distinguished Lecturer |
On Some of the Differences Between Z and
Z2 in Dynamics |
Around 1970 it was discovered that measure preserving actions of Z2 on probability spaces can have remarkably different
properties from Z-actions, i.e. from actions of single
transformations. Many of these differences can be described as 'rigidity
phenomena' and manifest themselves in the scarcity of certain objects
for Z2-actions which are abundant for Z-actions:
they may have unexpectedly few invariant probability measures, invariant
sets, isomorphisms or automorphisms.
This lecture will aim to explain--in some examples--both the reasons
and the ramifications of these rigidity phenomena.
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February 13, 2007 |
Olga Holtz
UC Berkeley |
Matrix Inequalities Galore |
The talk is about various inequalities involving matrix functions and many
contexts where they arise.
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March 8, 2007 |
Steve Doty
Loyola University, Chicago |
Diagram Algebras |
In 1937 Richard Brauer defined a finite dimensional algebra by means of a
combinatorial basis of certain graphs in order to describe the invariants of
classical groups acting on tensor powers of the vector representation. Starting
in the 1970s, various other examples of such "diagram" algebras have been
discovered by mathematical physicists and mathematicians. There are links to
knot theory (pun intended), physics, representation theory, and invariant
theory. There have been at least three conferences devoted to this topic since
2005. I'll try to give an overview of the main examples and some of the main
techniques people are using to study these algebras. There may be pictures.
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March 27, 2007 |
Guy Brousseau
Université Bordeaux 1
Ginger Warfield
University of Washington |
Didactics of Mathematics: From the Classroom to the Educational System |
Didactics (also known as Didactique) of Mathematics has been a lively research
field in France since the 1960's. This talk will begin with a brief summary of
the history and major concepts of the field, then give an illustration by
focussing on one particular topic: in the late seventies Brousseau, who founded
the field, applied some of its results as well as some of its analytical tools
to an analysis of the use of evaluation. Thirty years later predictions made
from that model appear to be confirmed. Recently he has been working on
extending the method to an analysis of other phenomena in the educational system
in general -- a study that applies equally well in France and the United States.
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March 30, 2007 |
Peter Lax
Courant Institute
PIMS 10th Anniversary Distinguished Lecturer |
A Phragmen-Lindelof and Saint Venant Principle in Harmonic Analysis |
Let S be a linear space of vector valued functions u(y) on
the half-line whose values belong to some Banach space. We suppose that S
is translation invariant; that is, if u(y) belongs to S, so
does u(y + t) for all t > 0. S is called
"interior compact" if the unit ball of S in the L1 norm
over a y-interval [a,b] is precompact in the L1
norm over any proper subinterval [a',b'].
THEOREM: Any function u(y) in a translation invariant, interior
compact space that is L1 on y > 0 decays exponentially
as y tends to infinity, and has an asymptotic expansion near infinity in
terms of exponential functions in y contained in S.
This result can be applied to solutions of elliptic equationsin a half cylinder.
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April 10, 2007 |
Maciej Zworski
University of California, Berkeley |
Quantum Chaos in Scattering |
Quantum resonances describe metastable states which in addition to energy/rate
of oscillations have decay rates. I will describe this concept using "real time"
MATLAB computations stressing clear manifestations of the quantum/classical
correspondence in the distribution of resonances. This correspondence becomes
more interesting in higher dimensions since the dynamics can then be chaotic. I
will explain how dynamical objects such as dimension of trapped sets and
topological pressure are related to the density of resonances and to bounds on
quantum decay rates. I will concentrate on (colourful) pictures and intuitions
rather than on the technical aspects of this (rather technical) subject.
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April 17, 2007 |
Carlos Kenig
University of Chicago
PIMS 10th Anniversary Distinguished Lecturer |
Recent Developments on the Well-posedness of Dispersive Equations |
We will review the progress in the last 20 years on the well-posedness theory of
dispersive equations, concentrating in the case of the Korteweg-de Vries
equation to illustrate it. We will then describe some of the current challenges
in the area, and biefly discuss recent joint works (mainly with Alex Ionescu) in
which progress has been made for the Benjamin-Ono equation, the Schrödinger map
system and the Kadomstev-Petviashvili I equation.
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April 19, 2007 |
Peter Winkler
Dartmouth College
PIMS 10th Anniversary Distinguished Lecturer |
Scheduling, Percolation, and the Worm Order |
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When can you schedule a multi-step process without having to take backward
steps? Critical are an old concept called "submodularity", a new structure
called the "worm order", and a variation of what physicists call "percolation".
With these tools we will attempt to update the computer system at UW, find a
lost child in the Cascades, and minimize water usage in Seattle, all without
backward steps.
Joint work in part with Graham Brightwell (LSE) and in part with Lizz Moseman
(Dartmouth). (This talk is designed to be accessible to undergraduates
interested in mathematics.)
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April 24, 2007 |
Richard Schoen
Stanford
PIMS 10th Anniversary Distinguished Lecturer |
The Sharp Isoperimetric Inequality on Minimal Submanifolds |
We will describe the history of the conjecture that the isoperimetric
inequality with the Rn constant should hold for a domain on a
minimal submanifold of dimension n in Euclidean space. We will then
describe a new method which proves the inequality in many cases for n=2,
and which we believe has the potential to handle the general two dimensional
case.
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May 9, 2007 |
Frances Kirwan
University of Oxford
PIMS 10th Anniversary Distinguished Lecturer |
Classification Problems in Algebraic Geometry |
This talk will attempt to give a survey (for non-experts) of a few of the
many fascinating problems and results concerning moduli spaces in algebraic
geometry, and in particular some moduli spaces associated to the simplest
objects of algebraic geometry: algebraic curves. These include moduli spaces of
bundles over curves and moduli spaces of 'stable maps' from curves to a fixed
target projective variety.
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May 11, 2007 |
Frances Kirwan
University of Oxford
PIMS 10th Anniversary Distinguished Lecturer |
Non-reductive Group Actions and Symplectic Implosion |
Given a linear action of a complex reductive group on a projective variety
X, geometric invariant theory provides us with an open subset of X
(the set of stable points) which has a well-behaved quotient, and a
compactification of this quotient which can be identified with a symplectic
reduction in the sense of Marsden and Weinstein. This talk will discuss an
analogue of this situation for groups which are not reductive, and its
relationship with symplectic implosion in the sense of Guillemin, Jeffrey and
Sjamaar.
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May 15, 2007 |
Susan Tolman
University of Illinois at Urbana-Champaign |
A Very Generalized Schubert Calculus |
Schubert calculus is the calculus of enumerative geometry. One of the main goals
is to find nice combinatorial descriptions of the structure constants of the
cohomology ring of the Grassmannian in terms of its natural geometric Schubert
basis. Our goal is to consider the same questions in the much broader context of
symplectic manifolds with Hamiltonian torus actions. We show that if there
exists an invariant Palais-Smale metric, then the structure constants can be
computed as a weighted sum over paths in a certain graph. (Joint work with R.
Goldin.)
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May 22, 2007 |
Shrawan Kumar
University of North Carolina
PIMS 10th Anniversary Distinguished Lecturer |
Eigenvalue Problem and a New Product in Cohomology of Flag Varieties |
This is a report on my joint work with P. Belkale. We define a new
commutative and associative product in the cohomology of any flag variety G/P
(which still satisfies the Poincaré duality) and use this product to
generate certain inequalities which solves the analog of the classical Hermitian
eigenvalue problem for any complex semisimple group G. Our recipe
provides considerable improvement, in general, over the set of inequalities
defined by Berenstein-Sjamaar. In fact, our set of inequalities form an
irredundant system of inequalities. The talk should be accessible to general
mathematical audience.
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July 12, 2007 |
Kari Astala
University of Helsinki
PIMS 10th Anniversary Distinguished Lecturer |
Mappings of Finite Distortion: Analysis in the Extreme |
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Degenerate structures arise naturally in many questions in PDE's and calculus
of variations as well as in their applications. Mappings of finite distortion
study how far the powerful tools of geometric analysis can here reach; the
theory of these maps has been developed in the last 7-8 years, and it is
surprising how precise information we can now achieve. We give an illustration
on these methods through two examples, one in nonlinear elasticity and the other
in inverse problems, in electric impedance tomography.
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