May 28, 2010 at 2:30pm
Electrical Engineering Building, Room 045 |
Jim Propp
University of Massachusetts Lowell |
Quasirandom Processes |
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Probability theory is concerned with
regularities in random processes, such as laws of large numbers and limit-shape
theorems. Recent work by researchers at the interface between probability and
combinatorics shows that many of these regularities apply, sometimes in
dramatically heightened form, to quasirandom systems: simple
deterministic systems whose microscopic behavior is designed to mimic the
average case behavior of random systems. Quasirandom processes often possess a
richness of structure not evident in the random processes that inspired them.
This talk will address the questions: Where do pictures like
http://rotor-router.mpi-inf.mpg.de/1Bio/?rotorseq=2 come from? And, what are
they telling us? View video
of this lecture.
View
slides from this lecture.
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May 7, 2010 at 2:30pm
Electrical Engineering Building, Room 045 |
Zhi-Quan (Tom) Luo
University of Minnesota |
The Optimization Work of Paul Tseng |
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View
video
of this lecture.
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April 30, 2010 at 2:30pm
Electrical Engineering Building, Room 045 |
Ravi Vakil
Stanford University |
Generalizing the Cross Ratio: The Moduli Space of n Points on the Projective Line Up to Projective Equivalence |
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Four ordered points on the projective line, up to projective equivalence, are
classified by the cross ratio, a notion introduced by Cayley. This theory can be
extended to more points, leading to one of the first important examples of an
invariant theory problem, studied by Kempe, Hilbert, and others. Instead of the
cross ratio (a point on the projective line), we get a point in a larger
projective space, and the equations necessarily satisfied by such points exhibit
classical combinatorial and geometric structure. For example, the case of six
points is intimately connected to the outer automorphism of S6.
We extend this picture to an arbitrary number of points, completely describing
the equations of the moduli space. This is joint work with Ben Howard, John
Millson, and Andrew Snowden. This talk is intended for a general mathematical
audience, and much of the talk will be spent discussing the problem, and an
elementary graphical means of understanding it. View
video
of this lecture.
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April 16, 2010 at 2:30pm
Electrical Engineering Building, Room 045 |
Neil Trudinger
Australian National University |
Weak Solution Concepts for Nonlinear Elliptic PDE and Associated Regularity |
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We introduce and explore some concepts of weak solution for particular
nonlinear partial differential equations.These notions have their roots in the fundamental
work of A. D. Aleksandrov on the Monge-Ampere measure. We also consider applications to
affine and complex geometry and to optimal transportation. View
video
of this lecture.
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February 5, 2010 at 2:30pm
Mary Gates Hall, Room 231 |
Carl de Boor
University of Wisconsin & UW (affiliate) |
Issues in Multivariate Polynomial Interpolation |
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While univariate polynomial interpolation has been a basic tool of scientific
computing for hundreds of years, multivariate polynomial interpolation is much
less understood. Already the question from which polynomial space to choose an
interpolant to given data has no obvious answer.
The talk presents, in some detail, one answer to this basic question, namely the
"least interpolant" of Amos Ron and the speaker which, among other nice
properties, is degree-reducing, then seeks some remedy for the resulting
discontinuity of the interpolant as a function of the interpolation sites, then
addresses the problem of a suitable representation of the interpolation error
and the nature of possible limits of interpolants as some of the interpolation
sites coalesce.
The last part of the talk is devoted to a more traditional setting, the
complementary problem of finding correct interpolation sites for a given
polynomial space, chiefly the space of polynomials of degree <= k for some k, and ends with a particular recipe for good
interpolation sites in the square, the Padua points.
References:
http://pages.cs.wisc.edu/~deboor/multiint. View
video
of this lecture.
View
slides from this lecture.
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January 29, 2010 at 2:30pm
Mary Gates Hall, Room 231 |
Lionel Levine
MIT & Microsoft Research |
Growth Rates and Explosions in Sandpiles |
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How do simple local interactions combine to produce complex large-scale
structure and patterns? The abelian sandpile model provides a beautiful test
case. I'll discuss a pair of conjectures about the scale invariance and
dimensional reduction of the patterns formed. A new perspective on sandpiles
views them as free boundary problems for the discrete Laplacian with an extra
integrality condition. The talk will contain theorems, conjectures, proofs and
pictures in about equal proportion.
Joint work with Anne Fey and Yuval Peres. View
video of this lecture.
View
slides from this lecture.
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January 15, 2010 at 2:30pm
Mary Gates Hall, Room 231 |
Alexander Holroyd
University of British Colubmia & Microsoft Research |
Random Sorting Networks |
Sorting a list of items is perhaps the most familiar problem of computer science. If one must
do this by swapping neighbouring pairs, the worst initial condition is when the
n items are
in reverse order, in which case n choose 2 swaps are needed. A sorting network is any
sequence of n choose 2 swaps which achieves this. We consider a uniformly random n-item
sorting network. Exact simulations and heuristic arguments have led to a wealth of
astonishing conjectures about the n→infinity limit. For instance, the half-time permutation
matrix is believed to be circularly symmetric, while the trajectories of items appear to
converge to random sine curves; the best known bounds on the permutation matrices and
trajectories are much weaker (but still non-trivial). The conjectures fit together into a
remarkable geometric picture. I'll also report on some recent progress on local sub-networks
and random sub-networks, both of which shed some new light on this picture.
Based on joint works with Omer Angel, Vadim Gorin, Dan Romik and Balint Virag. See
this site for pictures.
View
video
of this lecture.
View
slides from this lecture.
Simulation Software (Windows Zip File)
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November 20, 2009 at 2:30pm
Mary Gates Hall, Room 241 |
Gunther Uhlmann
University of Washington |
Cloaking and Transformation Optics |
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We describe recent theoretical and experimental progress on making objects
invisible to detection by electromagnetic waves, acoustic waves and quantum
waves. Maxwell's equations have transformation laws that allow for design of
electromagnetic materials that steer light around a hidden region, returning
it to its original path on the far side. Not only would observers be unaware
of the contents of the hidden region, they would not even be aware that
something was being hidden. The object, which would have no shadow, is said to
be cloaked. We recount some of the history of the subject and discuss some of
the mathematical issues involved. View
video
of this lecture.
View
slides from this lecture.
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November 6, 2009 at 2:30pm
Mary Gates Hall, Room 241 |
Bill Fulton
University of Michigan |
Character Formulas |
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In this expository talk, we will give a simple formula, with a simple
proof, for the equivariant euler characteristic of an equivariant
vector bundle on on complete, smooth variety with a torus action.
On homogeneous varieties this gives Weyl's character formula, and
on toric varieties it gives Brion's formula for lattice points in
polytopes. This is based on ideas of George Quart in the 1970s and
recent conversations with Bill Graham.
View
video
of this lecture.
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October 23, 2009 at 2:30pm
Mary Gates Hall, Room 241 |
Persi Diaconis
Stanford University |
Shuffling Cards and Adding Numbers |
When several large integers are added in the usual way 'carries' occur along the
way. It is natural to ask: 'About how many carries are there and how are they distributed
for typical numbers?' It turns out that these questions are intimately related to the
mathematics of the usual way we shuffle cards. I will explain the mathematics of
'carries' (they are cocycles!), shuffling and the connection. This is joint work with Jason Fulman.
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