Dyson's conjecture asserts that the constant term of certain
Laurent polynomial in the form of a product is the multinomial coefficients.

It was studied by many authors, including a proof by the Nobel prize
winner Wilson, an elegant recursive proof by Good, and a combinatorial
proof by Zeilberger. I will talk about an elementary approach by using
basic property of polynomials to several extensions of the Dyson constant
term identity.

A fundamental difficulty in statistical learning is the "curse of
dimensionality," where most learning problems become notoriously
difficult when the data are high dimensional. Even the simplest of
methods---the linear model---has proved to be interesting and
challenging to understand in the high dimensional setting, and has
attracted the recent attention of multiple communities, including
applied mathematics, signal processing, statistics, and machine
learning.

In this talk we present some recent work on several nonparametric
learning problems in the high dimensional setting. In particular, we
present theory and methods for estimating sparse regression functions,
additive models, and graphical models. For nonparametric regression,
we present a greedy algorithm based on thresholding derivatives that
achieves near optimal minimax rates of convergence. For additive
models, we present a functional version of methods based on L1
regularization for linear models. For graphical models, we present a
method for estimating the graph underlying an unknown graphical model
based only on observations.

The talk is based on work with Larry Wasserman, Pradeep Ravikumar, Han
Liu, and Martin Wainwright.

In 1954, John Milnor introduced the notion of link homotopy and
his invariants of links which he used to classify 3 component links up to
homotopy. In 1987 the speaker and XS Lin acheived the classification,
for any number of components, essentially by refining the Milnor invariants.

The Habegger-Lin classification scheme was extended to other equivalence
relations in Lin's thesis and to more general concordance-type relations
satisfying a list of 6 axioms. Axioms 1-4 are local, axiom 5 says that
any string link (or 'pure tangle' as in pure braid) has an inverse, while
axiom 6 says the equivalence relation on links is generated by isotopy and
the equivalence relation on string links (every string link yields a link
after 'closure').

In the early 90's Birman and Lin studied the work of Vassiliev on links
and described in simple terms the Vassiliev filtration. Bar-Natan adopted
their description as a definition of 'finite type' invariants and
eventually all this was tied back to the perturbative Chern Simons quantum
invariants via the Kontsevich Integral.

Early on, Lin suggested the Milnor invariants were of finite type, but
this is strictly true only of the string link invariants because Milnor's
invariants are only 'partially' defined, i.e. their indeterminacy depends
on the lower order invariants. The speaker and G. Masbaum actually gave
in 1997 a formula computing the Milnor string link invariants from the
Kontsevich Integral. The tree-like Feynman diagrams correspond to the
Milnor invariants.

The nagging problem that Vassiliev invariants of links are universally
defined, but Milnor invariants, which ultimately gave the link-homotopy
classification, are only partially defined, suggests that finite-type
invariants of links are deficient. It turns out that axiom 6 of the
aforementioned classification sheme is not satisfied so that Vassiliev
(finite type) invariants of links can and ought to be refined, as shown in
a recent preprint by the speaker and JB Meilhan.

See http://www.math.sciences.univ-nantes.fr/~habegger/) for the
aforementioned works. (For those interested in the non-perturbative CS
theory, one can also find at this address the work of BHMV on Topological
Quantum Field Theory derived from the Kauffman bracket, e.g. the Jones'
Polynomial.)

A more accurate subtitle for this talk would have been "When You Can't Get There From Here By Going
Thataway, You Got To Go Somewhere Else First,", but that was too long to fit. A subriemannian manifold is a sort of
space in which certain directions of travel are illegal. This can describe lots of problems involving systems with
too many degrees of freedom. I'll talk about several examples, including rolling balls, falling cats, Carthaginian
queens, and drunks with planimeters (if you don't know what a planimeter is, go ask John Eggers), and also about
Chow's Theorem, which says that maybe you can get there from here after all. If time permits, there might be some
applications to PDEs and some open problems mentioned. This talk will be accessible to anyone who has heard of a
smooth manifold.

The computation of the Hilbert class polynomial has
applications ranging from explicit class field theory to
cryptography. Several new algorithms to compute it have
been developed during the last 5 years, each having its pro's
and cons. In this talk we will present a significant speed
up of the `Chinese remainder theorem approach'. We will give
a detailed run time analysis of the new algorithm, using
tools from both analytic number theory and arithmetic
geometry. The resulting run time is almost optimal: one of
the bottlenecks is writing down the answer.

We will have three panelists who have recently found jobs: Steve Butler, post doc, UCLA, Maia Averett, Mills College, Oakland California, and Dave Clark Assistant Professor, Randolf Macon College, Ashland, Virginia They will describe their experiences applying for an academic job. Some of the questions they will answer are: How many applications should I send out? How do I prepare for an interview? What should I write in my cover letter and resume? What are important qualifications for a teaching job, postdoc job, tenure track research job? If you are soon to be on the job market, this is a terrific opportunity to find out the optimal way to apply for jobs.
The discussion will be followed by a question and answer period.