This is the organizational meeting for our topology learning semiar. The topic this quarter is formal group laws and their applications in topology. Nitu will give an introduction to the topic and we will distriubute the talks.

This lecture will talk about semidefinite programming (SDP) and
its applications in global polynomial optimization. Firstly, after
introducing SDP, we will how to represent k-elliptic curves by SDP.
Secondly, after an overview of the sum of squares (SOS) relaxation, which
can be reduced to SDP, we will present gradient SOS relaxation. While the
general SOS relaxation has a gap in finding the global minimum, the gradient
SOS relaxation can find the global minimum whenever a global minimizer
exists. Lastly, we will show how to exploit sparsity in SOS and its
applications in sensor network localization.

Fusion energy holds the promise of a clean, sustainable and safe
energy source for the future. While research in this field
has been ongoing for the last half century, much work
remains before it may prove a viable source of energy. In
this talk, I discuss some of the scientific and engineering
challenges remaining in fusion energy, and the role of
mathematics research in overcoming these obstacles. In
particular, I will discuss some of the mathematical models
used in studying fusion stability and refueling, how
solutions to those models may be approximated, and introduce
some model improvements to better simulate fusion processes.

We introduce the concept of automatic structure. Informally, these
are structures that can be defined in terms of automata. By automata
we mean any of the following machines: finite automata, tree automata,
Buchi automata, and Rabin automata.

Nerode and the speaker initiated the systematic study of automatic
structures in 94. An important property of automatic structures is
that these structures are closed under the first order interpretations
and have effective semantics. In particular, the first order theory
of any automatic structure is decidable.

The theory of automatic structures has become an active research
area in the last decade with new and exciting results. In this talk
we survey recent results in the area and outline some of the interesting
proofs. The talk will provide many examples.

Some of the results of the talk are published in LICS 01-04 and
STACS04 conference proceedings. Results are joint with Nerode,
Rubin, Stephan, and Nies.

I will begin by explaining the statement of the Homological
Mirror Symmetry conjecture for Fano toric varieties and outline how
Lefschetz fibrations have been used to prove the conjecture in some cases.
I will then show how Mikhalkin's flavour of tropical geometry can be used
to prove half of the homological mirror conjecture for all smooth
projective toric varieties (dropping the Fano condition!).

I will begin with the basics of the two subjects, with the goal of
explaining how each has been used as a tool in the other. The first half
will be devoted to foundational results, dating to the 1930s for potential
theory and the 1980s for complex dynamics. The second half will be
devoted to more recent developments.

K-theory is a valuable tool with applications in various areas of mathematics, including topology, geometry, algebraic geometry, and number theory. In this talk I will attempt to give a rough outline of what K-theory is about, in the algebraic as well as the geometric context. Given a ring R and a space X, I will define the groups K(R) and K(X), and explain the relation between the two. I also hope to outline some applications in various fields. This talk should be accessible and I hope interesting to students of both algebra and geometry alike.

Many mathematical objects come in continuous families, prompting the
desire to define a ``universal family'' that contains each such object
exactly once up to isomorphism. When this isn't possible (because the
family would be too bad to be worthwhile -- I'll talk about this
behavior),
we can try to come close, by including only ``stable'' objects.

Frequently the universal family is constructed by starting with a
bigger family that includes each object many times, then dividing
by a group action that implements the isomorphisms. There are two ways to
do this, one algebro-geometric (complex) and one symplecto-geometric
(real), and I'll give some idea of why they agree.

The main example will be the space of $N$ ordered points on the
Riemann sphere, modulo M\"obius transformations. These are unstable if two

Vanishing theorems for cohomology groups are one of the
essential tools of modern algebraic geometry, and have particularly
important applications in higher dimensional geometry. Under strong
positivity assumptions on line bundles, for example ampleness, there are
well-known "standard" vanishing theorems, like those of Kodaira, Nakano
and Kawamata-Viehweg. They have very useful partial analogues, called
Generic Vanishing Theorems - first discovered by Green and Lazarsfeld -
when the positivity hypotheses are weakened. I will describe all of the
above and their importance, and then explain that recent techniques based
on Fourier-Mukai functors and homological algebra can be used to widely
extend the context of generic vanishing, and relate it to standard
vanishing. As an application, I will explain how to generalize the results
of Green-Lazarsfeld to a version of Kodaira vanishing under weak
positivity hypotheses.

We describe various results on Galois groups of iterates of a given quadratic polynomial. Joint work with Rafe Jones (University of Wisconsin).

We consider sufficient conditions for regularity
of Leray-Hopf solutions of the Navier-Stokes equation. By a
result of Neustupa and Panel, a Leray-Hopf weak solution is
regular provided a single component of the velocity is
bounded. In this talk we will survey existing and present
new results on one component and one direction
regularity. We will also show global regularity for a class
of solutions of the Navier-Stokes equation in thin
domains. This is a joint work with M. Ziane.