One of the successes of tropical geometry is its applications to a number of different areas of recently developing mathematics. Among these are enumerative geometry, symplectic field theory, mirror symmetry, dimer models/random surfaces, amoebas and algas, instantons, cluster varieties, and tropical compactifications. While these fields appear quite diverse, we believe the common meeting ground of tropical geometry will provide a basis for fruitful interactions between participants.

The theory of holomorphic curves in symplectic manifolds leads
to rich algebraic structures. The study of these structures is
increasingly important both for understanding the theory itself, and
for actual computations and applications. The aim of the workshop
is to survey ongoing developments in the area. Some of the topics
of interest are: cohomological field theories; relative and tropical
Gromov-Witten invariants; Symplectic Field Theory (SFT) and connections
with string topology; theories of holomorphic curves with Lagrangian
boundary conditions, such as relative SFT, open Gromov-Witten theory,
and Fukaya categories.

This workshop will concentrate on tropical methods in Combinatorics
and Algebra. Some of the topics we expect to explore are
tropical ideas and methods in combinatorial linear algebra and in
combinatorial representation theory, as well as computational issues and applications of tropical methods in algebraic statistics.

The mathematical study of the dynamics of the Einstein equations forms a central part of both partial differential equations and geometry, and is intimately related to our current physical understanding of gravitational collapse.

This workshop is to lay the foundations for the upcoming program. Mini-courses comprising lectures and exercise/discussion sessions will cover the foundational aspects of tropical geometry as well as its connections with adjacent areas: symplectic geometry, several complex variables, algebraic geometry (in particular enumerative and computational aspects) and geometric combinatorics. The mini-courses will be augmented by research talks on current tropical develpoments to open the scene and set up new goals in the beginning semester.

The aim of this workshop is to introduce advanced graduate students and postdocs to tropical geometry. Various aspects of this multi-faceted field will be highlighted in two short-courses comprising lectures and exercise/discussion sessions as well as in research talks. The workshop will thus provide the participants with
an excellent introduction to the forthcoming events of the program. The scientific part will be complemented by a round table discussion on career issues of female mathematicians.

This workshop aims both to introduce
people to a broad swath of the field
and to frame its most important problems.
Each day will be organized around a
basic topic, such as how to count holomorphic
curves with boundary on a Lagrangian submanifold (which
leads to various versions of Floer theory)
or how to understand the general structure of
symplectic and contact manifolds.
There will also be an introduction to the
analytic and algebraic aspects of symplectic
field theory, and a discussion of some applications.

This will form a bridge between
the graduate student workshop which will just be ending and
the Introductory workshop. After some
elementary talks describing some of the main questions
in the field, there will be an extended discussion session
intended to explain basic concepts to those unfamiliar with the area.
There will also be an opportunity for young researchers in the field
to present their work, and an evening social event.

Symplectic and Contact Topology has undergone rapid and exciting growth
in the past few decades and is currently a rich subject, employing a variety of diverse techniques and touching on many areas of mathematics, such as algebraic and differential geometry, dynamical systems and low dimensional topology. This workshop is intended both for graduate students new to the
area and for those working in the field.
Lectures in the first week will introduce participants to basic topological, geometric and analytic techniques, including J-holomorphic curves. The second week will discuss applications to symplectic geometry and to 3-dimensional topology and knot theory. A variety of discussion
sessions in the afternoon will cater to the differing interests of the students. Participants may consider staying for the Connections for Women and/or the Introductory workshop to the year long Symplectic Geometry program that starts just after this workshop.

This workshop is part of the semester program on Algebraic Geometry, and
additional funding will be available for participants to attend the associated
"Introductory workshop: Classical algebraic geometry," January 26-30, 2009.