This workshop will introduce graduate students to several central ideas in the arithmetic of elliptic curves. Participants will join a project group that will focus mainly on one topic, possibly involving elliptic curves over number fields, complex or p-adic L-functions, Heegner points and Kolyvagin classes, Iwasawa theory, and the Birch and Swinnerton-Dyer conjecture. The workshop will emphasize the essential interplay of abstract mathematics with explicit computation, which has played a central role in number theory ever since Birch and Swinnerton-Dyer made their famous conjecture in the 1960s. Participants will use, and improve, the free open-source Python-based mathematical software system Sage (http://www.sagemath.org) for computational projects.

The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences. The academic portion of the program will be led by Dr. Edray Goins.

The National Council of Teachers of Mathematics has just released a new document, Focus in High School Mathematics: Reasoning and Sense-Making. The Council of Chief State School Officers and the National Governor’s Association have initiated a state led effort to produce Common Core State Standards, which they hope will move states toward national curricular coherence. The national scene is being transformed through stimulus money aimed at having states adopt common standards. This is a significant time for mathematicians to weigh in for coherence and a focus on thinking, understanding and sense-making. For this reason MSRI will host the seventh Critical Issues in Mathematics Education Workshop on this topic. Themes of the workshop will include international comparisons, the role of a coherent national curriculum in the teaching of mathematics, and the ways in which technology can be used to support reasoning and sense-making.

Sponsor: Hayashibara Foundation




Symplectic geometry originated as a mathematical language for Hamiltonian mechanics, but during the last 3 decades it witnessed both, spectacuar development of the mathematical theory and discovery of new connections and applications to physics. Meanwhile, non-commutative geometry naturally entered into this picture.

The first week of May 2010 coincides with the first year anniversary of Alan Weinstein's retirement from UC Berkeley; Weinstein has been one of the most influential figures in symplectic geometry, Poisson geometry and analysis in the past forty years. Weinstein's fundamental work inspired many others and led to the development of central concepts in symplectic and Poisson geometry, as well as to the establishment of symplectic geometry as an independent discipline within mathematics. This conference will be a forum to celebrate Weinstein's fundamental contributions to geometry and mathematics at large.

This workshop will focus on recent progress in central problems in
symplectic and contact topology and Hamiltonian dynamics such as
rigidity of Lagrangian submanifolds, algebra/topology/geometry of
symplectomorphism and contactomorphism groups, exotic symplectic and
contact structures, and existence of
periodic orbits of Hamiltonian systems and Reeb flows.
It will explain applications of the "large machines"
such as Floer Theory, Symplectic Field Theory and Fukaya categories,
showing where these machines do not yet provide satisfactory
answers. Special attention will also be paid to articulating
new problems and
directions, as well as to explaining
interactions between symplectic and contact
topology and other fields.

Link homology is a young and rapidly-developing area drawing on many branches of mathematics. The subject has its roots in representation theory, and it has benefitted from its interactions with low-dimensional, classical, and quantum topology and symplectic geometry. Indeed, several recent developments have underscored the close parallels between link homology and Floer homological invariants for low-dimensional manifolds.

This workshop will introduce the main branches in the study of knot homology theories. It will consist of three mini-courses, one on knot Floer homology and related topics; one on the various approaches to
Khovanov and Khovanov-Rozansky homology; and one on categorification on quantum groups. (There will also be several stand-alone lectures.) The techniques involved in the three branches are quite different; in
particular, Heegaard Floer theory is analytic in nature, with its origin in gauge theory and symplectic geometry, while both Khovanov homology and categorification are more algebraic in nature, with origins in representation theory and algebraic geometry. The workshop will provide an opportunity for graduate students and researchers
outside the field to gain entry, as well as for researchers working in one part of the field to learn about techniques and developments in other parts.

This 2-day workshop will serve as a prelude to the introductory workshop for the semester-long program on homology theories of knots and links. Survey talks in the mornings will position the work in Khovanov and Heegaard Floer homology in a broader context, focusing on:

1) applications to classical questions in low-dimensional topology, and
2) connections to contact and symplectic topology.

Research talks in the afternoons will highlight the range of current activity in the field. We plan a format of no more than four talks each day to allow ample time for presentation opportunities for younger researchers and formal and informal discussions.