Principal Investigator: Haynes R. Miller, Michael J. Hopkins
The Talbot program is a series of yearly workshop retreats bringing together graduate students, recent PhDs, and faculty mentors for an intense exploration of a topic of active mathematical research. The Talbot 2004 workshop was devoted to the Stolz-Teichner model of elliptic cohomology and was mentored by Stephan Stolz. The Talbot 2005 workshop will focus on the geometric Langlands program and be mentored by David Ben-Zvi. The geometric Langlands program is one of the most fertile areas of mathematics, integrating representation theory, topology, and algebraic geometry. The original conjectures of Langlands tie infinite-dimensional representation theory to the structure of Galois groups, with profound applications to number theory. From the geometric perspective deriving from the renowned theorem of Borel-Weil-Bott, one would like to classify such representations by equivariant sheaves on a parametrizing space. Drinfeld and Laumon, among others, adapted Langlands' original ideas for function fields to formulate a far reaching geometric generalization, now called the geometric Langlands program. Work in this area over the past twenty years has changed the face of modern representation theory. Recent advances, such as new geometric generalizations of the classical Satake isomorphism, make this an ideal time to gather young mathematicians to study these results, the techniques used to derive them, and newly exposed avenues for future research.
The Talbot workshops seek to introduce aspiring mathematicians to active areas of mathematical research, foster community and collaboration across subdisciplinary and institutional lines, and form pedagogical and research ties between established mathematicians and young researchers. The topic for Talbot 2004 concerns a central problem in mathematics deeply related to neighboring areas of science. For instance, the crystallographic study of proteins, quantum states of particles, and the structure of integer solutions to polynomial equations are all central to chemistry, physics, and cryptography/number theory; their deeper study requires probing the algebraic structures that govern aspects of their nature. Representation theory is precisely the field of mathematics devoted to this study. Further, the geometric Langlands program offers a unifying vision for the specific algebraic structures suited to each of these areas, which are called reflection groups, Lie groups, and Galois groups. This Talbot workshop will bring together graduate students with different specialties and from many universities to spend a week focused on this important subject. The participants and their mentor, David Ben-Zvi, will share a residence as well as lectures, discussions, and meals. This informal atmosphere, mixing fellowship and intellectual interest, will promote a concentrated study of the material and lay the foundation for future collaboration and research.
