05w5081 Progress in algebraic geometry inspired by physics

Arriving Saturday, October 8 and departing Thursday, October 13, 2005

Organizers: Jim Bryan (University of British Columbia), Michael Thaddeus (Columbia University), Ravi Vakil (Stanford University).

Confirmed Participants

Information for Participants

Final Report (PDF file)

Objectives

Our main goal, as for any successful workshop, is to foster lively interaction between the participants, leading to original ideas and new directions in research. Having said that, we also have a particular agenda: to bring together and set up discussion among several different groups whose expertise could lead to notable advances in algebraic geometry inspired by physics.



First, there are workers in algebraic geometry trained in the last 20 years, since the influence of string theory became a familiar part of the geometric landscape. They are immersed in problems of a physical origin, though they are not so often familiar with physics itself. A large number, for example, work in Gromov-Witten theory, which traces its lineage to the sigma-models of conformal field theory. While expert in algebraic geometry and in the details of problems of physical origin, they could benefit from taking a broader perspective. S. Katz frequently expresses his surprise that the younger generation of algebraic geometers is more familiar with the newfangled subject of quantum cohomology than with variation of Hodge structure, which is his generation's bread and butter. But in mirror symmetry, one is the A-model, and the other is the B-model. For a complete understanding of mirror symmetry, one needs to know both.



This brings us to our second group, those trained in gauge theory, Lie theory, and algebraic geometry before the string revolutions. This includes those whose study of Lie groups led them to algebraic problems on homogeneous spaces; those from the Atiyah school, which incorporated gauge theory into algebraic and differential geometry; and those classically trained in algebraic geometry itself. Their own research may not treat problems motivated by physics, but they have much to teach those of us who do about our own subject.



Next, there are a select few in cognate mathematical fields that are of particular relevance to the interaction between algebraic geometry and physics. For example, many of the structures of current interest in algebraic geometry, like moduli spaces of holomorphic curves, or orbifold cohomology, are analogues of broader constructions in symplectic geometry. The presence of a few leading symplectic geometers will enable participants to see these problems illuminated from a new angle. Likewise, a few experts in algebraic topology may help shed light on the deep connections that Floer homology should provide between stringy geometry and the topology of loop spaces.



Finally, there are the physicists themselves. While we do not intend to hold a workshop in string theory, which would be a very different event, we feel that the presence of a few outstanding researchers in quantum field theory and string theory would greatly enrich a meeting of mathematicians looking for physical motivation. They can tell us how to cut through mathematical details to the physical heart of the matter: what space of fields to use, what Lagrangian, what stationary phase computation. As M.F. Atiyah insists, the impact of physics on mathematics will be much greater if mathematicians engage with physics as deeply as possible, not only as a source of problems and formulas, but as a source of methods that we try our best to convert into rigorous mathematical arguments. Atiyah's own notion of a topological quantum field theory is an outstanding example of this. Inviting a few physicists might allow similar ideas to germinate at the workshop.



All of these groups will have much to teach each other. But we do not wish to suggest that this will be a meeting of a merely instructional or expository nature. Just the opposite: we expect to stimulate research developments of the highest order, and the most current interest. The achievements described in our overview of the subject area are remarkable examples of the intellectual ferment that have resulted, on very recent occasions, when the precision of mathematics is fertilized by the power of physics. We do not know what the future holds, and we suspect it will already be very different in two years. Just today, for example, we learned of a conjectural relationship between the Gromov-Witten theory of toric threefolds and certain models of 3-dimensional crystals in statistical mechanics. But we are confident that by 2005 the boundary between physics and algebraic geometry will be even busier and more productive. We are looking forward to the Banff workshop with excitement.