Interactions of Homotopy Theory and Algebraic Topology with Physics through Algebra and Geometry
Organized by John Morgan and Dennis Sullivan
October 1, 2014 – June 30, 2015
While activities will depend on the visitors for their specific focus, we expect them to be organized around several general themes: (i) rigorous approaches to perturbative quantum field theories, and especially to gauge theories using homological and homotopy-theoretic techniques, (ii) formal quantization, and (iii) TQFTs and infinity structures. Within those themes, a partial list of topics would include: BV algebras; operads; the Fukaya category; various compactifications of moduli spaces of stable curves and stable maps, as in string topology and contact homology and twisted K-theory. Our plan is to have activity spread over the entire academic year, rather than a more concentrated activity during one semester, with between 4 and 6 visitors in residence at any one time. In addition to Fukaya, Sullivan, and Morgan, who are permanently in residence at Stony Brook, those who have expressed interest in attending the program (for 3 weeks to a month, with a few longer visits) include Kevin Costello, Jacob Lurie, Dan Freed, Constantin Teleman, and Alberto Cattaneo.
Application for program is now closed.
As part of the Simons Center program, Interactions of Homotopy Theory and Algebraic Topology with Physics through Algebra and Geometry, John Morgan will give a series of lectures (8 to 10 lectures) on Sheaf Theory with applications to duality. The course will be aimed at intermediate graduate students and above. The only prerequisite is a basic course in algebraic topology.
These lectures will be on Fridays at 2:45pm, in the Simons Center seminar room, 313 beginning Friday October 3.
Title: A Topologist looks at Sheaf Theory
Abstract:
Sheaf theory has long been an essential tool in algebraic geometry, algebraic number theory, and complex analysis, but its inspiration comes directly from topology. This lecture course will emphasize these roots, hopefully making sheaf theory seem natural to those with a topological bent. The course will begin by covering the basic topics in sheaf theory describing the objects and the four basic maps of the theory and then will culminate with a discussion of Verdier duality, which generalizes Poincare duality.
This theory will then be applied to define a bordism theory, called duality bordism, whose coefficient group agrees with the Grothendieck group of chain complexes satisfying Poincare duality modulo those that sit as the boundary term in an exact sequence satisfying Lefschetz duality. This bordism group is the Pontryjagin dual homology theory to the cohomology theory associated with surgery theory. This means that a surgery problem is completely classified by evaluating surgery obstructions (signatures, and Arf invariants) of its restrictions to all possible duality bordism elements.
Direct analysis of this bordism theory allows one to identify it at odd primes with real K-theory and at the prime 2 with ordinary homology.