Factorizable Structures in Topology and Algebraic Geometry (15w5125)

Arriving in Banff, Alberta Sunday, August 9 and departing Friday August 14, 2015

Organizers

Gregory Arone (University of Virginia)

David Ayala (Montana State University)

(Northwestern University)

Dennis Gaitsgory (Harvard)

(Max Planck Institute for Mathematics)

Objectives

Beilinson & Drinfeld first introduced the idea of factorization algebras and factorization homology in the 1990s, in their conceptual reformulation of the mathematical structures of conformal field theory. There has been an explosion of activity around these ideas in the last five years, primarily in topology but also in algebraic geometry and quantum field theory. Because of the varied backgrounds of the researchers involved (for example, from operad theory, low-dimensional topology, vertex algebras, homotopy theory, geometric representation theory, K-theory), interaction is often indirect and intermittent. This workshop provides the opportunity for direct communication and the chance to share perspectives and insights arising from diverse mathematical contexts.We envision talks and collaboration around the following topics.> Applications to low-dimensional topology, particularly knot and 3-manifold invariants.> Deepening the relationship with the manifold calculus of Goodwillie-Weiss, with applications to embedding theory.> Foundational aspects, especially clarifying the relationship among the different flavors of factorization homology.> The utility of factorization techniques in quantum field theory, notably conformal field theory and gauge theory.> Improved computational techniques, notably with spectral sequences.This workshop comes at a time when there is heightened interest among several communities in the techniques and applications of factorization homology.First, the manifold calculus of Goodwillie-Weiss has prompted recent, productive activity by connecting the study of long knots, configuration spaces, and operad theory (notably, the little n-disks operad). There is an obvious yet undeveloped relationship between factorization homology and this embedding calculus. Namely, manifold calculus views an n-manifold as right module over the little n-disk operad, while factorization homology views a factorization algebra as a right comodule over this same operad. This workshop will be a chance to foster a deeper, systematic understanding of how these dual approaches relate.Second, computations in algebraic K-theory have been substantially enhanced through trace methods. For many situations, the naive target of trace maps is an instance factorization homology in dimension one, specifically the circle. Additional functorialities for factorization homology among finite sheeted covers of the circle suggest a structure reminiscent of the cyclotomic trace which approximates K-theory. Moreover, factorization homology provides a framework for studying higher traces from iterated K-theory. These observations are enticing as they relate K-theory and chromatic homotopy theory. This workshop can be a time to lay down some foundations of factorization homology prepared for use in stable homotopy theory.Third, the study of D-branes and surface operators has led theoretical physicists and their mathematical collaborators to a deepened appreciation for higher categorical methods in organizing and constructing field theories. Because the observables of a perturbative quantum field theory form a factorization algebra, there are immediate applications of factorization methods in physics, and indeed factorization algebras provide a unifying language for many approaches to quantum field theory. For example, Costello has recently constructed the Yangian, a central object in supersymmetric gauge theory for four-manifolds, as a factorization algebra. In a more topological direction, Ginot, Tradler, and Zeinalian are applying factorization methods to string topology and its higher analog, brane topology.