Perspectives on Knot Homology (21w5105)


(University of Virginia)

Mina Aganagic (University of Berkeley)

(University of Oregon)


The Banff International Research Station will host the "Perspectives on Knot Homology" workshop in Banff from May 16 to May 21, 2021.

Quantum polynomial invariants of links and tangles, such as the Jones polynomial and the HOMFLY-PT polynomial, have played a central role in many areas of mathematics and physics over the last 3 decades. Knot homology is a far reaching generalization of polynomial invariants, which is still being developed. Homological knot invariants are referred to as categorification of the polynomial invariants -- polynomial invariants arise as the (graded) Euler characteristic of a homology theory. There are deeper structures which become manifest at the categorified level -- just like in transition from the Euler characteristic to homology in basic algebraic topology. In particular, one can speak of maps between vector spaces, but not between numbers or polynomials. The appropriate notion of a “map” between two knots is a surface cobordism in 4-space. In several cases, these give rise to maps on homology of the boundary links - a feature not available at the level of the Euler characteristic.

Many examples of homological invariants of knots and links were constructed over the last two decades, using a wide variety of methods: diagrammatic and geometric representation theory, gauge theory, and symplectic geometry. String theory and quantum field theories have led to influential predictions about the structure of the homological invariants, and provide general outlines of how they should arise from physics. While there is a uniform mathematical construction of quantum link and three-manifold invariants, we are yet to discover a uniform approach to their homological generalization.

The aim of this workshop is to bring together experts on many developing perspectives on knot homology (physical, geometric and algebraic) to draw connections between them, and to explore applications.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).