3. Postdoctoral Program - MSRI

REPORT ON THE MSRI WORKSHOP “HOMOLOGY THEORIES
OF KNOTS AND LINKS”
Organizers
• Mikhail Khovanov (Columbia University)
• Peter Ozsváth (Columbia University)
• Lev Rozansky (UNC)
• Zoltán Szabó (Princeton University)
• Dylan Thurston (Columbia University/Barnard)
1. Scientific description
Link homology is a new source tools for studying low-dimensional phenomena. Although
its goal is to explore the topology of familiar low-dimensional objects – knots,
links, and indeed three- and four-manifolds – this rapidly-developing subject draws
on many seemingly unrelated branches of mathematics. The field is driven primarily
by three currents in mathematics: representation theory, gauge theory, and symplectic
geometry. These three currents have lead, respectively, to Khovanov homology
and other “categorifications”; forms of gauge-theoretic Floer homology including instanton
Floer homology (using anti-self-dual connections), and more recently Floer
homology for Seiberg-Witten monopoles; and finally, Heegaard Floer homology, along
with its other variants for knots, links, and sutured manifolds.
This new discipline is at a critical moment in its development. Categorification
has seen a broad expansion as a subject. It is now solidly linked to homological
algebra of rings and differential graded rings. Relations have been found between
link homology and algebraic geometry, including derived categories of sheaves on
suitable quiver varieties and convolution varieties in affine Grassmannians. A more
direct connection between categorification and the Langlands program is likely to be
found in the near future.
Various calculational techniques have rendered aspects of Heegaard Floer homology
to be combinatorially describable (a goal which has so far eluded its gauge-theoretic
predecessors). Various relationships have been discovered relating categorifications
with their more geometrically-defined cousins (typically formulated as spectral sequence
from categorified invariants to gauge-theoretic or symplectically defined invariants).
Finally, continuing the thread unifying gauge theory and symplectic geometry
initiated by Taubes (in his proof that Seiberg-Witten invariants count certain
Gromov invariants), the close relationship between Heegaard Floer homology and
Seiberg-Witten theory is well on its way from being a conjecture to a theorem. In
addition to these various exciting developments within the subject of link homology,
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