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Homology theories in low dimensional topology

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

Figure from Ken Baker's Sketches of Topology blog
Programme
9th January 2017 to 30th June 2017

Organisers: Andrew Lobb (Durham), Dorothy Buck (Imperial College London), Jacob Rasmussen (Cambridge), Liam Watson (Glasgow)

Programme Theme

The subject of homology theories in low-dimensional topology dates back to the work of Floer, and gained new breadth and vigour with the introduction of Khovanov homology fifteen years ago. Since then, these theories have had a far-reaching impact both in topology and more widely with analysts, algebraists, geometers, and physicists all contributing to and benefiting from their development. This programme will follow three broad themes:

  1. The meaning of Floer homology: The programme will pursue and uncover intimate connections of Floer homology with underlying topological data. For example: the connection of Floer homology with the fundamental group by way of orderable groups, foliations, and related structures.
  2. The meaning of quantum knot homologies: In comparison to Floer homology, the geometrical meaning of quantum knot homologies such as Khovanov homology remains relatively obscure. One natural place to look for such meaning is in physics, which has played an important role in this field ever since Witten interpreted the Jones polynomial in terms of Chern-Simons theory.
  3. Quantum 3-manifold invariants:  We hope to make significant progress towards, for example, lifting the numerical Reshetikhin-Turaev 3-manifold invariants to homological invariants.  This should then provide a quantum counterpart to analytic invariants of smooth 4-manifolds.

Tying these three themes together will be a central focus of the programme. It has become apparent in recent years that one should think of quantum invariants as first-order approximations to Floer invariants. The programme aims to make relationships such as these more precise and general.

 

Figure from Ken Baker's Sketches of Topology blog

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons