FloerThInLowDimProject reference: 268389
Funded under :
Floer theoretical invariants of low dimensional manifolds
Total cost:EUR 100 000
EU contribution:EUR 100 000
Coordinated in:United Kingdom
Call for proposal:FP7-PEOPLE-2010-RGSee other projects for this call
Funding scheme:MC-IRG - International Re-integration Grants (IRG)
My main research interests are in low dimensional topology, symplectic geometry and gauge theory. Over the past 20 years, these fields has seen an explosion of activity due to its relevance to string theory.
As part of my PhD thesis, I proved an equivalence between two 3-manifold invariants coming from Floer theory. These are Perutz's Lagrangian matching invariants and Ozsvath and Szabo's Heegaard Floer theory. Although, Heegaard Floer theory has been studied extensively, Lagrangian matching invariants is a relatively recent theory and it remains to be explored more thoroughly. The set-up of Lagrangian matching invariants gives more emphasis on symplectic techniques, and this offers a different approach to Heegaard Floer theory. My goal is to explore these invariants in more depth and bring in new symplectic techniques to the study of 3-manifolds. As a concrete project along these lines, I have been working with Perutz in extending these invariants to bordered three manifolds for which we apply techniques used in the study of Fukaya categories of symplectic manifolds. As a byproduct, we obtain categorical mapping class group actions.
Another main part of my research is the study of Fukaya categories of Lefschetz fibration on the Hilbert schemes of the A_n type Milnor fibre, a special type quiver variety. This involves Floer theoretic calculations of non-compact Lagrangian submanifolds. The applications of this research has deep connections with conjectures involving the relation of the Fukaya category to geometric representation theory, in particular to Khovanov's combinatorial link invariants.
In addition to the projects described above, I am interested in various structures in low dimensional topology. For example, I proved that every smooth 4-manifold admits a broken Lefschetz fibration. This gives a new calculus of 4-manifolds, which I plan to apply to solve old conjectures about 4-manifolds.
EU contribution: EUR 33 333,33
WC2R 2LS LONDON
EU contribution: EUR 66 666,67
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