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Low Dimensional Topology in Budapest

Objective

"Heegaard Floer theory. In this project (in collaboration with P. Ozsváth and Z. Szabó) we plan to extend our earlier results computing various versions of Heegaard Floer homologies purely combinatorially. We also plan to find combinatorial definitions of these invariants (as graded groups). Such results will potentially lead to a combinatorial description of 4-dimensional Heegaard Floer (mixed) invariants, conjecturally equivalent to Seiberg-Witten invariants of smooth 4-manifolds. In particular, we hope to find a combinatorial proof of Donaldson’s diagonalizability theorem, and find relations between the Heegaard Floer and the fundamental groups of a 3-manifold.

Contact topology. Using Heegaard Floer theory and contact surgery, a systematic study of existence of tight contact structures on 3-manifolds is planned. Similar techniques also apply in studying Legendrian and transverse knots in contact 3-manifolds. In particular, the verification of the existence of tight structures on 3-manifolds given by surgery on a knot (with high enough framing) in the 3-sphere is proposed. Using the Legendrian invariant of knots, Legendrian and transverse simplicity can be conveniently studied. The ideas detailed in this part are planned to be carried out partly in collaboration with Paolo Lisca, Vera Vértesi and Hansjörg Geiges.

Exotic 4-manifolds. Extending our previous results, we plan to investigate the existence of exotic smooth structures on 4-manifolds with small Euler characteristics, such as the complex projective plane CP2, its blow-up CP2#CP2-bar, the product of two complex projective lines CP1×CP1 and ultimately the 4-dimensional sphere S4. We plan to investigate the effect of the Gluck transformation. Possible extensions of the rational blow down procedure (successful in producing exotic structures) will be also studied. We plan collaborations with Zoltán Szabó, Daniel Nash and Mohan Bhupal in these questions."

Field of science

  • /medical and health sciences/clinical medicine/surgery
  • /natural sciences/mathematics/pure mathematics/topology

Call for proposal

ERC-2011-ADG_20110209
See other projects for this call

Funding Scheme

ERC-AG - ERC Advanced Grant

Host institution

RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Address
Realtanoda Street 13-15
1053 Budapest
Hungary
Activity type
Other
EU contribution
€ 1 208 980
Principal investigator
Andras Istvan Stipsicz (Prof.)
Administrative Contact
Peter Pal Palfy (Prof.)

Beneficiaries (1)

RENYI ALFRED MATEMATIKAI KUTATOINTEZET
Hungary
EU contribution
€ 1 208 980
Address
Realtanoda Street 13-15
1053 Budapest
Activity type
Other
Principal investigator
Andras Istvan Stipsicz (Prof.)
Administrative Contact
Peter Pal Palfy (Prof.)