Adam Levine : Heegaard Floer Homology and Closed Exotic 4-Manifolds
- Geometry and Topology ( 171 Views )We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.
Vera Vértesi : Knots in contact 3--manifolds
- Geometry and Topology ( 168 Views )In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsváth-Szabó-Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsváth-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vértesi).
Goncalo Oliveira : Monopoles in Higher Dimensions
- Geometry and Topology ( 130 Views )The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G_2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.
Tye Lidman : Positive-definite symplectic four-manifolds
- Geometry and Topology ( 107 Views )We will prove that certain simply-connected four-manifolds with positive-definite intersection forms cannot admit symplectic structures. This is related to the existence of so-called perfect Morse functions. This is joint work with Jennifer Hom.
Justin Sawon : Lagrangian fibrations by Prym surfaces
- Geometry and Topology ( 0 Views )Holomorphic symplectic manifolds (aka hyperkahler manifolds) are complex analogues of real symplectic manifolds. They have a rich geometric structure, though few compact examples are known. In this talk I will describe attempts to construct and classify holomorphic symplectic manifolds that also admit a holomorphic fibration. In particular, we will consider examples in four dimensions that are fibred by abelian surfaces known as Prym varieties.