Georgia Institute of Technology College of Sciences

Zoom Link- https://gatech.zoom.us/j/97563537012?pwd=dlBVUVh2ZDNwdDRrajdQcDltMmRaUT09 (Meeting ID: 975 6353 7012 Passcode: 525012)

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

Fiber sums and the rational blowdown have been very useful tools in constructing smooth, closed, oriented 4-manifolds. Applying these tools to genus g>1 Lefschetz fibrations with clustered nodal fibers, we will construct symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line, providing a symplectic extension of classical works populating the complex geography plane with holomorphic Lefschetz fibrations.

We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb. 

Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology.

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on CAT(0) cube complexes.

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5. 

The contact invariant, introduced by Kronheimer and Mrowka,
is an element in the monopole Floer homology of a 3-manifold which is
canonically attached to a contact structure. I will describe an
application of monopole Floer homology and the contact invariant to
study the topology of spaces of contact structures and
contactomorphisms on 3-manifolds. The main new tool is a version of
the contact invariant for families of contact structures.