Autumn Term 2012

Jake Rasmussen (Cambridge). Khovanov homology and torus knots. Friday October 5th. Huxley 130, 1.30-2.30pm.

Khovanov homology is an invariant of knots in S^3 which generalizes the Jones polynomial. Although it is quite easy to define, it has many subtle connections with algebraic geometry and representation theory. I’ll discuss a conjecture which relates the Khovanov homology of torus knots with the geometry of the Hilbert scheme of points on a singular curve and the representation theory of the rational Cherednik algebra. Joint work with Eugene Gorsky, Alexei Oblomkov, and Vivek Shende.

Mark Haskins (Imperial). Recent progress in G_2 geometry. Friday October 12th. Huxley 130, 1.30-2.30pm.

András Juhász (Imperial). Naturality of Heegaard Floer homology. Friday October 19th. Huxley 130, 1.30-2.30pm.

Previously, the Heegaard Floer invariants of 3-manifolds and links have only been defined up to isomorphism. This suffices for many applications, but to be able to talk about diffeomorphism or cobordism maps on such groups one needs a functorial construction. We will provide the missing ingredient for functoriality, and define the action of the based mapping class group. Our approach relies on the bifurcation analysis of 2-parameter families of gradient vector fields. This is joint work with Dylan Thurston and Peter Ozsváth.

Zheng Hua (Kansas). Spin structures on moduli spaces of sheaves on Calabi-Yau threefolds. Friday October 26th. Huxley 130, 1.30-2.30pm.

Given a CY manifold X of complex dimension three, we consider the moduli space of coherent sheaves on X. This is a set of Artin stacks indexed by the elements in topological K_0 of X. These moduli spaces have a shifted symplectic structure in the sense of derived algebraic geometry. Kontsevich and Soibelman conjectured that there exists a consistent choice of spin structure on these moduli spaces that are compatible under taking short exact sequences of sheaves. These spin structures play an important role in Donaldson-Thomas theory. We will discuss some recent progress on this.

Rob Kirby (UC Berkeley). Morse 2-functions and trisections of 4-manifolds. Friday November 2nd. Huxley 130, 1.30-2.30pm.

In joint work with David Gay, we show existence and uniqueness of Morse 2-functions (also known as broken Lefschetz fibrations), and then existence and uniqueness (up to stabilization) of trisections of 4-manifolds (analogous to Heegaard splittings of 3-manifolds).

Yu-jong Tzeng (Harvard). On generalizations of Göttsche’s conjecture. Friday November 9th. Huxley 130, 1.30-2.30pm.

Consider a smooth projective surface S and a sufficiently ample line bundle L. Göttsche’s conjecture states for any r, that the number of r-nodal curves in the linear system |L| is a universal polynomial of Chern numbers of S and L. Göttsche’s conjecture now has several proofs and many generalizations. In this talk, I will explain the generalization of Göttsche’s conjecture for higher dimensional subvarieties with more complicated singularities, the refined curve counting invariants conjectured by Göttsche and Shende, and some computation results.

Sebastian Goette (Freiburg). Diffeomorphism types and eta invariants. Friday November 16th. Huxley 130, 1.30-2.30pm.

Kreck and Stolz have shown how to determine the diffeomorphism type of a particular class of manifolds using secondary differential-topological invariants. Together with Crowley, we have found similar invariants for highly connected seven-manifolds. These invariants can be represented – sometimes even computed – using eta invariants. As an application, we determine the diffeomorphism type of a particular seven-manifold of positive sectional curvature.

Chris Wendl (UCL). Spinal open books and algebraic torsion in contact 3-manifolds. Friday November 23rd. Huxley 130, 1.30-2.30pm.

By the Giroux correspondence, contact structures on a closed manifold can be understood in terms of open book decompositions that support them. A “spinal” open book is a more general notion that also supports contact structures, and arises naturally e.g. on the boundary of a Lefschetz fibration whose fibers and base are both oriented surfaces with boundary. One can learn much about symplectic fillings by studying spinal open books: for instance, using holomorphic curve methods, we can classify the symplectic fillings of S^1-invariant contact structures on any circle bundle over a surface (joint work with Sam Lisi and Jeremy Van Horn-Morris). One can also use them to compute an invariant that lives in Symplectic Field Theory and measures the “degree of tightness” of a contact manifold (joint work with Janko Latschev).

Yuji Odaka (Imperial). Towards compact algebraic moduli of singular cscK varieties. Friday November 30th. Huxley 130, 1.30-2.30pm.

The speaker proposes a conjecture that moduli variety of K-polystable polarized varieties exist, whose components are projective (K-moduli). This unifies following backgrounds with different origins: (1) General type (K>0) case: the construction, originally via birational geometry (Kollár—Shepherd-Barron etc.). (2) Infinite dimensional Kempf-Ness picture on space of complex structures (Fujiki, Donaldson). (3) Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds (Donaldson-Sun). (4) Generalized Weil-Petersson metrics (Fujiki-Schumacher). We roughly discuss conceivable approach to solve the conjecture, while putting focus on concrete examples with pictures. They will include at least del Pezzo case from the viewpoint (3) (j.w.w. Spotti, Sun) and if time permits some from (1) and others.

Simon Donaldson (Imperial). Existence of Kähler-Einstein metrics on Fano manifolds. Friday December 7th. Huxley 130, 1.30-2.30pm.

The talk will be based on recent joint work with Chen and Sun on the existence of Kähler-Einstein metrics on “stable” Fano manifolds. We will discuss some aspects of the proof and prospects for further work in this direction.

Junwu Tu (Oregon). Mirror construction via gluing of Maurer-Cartan moduli. Friday December 14th. Huxley 130, 1.30-2.30pm.

In this talk I will describe an algebraic approach to solve the reconstruction problem in mirror symmetry. In the end we get a rigid analytic space whose transition functions involve counting invariants of pseudo-holomorphic disks.

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