Spring Term 2017

Dave Morrison (UCSB). Degenerations of K3 surfaces, gravitational instantons, and M-theory. Friday 13th Jan., 1:30-2:30pm. Huxley 213.

Abstract: The detailed study of degenerations of K3 surfaces as complex manifolds goes back more than forty years and is fairly complete. Much less is known about the analogous problem in differential geometry of finding Gromov–Hausdorff limits for sequences of Ricci-flat metrics on the K3 manifold. I will review recent work of H.-J. Hein and G. Chen–X. Chen on gravitational instantons with curvature decay, and descirbe applications to the K3 degeneration problem. M-theory suggests an additional geometric structure to add, and I will give a conjectural sketch of how that structure should clarify the limiting behavior.

Michel van Garrel (KIAS). From local to relative Gromov-Witten invariants via log geometry. Friday 20th Jan., 1:30-2:30pm. Huxley 341.

Abstract: Let X be a smooth projective variety and let L be a line bundle corresponding to a smooth ample divisor D. In this joint work with Graber and Ruddat, we show that the genus zero local Gromov-Witten invariants of L are maximally tangent relative Gromov-Witten invariants of the pair (X,D). This generalizes an old formula of Takahashi. The key technical ingredient is the theory of log stable maps by Gross and Siebert.

Dan Pomerleano (Imperial College). Two, infinity and beyond . Friday 27th Jan., 1:30-2:30pm. Huxley 341.

Abstract: I will sketch a proof that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. Key ingredients in the proof are the isomorphism between embedded contact homology and Seiberg-Witten Floer cohomology as proven by Taubes, an identity recovering the contact volume from the lengths of certain Reeb orbit sets, and the theory of global surfaces of section as developed by Hofer-Wysocki-Zehnder. This is joint work with Daniel Cristofaro-Gardiner and Michael Hutchings.

Jason Lotay (University College London). Invariant G_2-instantons. Friday 3rd Feb., 1:30-2:30pm. Huxley 341.

Abstract: Since Donaldson-Thomas proposed a programme for studying gauge theory in higher dimensions, there has been significant interest in understanding special Yang-Mills connections in Ricci-flat 7-manifolds with holonomy G_2 called G_2-instantons. However, still relatively little is known about these connections, so we begin the systematic study of G_2-instantons in the SU(2)^2-invariant setting. We provide existence, non-existence and classification results, and exhibit explicit sequences of G_2-instantons where “bubbling” and “removable singularity” phenomena occur in the limit. This is joint work with Goncalo Oliveira (Duke).

Zsolt Patakfalvi (École polytechnique fédérale de Lausanne). Characterization of (ordinary) abelian varieties in positive characteristic. Friday 10th Feb., 1:30-2:30pm. Huxley 341.

Abstract: I am going to present a joint work (partially in progress) with Christopher Hacon aiming to give a birational characterization of (ordinary) abelian varieties in positive characteristic using birational invariants, such as the first Betti number and the Kodaira dimension. Over the complex number this was shown by Kawamata in ’81, where the precise statement was that a smooth projective variety is birational to an abelian variety if and only if the first Betti number equals twice the dimension, and the Kodaira dimension is zero. I will explain the importance of such results, the reasons why the positive characteristic versions are harder and why they became approachable only recently, as well as our results that solve the above problem for ordinary abelian varieties, and solve half of the problem for general abelian varieties.

Thibaut Delcroix (École normale supérieure – Paris). K-stability of Fano spherical varieties. Friday 17th Feb., 1:30-2:30pm. Huxley 341.

Abstract: The resolution of the Yau-Tian-Donaldson conjecture for Fano manifolds, that is, the equivalence of the existence of Kähler-Einstein metrics with K-stability, raises the question of determining when a given Fano manifold is K-stable.
I will present a combinatorial criterion of K-stability for Fano spherical manifolds. These form a very large class of almost-homogeneous manifolds, containing toric manifolds, homogeneous toric bundles, and classes of manifolds for which the Kähler-Einstein existence question was not solved yet, for example equivariant compactifications of (complex) symmetric spaces.

Alexandru Dimca (Université de Nice). On a theorem of Francesco Severi on nodal surfaces. Friday 24th Feb., 1:30-2:30pm. Huxley 341.

Abstract: The singular points of a nodal hypersurface of degree are in general position with respect to linear systems of a certain degree.
In my talk I will discuss the plane curve case, going back perhaps to the 19th century, and a result of F. Severi (1946) concerning the nodal surfaces. Related results by D.M. Burns and J. M. Wahl (1974), A. Nobile (1986) and R. Kloosterman (2016) on the deformation of nodal surfaces will also be discussed.

Jean-Pierre Demailly (Université de Grenoble). Compact Kähler manifolds with numerically effective anticanonical bundles. Friday 3rd Mar., 1:30-2:30pm. Huxley 341.

Abstract: The goal of the talk will be to present a few results describing the structure of compact manifolds possessing a nonnegative Ricci curvature, or more generally, a numerically effective anticanonical bundles. (This is based on joint work with Campana and Peternell; we will also present very recent results due to Junyan Cao).

Florin Ambro (Institute of Mathematics of the Romanian Academy). Curves with ordinary singularities. Friday 10th Mar., 1:30-2:30pm. Huxley 341.

Abstract: I will discuss the classification of projective curves with ordinary singularities (simplest kind),
in a way parallel to the classification of projective curves with no singularities.

Kevin McGerty (University of Oxford). Kirwan surjectivity for quiver varieties. Friday 17th Mar., 1:30-2:30pm. Huxley 341.

Abstract: A classical result of Kirwan proves that cohomology ring of a quotient stack surjects onto the cohomology of an associated GIT quotient via the natural restriction map. In many cases the cohomology of the quotient stack is easy to compute so this often yields, for example, generators for the cohomology ring of the GIT quotient. In the symplectic case, it is natural to ask whether a similar result holds for (algebraic) symplectic quotients. Although this surjectivity is thought to fail in general, it is expected to hold in many cases of interest. In recent work with Tom Nevins (UIUC) we establish this surjectivity result for Nakajima’s quiver varieties. An important role is played by a new compactification of quiver varieties which arises from the study of graded representations of the preprojective algebra.

Please note the unusual location:
Elisenda Grigsby (Boston College). Annular Khovanov-Lee homology, braids and cobordisms. Friday 24th Mar., 1:30-2:30pm. Huxley 642.

Abstract: Khovanov homology associates to a knot K in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex. Using a deformation of Khovanov’s complex, due to Lee, Rasmussen defined an integer-valued knot invariant he called s(K) that gives a lower bound on the 4-ball genus of knots, sharp for knots that can be realized as quasipositive braid closures.
On the other hand, when K is a braid closure, its Khovanov complex can itself be realized in a natural way as a deformation of a triply-graded complex, defined by Asaeda-Przytycki-Sikora, further studied by L. Roberts, and now known as the (sutured) annular Khovanov complex.
In this talk, I will describe joint work with Tony Licata and Stephan Wehrli that unifies these two constructions. We obtain a version of the theory particularly well-suited to studying braided cobordisms and braids viewed as mapping classes of the punctured disk.

Allen Knutson (Cornell Univesity). Juggling on the Grassmannian. Friday 7th Apr., 1:30-2:30pm. Huxley 139.

Abstract: The “Bruhat decomposition” of the Grassmannian into cells is well-studied and extremely well-behaved. The common refinement of all n! permutations of it, the “matroid decomposition”, is famously awful.
Lusztig introduced a decomposition in between these for studying positive real geometry, which Postnikov recognized as the refinement of the n cyclic shifts of the Bruhat. I’ll explain how it’s just as good as the Bruhat and in some ways better, and how it naturally arises from characteristic p geometry and from Poisson geometry; also I’ll index its strata by “bounded juggling patterns”. Finally I’ll talk about recent results on the flag manifold. This work is joint with Thomas Lam and David Speyer.