Autumn Term 2014

Yoshonori Gongyo (Imperial College). Pluricanonical representation, Extension, and the abundance conjecture. Friday Oct 10th, Huxley 130, 1.30-2.30pm.

Abstract:I will talk about an approach to the abundance conjecture. In particular, I will discuss about the finiteness of the pluricanonichal representation and a version of an extension theorem for adjoint bundles. These are joint works with O.Fujino and S. Matsumura.

Ken Baker (University of Miami). Unifying unexpected exceptional Dehn surgeries. Friday Oct 17th, Huxley 130, 1.30-2.30pm.

Abstract:  This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.  Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families.  In this talk we’ll explicitly describe our construction and discuss related applications of the technique.

Note also the event Groups and Geometry in the South East on Friday Oct. 17th.

Boris Dubrovin (SISSA, Trieste). On quantum integrable systems, symplectic field theory, and Schur polynomials. Friday Oct 24th, Huxley 130, 1.30-2.30pm.

Abstract: We will consider quantisation of Hopf equation considered as an integrable Hamiltonian system with infinite number of degrees of freedom. Eigenvectors of the commuting quantum Hamiltonians are given by Schur polynomials. Connections of the quantisation problem with symplectic field theory will be explained.

Gabriele Mondello (Università di Roma “Sapienza”). On the Dolbeault cohomological dimension of the moduli space of Riemann surfaces. Friday Oct 31st, Huxley 130, 1.30-2.30pm.

Abstract: The moduli space M_g of Riemann surfaces of genus g is (up to a finite étale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension (i.e. the highest k such that H^{0,k}(M_g,E) does not vanish for some holomorphic vector bundle E on M_g). The conjecturally optimal bound is g-2, which is verified for g=2,3,4,5.
I can prove that such dimension is at most 2g-2. The key point is to show that the Dolbeault cohomological dimension of each stratum of translation surfaces is at most g (still non-optimal bound).
In order to do that, I produce an exhaustion function, whose complex Hessian has controlled index: in the construction of such a function basic geometric properties of translation surfaces come into play.

Jonny Evans (UCL).  Floer Theory of Homogeneous Lagrangian Submanifolds. Friday Nov 7th, Huxley 130, 1.30-2.30pm.

Abstract: Joint work with YankI Lekili. If one is interested in studying Lagrangian submanifolds and Fukaya categories, Lagrangian orbits of Hamiltonian group actions form a rich class of examples where one can hope to say something about the Floer cohomology. This includes toric fibres as well as nonabelian orbits such as RP^n in CP^n and more exotic examples like E_6/F_4.Z/3 in CP^26 or the Chiang Lagrangian in CP^3 (if CP^3 is the space of triples of points in S^2 then the Chiang Lagrangian is the subset of triples with centre of mass at the origin). I will discuss some approaches to studying the Floer theory of homogeneous Lagrangians and some examples where we can calculate everything.

Blaine Lawson (Stony Brook). Potential Theory for Nonlinear PDE’s. Friday Nov 14th, Huxley 130, 1.30-2.30pm.

Abstract: There is an interesting potential theory  associated to each degenerate elliptic, fully nonlinear equation of the form $f(D^2u) = 0$. For the standard complex Monge-Ampère equation, it is just the classical pluripotential theory.  I will explain how these theories are defined in general.
Fundamental to the analysis  is a new invariant of such equations,  called the Riesz characteristic, which governs asymptotic structures.
The notions of tangents to subsolutions and densities will  be introduced. Results concerning existence and uniqueness of tangents, the structure of sets of high density points, and the  regularity  of  subsolutions for  certain Riesz characteristics, will be discussed.  The Dirichlet problem with prescribed asymptotics will be treated. I will also  touch upon the question of removable singularities.
Examples include real, complex and quaternionic Hessian equations, the p-convexity equation, and equations from calibrated geometry. In particular, this establishes a potential theory on every calibrated manifold.

Michael McQuillan (Università di Roma “Tor Vergata”). 2-Galois theory. Friday Nov 21st, Huxley 130, 1.30-2.30pm.

Abstract: A theorem of Whitehead asserts that the topological 2-type of a (connected) space is uniquely characterised by the triple (\pi_1, \pi_2, k_3), where the \pi_i, i\leq 2 are the homotopy groups \pi_i, i\leq 2, k_3 is the Postnikov class \in H^3(pi_1, \pi_2), and, indeed all such triples may be realised. Such triples are synomous with a 2-group, \Pi_2, i.e. a group `object’ in the category of categories, which plays the same role for 2-types as the fundamental group does for 1-types. In particular, there is a 2-Galois correspondence between the 2-category of champs which are etale fibrations over a space and \Pi_2 equivariant groupoids generalising the usual 1-Galois correspondence between spaces which are etale fibrations over a given space and \pi_1 equivariant sets.
The talk will explain the pro-finite analogue of this correspondence, so, albeit only for the 2-type, a much simpler and more generally valid description of the etale homotopy than that of Artin-Mazur.

Weiyi Zhang (University of Warwick). Geometric structures, Gromov norm and Kodaira dimensions. Friday Nov 28th, Huxley 130, 1.30-2.30pm.

Abstract: Kodaira dimension provides a very successful classification scheme for complex manifolds. The notion was extended to symplectic 4-manifolds. In this talk, we will define the Kodaira dimension for 3-manifolds through Thurston’s eight geometries. This is compatible with other Kodaira dimensions in the sense of “additivity”. We will then explore the relations of geometric structures and mapping orders with various Kodaira dimensions and other invariants like Gromov norm.

Nero Budur (University of Leuven). Differential graded Lie algebra pairs. Friday Dec 5th, Huxley 130, 1.30-2.30pm.

Abstract: In practice, any deformation problem over fields of characteristic zero is governed by a differential graded Lie algebra (DGLA). Following Deligne, Goldman-Millson and Simpson described via DGLAs the local structure of moduli spaces for various geometric situations. Given an object with a notion of cohomology theory, how can one describe all its deformations subject to cohomology constraints? We will present an approach via DGLA pairs. As applications, we discuss the structure of cohomology jump loci of vector bundles and of local systems. This is joint work with Botong Wang.

Richard Thomas (Imperial College). The Katz-Klemm-Vafa formula. Friday Dec 12th, Huxley 130, 1.30-2.30pm.

Abstract: I will describe joint work with Rahul Pandharipande proving the Katz-Klemm-Vafa conjecture. This expresses the Gromov-Witten theory of K3 surfaces and K3-fibred 3-folds — in all genera and for all multiple covers — in terms of modular forms. We use Pandharipande-Pixton’s proof of the MNOP conjecture for many 3-folds and a new result on stable pairs and Noether-Lefschetz loci to translate the problem into a computation of stable pair invariants. This is carried out using a degeneration formula and a vanishing result. I will explain all of the above words…