Autumn Term 2021

Mehdi Yazdi (King’s College). The fully marked surface theorem. Friday 15th Oct, 1:30-2:30pm. Huxley 140.

Abstract: In his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a codimension-1 foliation of a compact 3-manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. We give a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation G such that S is homologous to a union of compact leaves and such that the plane field of G is homotopic to that of F. In particular, F and G have the same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. My previous work, together with our main result, gives a negative answer to Thurston’s conjecture. We mention how Thurston’s conjecture leads to natural open questions on contact structures, flows, as well as representations into the group of homeomorphisms of the circle.
This is joint work with David Gabai.

Lewis Topley (University of Bath). Finite W-algebras and the orbit method. Friday 22nd Oct, 1:30-2:30pm. Huxley 140.

Abstract: Recently Losev has described a version of the orbit method for semisimple Lie algebras, which is a natural map from the set of coadjoint orbits to the space of primitive ideals of the enveloping algebra. The construction involves classifying the quantizations of conic symplectic singularities and considering the affinizations of nilpotent covers which arise from generalized Springer bundles. I will begin this talk by attempting to survey parts of his construction. The finite W-algebra is a natural quantization of the transverse slice to a nilpotent orbit in a semisimple Lie algebra and, although it is a priori unrelated to Losev’s orbit method map, he conjectured that the primitive ideals which appear in the orbit method correspondence can be characterised in terms of the representation theory of finite W-algebras. In the later part of the talk I will explain my proof of his conjecture using techniques from Poisson algebraic geometry, along with the theory of shifted Yangians.

Ruadhaí Dervan (University of Cambridge). Stability conditions for polarised varieties. Friday 29th Oct, 1:30-2:30pm. Huxley 140.

Abstract: A central theme of complex geometry is the relationship between differential-geometric PDEs and algebro-geometric notions of stability. Examples include Hermitian Yang-Mills connections and Kähler-Einstein metrics on the PDE side, and slope stability and K-stability on the algebro-geometric side. I will describe a general framework associating geometric PDEs on complex manifolds to notions of stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as an analogue in the setting of varieties of Bridgeland’s stability conditions on triangulated categories.

Fabian Haiden (University of Oxford). TBA. Friday 5th Nov, 1:30-2:30pm. Huxley 140.

Nikon Kurnosov (University College London). TBA. Friday 12th Nov, 1:30-2:30pm. Huxley 140.

Alastair Craw (University of Bath). TBA. Friday 19th Nov, 1:30-2:30pm. Huxley 140.

Jeff Carlson (Imperial College). TBA. Friday 26th Nov, 1:30-2:30pm. Huxley 140.

Selim Ghazouani (Imperial College). TBA. Friday 3rd Dec, 1:30-2:30pm. Huxley 140.

John Nicholson (Imperial College). TBA. Friday 10th Dec, 1:30-2:30pm. Huxley 140.