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). Motivic DT invariants of quadratic differentials. Friday 5th Nov, 1:30-2:30pm. Huxley 140.
Abstract: The problem of counting saddle connections and closed loops on Riemann surfaces with quadratic differential (equivalently: half-translation surfaces) can be, somewhat surprisingly, reformulated in terms of counting semistable objects in a 3-d Calabi-Yau category with stability condition. Here “counting” happens within the powerful framework of motivic Donaldson-Thomas theory as developed by Kontsevich-Soibelman, Joyce, and others. For meromorphic quadratic differentials with simple zeros, this reformulation is due to the work of Bridgeland-Smith. The case of quadratic differentials without higher order poles – in particular holomorphic ones – requires different methods and was solved in my recent preprint arXiv:2104.06018. As an application, counts of saddle connections and closed loops are related by the wall-crossing formula as one moves around in the moduli space. The talk will aim to be a gentle introduction to this circle of ideas.
Nikon Kurnosov (University College London). Geometry of Bogomolov-Guan manifolds. Friday 12th Nov, 1:30-2:30pm. Huxley 140.
Abstract: Constructing new compact holomorphic symplectic manifolds has always been an interesting question. I will describe Bogomolov-Guan approach to construct non-Kähler holomorphic symplectic simply-connected manifold. It emphasizes the analogy with the Kodaira example of non-Kähler symplectic surfaces. In this talk we will discuss the geometry of such manifolds: their submanifolds and algebraic reduction to P^n; and the deformation theory of them, which is quite similar to that of the hyperkähler case.
The seminar on 19th Nov has been canceled.
Selim Ghazouani (Imperial College). Existence of closed geodesics for affine structures on surfaces. Friday 26th Nov, 1:30-2:30pm. Huxley 140.
Abstract: It is a classical result in Riemannian geometry that every compact manifold carries closed geodesics. In this talk we will discuss the analogous question for affine manifolds. We will quickly turn to the case of surfaces with singularities, and try to highlight the links to Teichmüller theory and classical dynamical systems. If time permits, I will allow myself to advertise new results joint with A. Boulanger and G. Tahar.
Jeff Carlson (Imperial College). The topology of the Gelfand–Zeitlin fiber. Friday 3rd Dec, 1:30-2:30pm. Huxley 140.
Abstract: Gelfand–Zeitlin systems are a well-known family of examples in symplectic geometry, singular Lagrangian torus fibrations whose total spaces are coadjoint orbits of an action of a unitary or special orthogonal group and whose base spaces are certain convex polytopes. They are easily defined in terms of matrices and their truncations, but do not fit into the familiar framework of integrable systems with nondegenerate singularities, and hence are studied as an edge case.
It is known due to work of Cho, Kim, and Oh that the fibers of these systems are determined as iterated pullbacks by the combinatorics of joint eigenvalues of systems of truncated matrices, but the resulting expressions can be rather inexplicit. We provide a new interpretation of Gelfand–Zeitlin fibers as balanced products of Lie groups (or biquotients), and pursue these viewpoints to a determination of their cohomology rings and low-dimensional homotopy groups which can be read transparently off of the combinatorics.
This all represents joint work with Jeremy Lane.
Rachel Webb (UC Berkeley). Abelianization and quantum lefschetz for orbifold I-functions. Friday 10th Dec, 1:30-2:30pm. Huxley 140.
Abstract: Let G be a connected reductive group with maximal torus T, and let V and E be two representations of G. Then E defines a vector bundle on the orbifold V//G; let X//G be the zero locus of a regular section. The quasimap I-function of X//G encodes the geometry of maps from P^1 to X//G and is related to Gromov-Witten invariants of X//G. By directly analyzing these maps from P^1, we explain how to relate the I-function of X//G to that of V//T. Our formulas validify a mirror symmetry computation of Oneto-Petracci that relates the quantum period of X//G to a certain Laurent polynomial defined by a Fano polytope. Finally, we describe a large class of examples to which our formulas apply, examples that are an orbifold analog of quiver flag varieties.