Ekaterina Amerik (Paris and HSE). Isotropic boundary of the ample cone. Friday 26th September, 1:30-2:30pm. Huxley 130.

Abstract: This is a joint work with M. Verbitsky and A. Soldatenkov. It follows from an old result of Kovacs that the ample cone of a projective K3 surface of Picard number at least three is either “round” (equal to the positive cone), or has “no round part”, that is the boundary of the ample cone does not contain any open subset of the isotropic cone (and is actually nowhere dense in the isotropic cone). With a help of hyperbolic geometry, this easily generalizes to the hyperkähler case. We pursue this a bit further and prove that any real analytic curve in the projectivization of the isotropic boundary of the ample cone of a projective hyperkähler manifold lies in a sphere contained in this isotropic boundary. We discuss how the union of maximal such spheres, the “Apollonian carpet”, equal to the union of all analytic curves on the projectivized isotropic boundary, can look like. Actually the phenomena discussed are already visible in the world of K3 surfaces.

Sungkyung Kang (Cambridge). An exotic Dehn twist after two stabilizations. Friday 3rd October, 1:30-2:30pm. Huxley 410.

Abstract: Unlike in higher dimensions, most exotic phenomena on simply-connected 4-manifolds are unstable; they become non-exotic after finitely many stabilizations. While we now know that some of them survive one stabilization, nothing is known about their behavior when we stabilize them more than once. In this talk, we present the first example of an exotic diffeomorphism on a smooth contractible 4-manifold, given as a boundary Dehn twist along its (nontrivial) boundary, which stays exotic after two stabilizations. This is an ongoing joint work with JungHwan Park and Masaki Taniguchi.

Kimoi Kemboi (Princeton). The Fano of lines and the Kuznetsov component of cubic fourfolds. Friday 10th October, 1:30-2:30pm. Huxley 658 and Online.

Abstract: A smooth cubic fourfold gives rise to two kinds of hyperkähler fourfolds: one is classical –the Fano variety of lines on the cubic; and the other is “non-commutative” –arising from the symmetric square of the Kuznetsov component. Galkin conjectured that these two objects should be derived equivalent. In this talk, I’ll explain a proof of this conjecture using ideas from matrix factorizations and window categories. This is joint work with Ed Segal.

Daniil Mamaev (LSGNT). Relative Wrapped Fukaya Categories of Surfaces. Friday 17th October, 12:45-13:45. Huxley 130.

Abstract: For a symplectic manifold M the counts of holomorphic discs with Lagrangian boundary conditions can be packaged into an A-infinity category, called Fukaya category. These counts are usually valued in a field and hence all “homological complexity” of the category comes from M. To obtain a relative Fukaya category we replace the coefficient field with a commutative ring R, which allows to non-trivially weight the disc counts by elements of R. The resulting category can be thought of as a family of categories over Spec R, it combines information about M with non-triviality of the family. In the context of homological mirror symmetry this allows to construct low-dimensional mirrors to spaces of higher dimension. I will give examples of this phenomena and sketch a construction for relative wrapped Fukaya categories of (real) surfaces, where all counts are explicit. The construction is geometric and relies on a new version of Chekanov-Eliashberg algebra.

Aleksander Doan (UCL). A Morse complex for the homology of vanishing cycles. Friday 24th October, 12:45-13:45. Huxley 130.

Abstract: This talk is based on joint work with  Juan Muñoz-Echániz. Motivated by the emerging framework of holomorphic Floer theories we provide a finite-dimensional model for such theories by constructing Morse homology groups associated with any regular function on a smooth complex algebraic variety. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and—in the absence of bifurcations at infinity—recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry.  

Riccardo Ontani (Imperial). Counting maps to flag varieties. Friday 31st October, 12:45-13:45. Huxley 130.

Abstract: Quasimap spaces provide tractable compactifications of spaces of maps from a curve into targets that are presented as quotients by group actions. For flag varieties, the relevant quasimap spaces are hyperquot schemes, which parametrise chains of subsheaves of the trivial bundle on the curve. I’ll present a closed, all-genus formula for virtual integrals of tautological classes on these spaces, and discuss conditions under which the resulting virtual counts are genuinely enumerative, recovering the actual number of maps to the flag variety. This talk is based on joint work with Shubham Sinha and Weihong Xu.

Silvia Sabatini (University of Cologne). Positive monotone symplectic manifolds with symmetries and GKM spaces. Friday 7th November, 12:45-13:45. Huxley 130 and Online.

Abstract: Positive monotone symplectic manifolds are the symplectic analogues of Fano varieties, namely they are compact symplectic manifolds for which the first Chern class equals the cohomology class of the symplectic form. In dimension 6, if the positive monotone symplectic manifold is acted on by a circle in a Hamiltonian way, a conjecture of Fine and Panov asserts that it is diffeomorphic to a Fano variety. In this talk I will report on recent results about positive monotone symplectic manifolds endowed with some special Hamiltonian actions of a torus, called GKM, showing that one can prove several finiteness results, which however only use the existence of the action and of an almost complex structure. 

Daniel Huybrechts (University of Bonn). Friday 14th November, 12.45-13.45.

Viveka Erlandsson (University of Bristol). Friday 21st November, 12:45-13:45. Huxley 130.

Erwan Brugallé (Université de Nantes). Welschinger-Witt invariants. Friday 28th November, 12:45-13:45. Huxley 130.

Abstract: Quadratic enumerative geometry is a rapidly expanding field of research. Based on A^1-homotopy theory, it aims to generalize many enumerative problems and computations that are already known over the fields C and R. It turns out that there also exists an alternative method to produce quadratic invariants out of complex and real invariants. It is a quite formal recipe based on Witt invariants rather than A^1-homotopy and geometry. As such, this latter quadratic invariants may be thought as « virtual » enumerative invariants. It turns out that in several cases, these virtual enumerative invariants recover (and extend) the geometric A^1-homotopic enumerative invariants. I will illustrate this phenomenon with the classical problem of enumerating rational curves in the projective plane, and if time permit in any rational surfaces. This is a joint work with Johannes Rau and Kirsten Wickelgren.

Javier Fernandez de Bobadilla (Basque Center for Applied Mathematics). A’Campo spaces and Lagrangian torus fibrations. Friday 5th December, 12:45-13:45. Huxley 130.

Abstract: A’Campo spaces are a hybrid geometry construction recently introduced by T. Pelka and myself. It replaces the central fibre of a normal crossings degeneration by a “radius 0 Milnor fibration” by means of a tropical blow up of the Kato-Nakayama space of the divisorial log-structure. The main advantage over other hybrid geometry constructions is that A’Campo space is a smooth manifold with boundary, with a smooth submersion to the real oriented blow up of the disc; this allows to endow them with a symplectic form. I will explain how to use this construction to produce Lagrangian torus fibrations for any maximal Calabi-Yau degenerations. This fibrations are the arquimedean analogues of the affinoid torus fibrations produced by Nicaise, Xu and Yu.