Summer Term 2023

Will Donovan (Tsinghua University). Homological comparison of crepant resolutions and smooth ambient spaces. Friday 28th Apr, 1:30-2:30pm. Huxley 340.

Abstract: Given a singularity with a crepant resolution, a symmetry of the derived category of coherent sheaves on the resolution may often be constructed, with applications to homological mirror symmetry and enumerative geometry. I relate such constructions to the derived category of a smooth ambient space for the given singularity. This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.

Laura Wakelin (Imperial College). The Dehn surgery characterisation of Whitehead doubles. Friday 5th May, 1:30-2:30pm. Huxley 340.

Abstract: A slope p/q is said to be characterising for a knot K in the 3-sphere if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. In this talk, I will discuss how JSJ decompositions and hyperbolic techniques can be used to find characterising slopes for a special class of satellite knots which includes Whitehead doubles. Time permitting, I will also exhibit a family of pairs of satellite knots sharing a non-characterising slope of the form 1/q.

Noah Porcelli (Imperial College). Floer theory and framed cobordisms between exact Lagrangians. Friday 12th May, 1:30-2:30pm. Huxley 340.

Abstract: Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more- we show it can detect smooth structures of certain Lagrangians and in particular recover and extend a result of Abouzaid-Alvarez-Gavela-Courte-Kragh about Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large’s recent construction of a stable-homotopical enrichment of Lagrangian Floer theory. This is based on joint work-in-progress with Ivan Smith.

Umut Varolgunes (Bogazici University). Involutive covers of symplectic manifolds and closed string mirror symmetry. Friday 19th May, 1:30-2:30pm. Huxley 340.

Abstract: Consider a closed graded symplectic manifold M with a finite involutive cover (notion will be reviewed). This gives a canonical spectral sequence that starts from the relative SH of the cover and converges to the quantum cohomology of M. I will discuss the compatibility of this SS with various algebraic structures, the consequences of degeneration at the earliest reasonable page and what it all means in the mirror symmetry context. If time permits I will outline a local to global computation of the A-side Yukawa coupling that is a reinterpretation of its equivalence to the B-side Yukawa coupling in mirror symmetry to illustrate the technique.

Cristina Palmer-Angel (University of Geneva). Globalising Jones and Alexander polynomials via configurations on arcs and ovals in the disc. Friday 26th May, 1:30-2:30pm. Huxley 340.

Abstract: Jones and Alexander polynomials are two important knot invariants and our aim is to see them both from a unified model constructed in a configuration space. More precisely, we present a common topological viewpoint which sees both invariants, based on configurations on ovals and arcs in the punctured disc. The model is constructed from a graded intersection between two explicit Lagrangians in a configuration space. Tt is a polynomial in two variables, recovering the Jones and Alexander polynomials through specialisations of coefficients. Then, we prove that the intersection before specialisation is (up to a quotient) an invariant which globalises these two invariants, given by an explicit interpolation between the Jones polynomial and Alexander polynomial. We also show how to obtain the quantum generalisation, coloured Jones and coloured Alexander polynomials, from a graded intersection between two Lagrangians in a symmetric power of a surface.

Justin Sawon (University of North Carolina at Chapel Hill). Lagrangian fibrations in four dimensions. Friday 2nd June, 1:30-2:30pm. Huxley 340.

Abstract: We consider Lagrangian fibrations by abelian surfaces over the complex projective plane, with total space holomorphic symplectic manifolds and orbifolds. There are examples whose fibres are (1,d) polarized for d=1 to 4. We recall some classification results of Markushevich and Kamenova in the principally polarized case, and a new classification result in the (1,2) polarized case (joint work with Xuqiang Qin). We also describe restrictions on the polarization; indeed, `most’ polarizations are not possible.

No seminar on 9th of June because of the Symplectic Topology Workshop.

Nikolaos Tziolas (University of Cyprus). Automorphism group schemes of surfaces of general type in positive characteristic. Friday 16th June, 1:30-2:30pm. Huxley 340.

Abstract: The automorphism groups scheme of an algebraic variety is a fundamental invariant of an algebraic variety and its structure is intimately related to its moduli theory. In this talk I will present some results about the structure of the automorphism group scheme of a minimal surface of general type defined over an algebraically closed field of positive characteristic. In particular I will emphasize on certain differences between the characteristic zero case and the positive characteristic case, like the existence of surfaces of general type with non reduced automorphism group scheme, a situation that appears exclusively in positive characteristic.

Please note the unusual room:
Maksym Fedorchuk (Boston College). What is a good model of a family of smooth quartic 3-folds over a punctured disk? Friday 23rd June, 1:30-2:30pm. Huxley 140.

Abstract: Consider a family of smooth quartic 3-folds over a punctured disk. How can we complete it without a base change? If we think of a quartic 3-fold as a quartic in P^4, then Kollár’s stability for hypersurfaces gives one way to complete any such family. If we think of it as a smooth Fano 3-fold with Picard rank 1 and anticanonical volume 4, then it is a (2,4)-complete intersection in P(1^5,2) and the generalization of Kollár’s stability for such complete intersections that we developed in joint work with Abban and Krylov gives another (and slightly better) completion to the original family. However, neither of these approaches gives a satisfactory answer. In this talk, I will describe the work in progress on finding a correct Kollár stability setup for this question that leaves the world of complete intersections. If time allows, I will draw analogies with the recently studied K-moduli space of quartic 3-folds.