Spring Term 2018

Mark Powell (Durham University). Stable diffeomorphism of 4-manifolds. Friday 12th Jan., 1:30-2:30pm. Huxley 139.

Abstract: We say that two 4-manifolds are CP^ – stably diffeomorphic if one can add copies of the complex projective plane, with either orientations, to both manifolds until they become diffeomorphic. I will explain how to determine whether two manifold agree modulo this equivalence relation. I may also discuss the related question where stabilisation by copies of S^2 x S^2 is permitted instead.

Jorge Pereira (IMPA). Deformation of rational curves along foliations. Friday 19th Jan., 1:30-2:30pm. Huxley 139.

Abstract: Foliations on an algebraic variety X naturally determine foliations on the spaces of morphisms to X. In general, these foliations are trivial, i.e. they are nothing but foliations by points. When nontrivial they are expected to impose strong constraints on the original foliation. The talk will review situations where such expectations have been turned into results.

Alessandro Chiodo (Institut de Mathématiques de Jussieu). Mirror pairs and automorphisms. Friday 26th Jan., 1:30-2:30pm. Huxley 139.

Abstract: The first occurrences of mirror symmetry paired two Calabi–Yau threefolds X and X’ and set a relation between H^{2,1} of X and H^{1,1} of X’. We say that H*(X) “mirrors” H*(X’). In this talk we look at some pairs of Calabi-Yau orbifolds with two automorphisms x and x’ of the same order N. Within X, we consider the fixed space X_m with respect to any power x^m of x. We also consider the decomposition E_0+E_1+…+E_{N-1} of the Hodge structures of X_m as a Z/N-representation with respect to the action by x. We will present several statements involving mirrors of fixed loci and of fixed cycles. For instance, for N=m*l (with m and l proper divisors of N), the eigenspace E_l of the cohomology of X_m “mirrors” the eigenspace E_m of the cohomology of X’_l. This is work in collaboration with Kalashnikov and Veniani.

Ailsa Keating (University of Cambridge). On symplectic stabilisations and mapping classes. Friday 2nd Feb., 1:30-2:30pm. Huxley 139.

Abstract: In real dimension two, the symplectic mapping class group of a surface agrees with its `classical’ mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, …,v_k and V_1, …,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

Navid Nabijou (Imperial College). Relative quasimaps and a Lefschetz-type formula. Friday 9th Feb., 1:30-2:30pm. Huxley 139.

Abstract: The theory of stable quasimaps is an important tool in modern enumerative geometry, providing an alternative system of curve counts to the usual Gromov-Witten invariants. In joint work with Luca Battistella, we define moduli spaces of relative stable quasimaps to a pair (X,Y), where Y is a hyperplane section in X. Intuitively these spaces parametrise quasimaps in X with specified orders of tangency with Y, and can be used to define relative quasimap invariants. By investigating these moduli spaces we obtain a Lefschetz-type formula, expressing certain quasimap invariants of Y in terms of the invariants of X. Since the I-function from mirror symmetry is equal to a generating function for these quasimap invariants, this result can be viewed as a “quantum Lefschetz theorem for I-functions.” It also agrees with an earlier formula obtained by Ciocan-Fontanine and Kim.

Morgan Brown (University of Miami). The Skeleton of a Product of Degenerations. Friday 16th Feb., 1:30-2:30pm. Huxley 139.

Abstract: The essential skeleton is an invariant of a degeneration that appears in both Berkovich geometry and minimal model theory. I will show that for degenerations with a semistable model, the essential skeleton of a product of degenerations is the product of their skeleta.
As an application, we are able to describe the homeomorphism type of some degenerations of hyperkähler varieties, both for a Hilbert scheme of degenerating K3 surfaces and for a Kummer variety associated to a degeneration of an abelian surface. This is joint work with Enrica Mazzon.

Yang Li (Imperial College). Mukai duality in G2 geometry. Friday 23rd Feb., 1:30-2:30pm. Huxley 139.

Abstract: G2 geometry can be thought of as the real 7-dimensional cousin of Calabi-Yau 3-folds or hyperkahler K3 surfaces, and its importance arises from Riemannian geometry, submanifold theory, gauge theory and physics. A recent proposal of Donaldson suggests studying G2 manifolds with a coassociative K3 fibration from the formal adiabatic limiting viewpoint, where the fibre size is formally taken to zero. It is then interesting to ask if Donaldson’s adiabatic fibrations can be dualised by replacing its K3 fibres with their Mukai dual K3 surfaces, what geometric structures we can get on the new fibration, and how geometries (associative submanifolds, gauge theory) on both sides are related. This talk aims to provide a positive answer to some of these questions by working over a local base, thus giving a mathematical interpretations to certain physical speculations of Gukov-Yau-Zaslow. For algebraic geometers, this is formally analogous to some aspects of relative Fourier-Mukai partner.

Yankı Lekili (King’s College). Homological mirror symmetry for K3 surfaces via moduli of A_\infty structures. Friday 2nd Mar., 1:30-2:30pm. Huxley 139.

Abstract: Around 2010 in joint work with Perutz, as a by-product of our proof of homological mirror symmetry for the once-punctured torus, we identified moduli of elliptic curves P(4,6) with a moduli of A_\infty structures on a finite-dimensional graded algebra. Generalizations of this story that covers other moduli of curves were subsequently pursued by Polishchuk. Again, inspired by mirror symmetry, in joint work with Polishchuk, we exhibited Smyth’s modular compactifications of M_{1,n} as moduli of A_\infty structures (and revealed new properties of these moduli spaces that were previously unknown). My talk will begin by surveying these results and pinpointing their common features.
In an ongoing work with Ueda, I am working on a higher dimensional generalization of this story. I will give sample examples in dimension two, identifying certain lattice polarized moduli of K3 surfaces with moduli of A_\infty structures. A proof of homological mirror symmetry for these K3 surfaces is on the horizon. (This is a different proof than Seidel’s, and Sheridan-Smith’s and covers some non-overlapping examples.)

Camille Laurent-Gengoux (Université de Lorraine). Symplectic groupoids and symplectic resolutions. Friday 9th Mar., 1:30-2:30pm. Huxley 139.

Abstract: Poisson brackets are a very natural object in mathematical physics, in particular in Hamiltonian mechanics.
The most regular and smooth example of Poisson brackets are those associated to symplectic manifolds.
However, Poisson structures do not need to be symplectic. They can be defined on a singular set, like an algebraic variety, but they can also have singularities of their own, even at smooth points. Several schools of mathematics have addressed the following question: can one replace a “bad” singular Poisson structure by a “nice” smooth symplectic one? The question makes sense in particular in algebraic geometry, where it is tempting to look for an equivalent to the Hironaka’s theorem for Poisson algebraic varieties.
There is no such a result in general, but algebraic Poisson varieties that do admit such resolutions have a rich geometry.
But the question also makes sense in the theory of integrable systems, where the relevant object was shown a long time ago to be the symplectic Lie groupoid that “integrates” the Lie bracket that a Poisson bracket is. We shall try to connect both manners to make a Poisson structure sympletic.
A list of surprisingly simple conjectures will be given.

Sam Payne (Yale University). Tropical curves, graph complexes, and top weight cohomology of M_g. Friday 16th Mar., 1:30-2:30pm. Huxley 139.

Abstract: I will discuss the topology of a space of stable tropical curves of genus g with volume 1. The reduced rational homology of this space is canonically identified with the top weight cohomology of M_g and also with the homology of Kontsevich’s graph complex. As one application, we show that H^{4g-6}(M_g) is nonzero for infinitely many g. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of a recent theorem of Willwacher, that homology of the graph complex vanishes in negative degrees, using the identifications above and known vanishing results for M_g. Joint work with M. Chan and S. Galatius.

Charles Favre (École Polytechnique). Degeneration of Monge-Ampere measures. Friday 23rd Mar., 1:30-2:30pm. Huxley 139.

Abstract: We shall explain how the construction of a hybrid space by Berkovich enables one to understand the degeneration of natural measures associated to one-parameter families of dynamical systems of various sorts.