SR1

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative mixed Hodge structure on the cohomology of the Fukaya category. I will discuss computable Hodge theoretic invariants arising from twist functors, and from geometric extensions. I will also explain how the instanton-corrected Chern-Simons theory fits in the framework of normal functions in non-commutative Hodge theory and will give applications to explicit descriptions of quantum Lagrangian branes. This is a joint work with L. Katzarkov and M. Kontsevich.

There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used

as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties.

The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles.

I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities.

I will talk about a recent paper of Huh, who, building on a wealth of pretty geometry and topology, has given a proof of a conjecture dating back to 1968 regarding the chromatic polynomial (the polynomial that determines how many ways there are of colouring the vertices of a graph with n colours in such a way that no vertices which are joined by an edge have the same colour). I will mainly talk about the way in which a problem that is explicitly a combinatorics problem came to be encoded in algebraic geometry, and give an overview of the geometry and topology that goes into the solution. The talk should be accessible to everyone: no stacks, I promise.