L4

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this work we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

This work was done in collaboration with G. Barrenechea, M. Bosy (Univ. Strathclyde) and F. Nataf, P-H Tournier (Univ of Paris VI)

Symplectic reduction is the natural quotient construction for symplectic manifolds. Given a free and proper action of a Lie group G on a symplectic manifold M, this process produces a new symplectic manifold of dimension dim(M) - 2 dim(G). For non-free actions, however, the result is usually fairly singular. But Sjamaar-Lerman (1991) showed that the singularities can be understood quite precisely: symplectic reductions by non-free actions are partitioned into smooth symplectic manifolds, and these manifolds fit nicely together in the sense that they form a stratification.

Symplectic reduction has an analogue in hyperkähler geometry, which has been a very important tool for constructing new examples of these special manifolds. In this talk, I will explain how Sjamaar-Lerman’s results can be extended to this setting, namely, hyperkähler quotients by non-free actions are stratified
spaces whose strata are hyperkähler.

After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy- Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow. Joint work with F. Cavalletti.

We develop a structure  theory for del Pezzo surfaces that are regular but geometrically non-normal, based on work of Reid, but now independence on the p-degree of the ground field. This leads to existence results, as well as non-existence results for ground fields  of p-degree one. In turn, we  settle questions arising from Koll'ar's analysis on the structure of Mori fiber spaces in dimension three. This is joint work with Andrea Fanelli.

3d N=4 supersymmetric gauge theories provide a method for constructing HyperK\”ahler singularities, known as the Coulomb branch.
This method is complementary to the more traditional way of construction using HyperK\”ahler quotients, known in physics as the “Higgs branch”.
Out of all possible gauge theories there is an interesting subclass of quiver varieties, where the Coulomb branch has been studied in some detail.
Some examples are moduli spaces of classical and exceptional instantons and closures of nilpotent orbits. An interesting feature of Coulomb and Higgs branches is the phenomenon of "3d mirror symmetry” where for a pair of gauge theories, the Higgs branch and Coulomb branch exchange.
There is a large class of “mirror pairs” which I will discuss in some detail.

A topic of recent interest is the notion of implosions. I will argue that there is a simple operation on the quiver which leads to implosion. In other words, given a quiver such that its Coulomb branch is moduli space A, a simple operation of the quiver (making a bouquet) provides the implosion of A.
This has been tested on closures of nilpotent orbits of A type and on nilpotent cones of orthogonal groups and found to agree with the expected results.
If time permits, I will discuss isometries of Coulomb branches

For many moduli problems, in order to construct a moduli space as a geometric invariant theory quotient, one needs to impose a notion of (semi)stability. Using recent results in non-reductive geometric invariant theory, we explain how to stratify certain moduli stacks in such a way that each stratum admits a coarse moduli space which is constructed as a geometric quotient of an action of a linear algebraic group with internally graded unipotent radical. As many stacks are
naturally filtered by quotient stacks, this involves describing how to stratify certain quotient stacks. Even for quotient stacks for reductive group actions, we see that non-reductive GIT is required to construct the coarse moduli spaces of the higher strata. We illustrate this point by studying the example of the moduli stack of coherent sheaves over a projective scheme. This is joint work with G. Berczi, J. Jackson and F. Kirwan.

I will talk about recent work, joint with M. Gröchenig and D. Wyss, on two related results involving the cohomology of moduli spaces of Higgs bundles. The first is a positive answer to a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The second is a new proof of Ngô's geometric stabilization theorem which appears in the proof of the fundamental lemma. I will give an introduction to these theorems and outline our argument, which, inspired by work of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.