Eckhard Meinrenken (Toronto)
On multiple clean intersections abstract
Abstract:
Two submanifolds N_1,N_2 of a manifold intersect cleanly if their intersection is a submanifold, with tangent bundle T(N_1\\cap N_2)=TN_1\\cap TN_2. This then implies that locally, the intersection is modelled by the intersection of linear subspaces. What is the right notion of clean intersection if more than two submanifolds are involved? I will explain that there is a good generalization, and present some applications to symplectic geometry and simplicial manifolds. (Joint work with Kain Dineen.)
11:30 • Université de Genève, Conseil Général 7-9, Room 1-15
Xiaomeng Xu (Beijing)
Quantization of Riemann-Hilbert-Birkhoff maps abstract
Abstract:
This talk introduces quantum Stokes matrices of meromorphic linear systems of ordinary differential equations with a pole of order $p+1$ and noncommutative coefficients. It then proves that these quantum Stokes matrices satisfy natural quantum exchange relations. These relations allow us to interpret the quantum Stokes matrices as an associative algebra homomorphism, which may be viewed as a quantization of the Riemann-Hilbert-Birkhoff map, regarded as a Poisson map, for meromorphic connections.
14:30 • Université de Genève, Conseil Général 7-9, Room 1-15
Adrien Busnot Laurent (INRIA Rennes)
An overview of recent algebraic and numerical results on aromatic structures abstract
Abstract:
Aromatic Butcher series are the natural extension of the standard B-series that allow to compute the divergence of a Taylor expansion over the jet space of a given ODE. First used as a tool for the study of volume-preserving numerical integrators, they were then studied for their numerous algebraic properties.In this talk, we will introduce aromatic structures step by step with freeness results and applications. Then, we shall present the aromatic bicomplex, in the spirit of the variational bicomplex in variational calculus, characterise Ker(Div) on aromatic trees, derive negative results on volume-preserving methods, and formulate the notion of symmetries and Noether theorems for tree-like structures. We will extend our approach to the preservation of measures for stochastic dynamics with a focus on the algebraic structure. If time allows, we will give an overview of related algebraic structures, that are aromatic multi-indices, Hopf algebroids, and planar aromatic trees, and discuss how they are used to tackle the open problems of geometric integration concerning measure-preservation.
15:00 • Université de Genève, Conseil Général 7-9, Room 1-05