Mathematics

Course material

After an introduction to toric geometry, where the main tool will be algebraic geometry, the course will explore the application of techniques from symplectic geometry, such as moment maps and symplectic quotients.

Syllabus:

The course aims at offering a self-contained introduction to complex differential geometry. The focus will be on showing how complex geometry affords powerful methods to study Riemannian notions, in particular the Ricci curvature. Thus we will start with basic notions of Riemannian geometry, such as curvature and harmonic theory, and then see how these take a special form for the class of compact Kähler manifolds.

The course is centered on the Hamiltonian aspects of integrable systems of Ordinary and Partial Differential Equations, with a focus on the geometrical side. After having reviewed the relevant notions of symplectic and Poisson geometry the following issues will be discussed

i) Group actions on Poisson manifolds and the Marsden-Weinstein reduction theorem.

ii) Distributions and the Marsden-Ratiu and Dirac reduction schemes.

iii) Lie-Poisson structures on duals of Lie algebras.

iv) Bihamiltonian structures

The main goal of the course is to provide a concise introduction to 4-manifold topology. A list of topics to be covered is as follows.

The course will discuss the mathematics of many-body quantum mechanics, with a focus on the rigorous derivation of effective theories for complex quantum systems. Topics to be covered include:

-) Introduction to quantum mechanics. The hydrogenic atom. Uncertainty principles, stability of matter of the first kind.

-) Bosons and fermions, density matrices. Introduction to large Coulomb systems, as models for atoms and molecules.

-) Lieb-Thirring inequalities, semiclassical approximations.

The course focuses on the latest ’layer’ Riemannian and Spin of Noncommutative Geometry (NCG). Its central concept, due to A. Connes, is ’spectral triple’ which consists of an algebra of operators on a Hilbert space and an analogue of the Dirac operator. A prototype is the canonical spectral triple of a Riemannian spin manifold which will be described starting with the basic notions of multi-linear algebra and differential geometry.

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of classical localisation formulae in equivariant cohomology. It starts with an introduction to these latter subjects (Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula) and their extension on supermanifolds. It then continues supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex.