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Geometry and Mathematical Physics

∙ Integrable systems in relation with differential, algebraic and symplectic geometry, as well as with the theory of random matrices, special functions and nonlinear waves, Frobenius manifolds • Deformation theory, moduli spaces of sheaves and of curves, in relation with supersymmetric gauge theories, strings, Gromov-Witten invariants, orbifolds and automorphisms
• Quantum groups, noncommutative Riemannian and spin geometry, applications to models in mathematical physics
• Mathematical methods of quantum mechanics
• Mathematical aspects of quantum Field Theory and String 
Theory
• Symplectic geometry, sub-riemannian geometry

• Geometry of quantum fields and strings

Differential Geometry

In the course we will cover the main ideas from Riemannian geometry (distances, volumes, tubes, ...) with a point of view on metric and topological properties of real algebraic varieties.  The list of topics to be covered includes:

Random matrices

We will discuss the main facts of random matrix theory from the viewpoint of mathematical analysis. We will introduce the Gaussian Unitary Ensemble as the main example and study its properties. Its properties are shared by many other ensembles of random matrices due to universality. We will emphasise the connection to orthogonal polynomials and asymptotic analysis (steepest descent and Riemann-Hilbert problem methods).

Rooms:

  • Thursdays room 134
  • Fridays room 137

Quiver varieties

Programme

  • Basic definitions
  • Gabriel’s theorem
  • Double quivers
  • Quiver representations
  • Quiver varieties
  • Hilbert schemes of points on complex surfaces as quiver varieties
  • The ADHM description of the moduli space of instantons on R4
  • Instantons on ALE spaces and the geometric McKay correspondence

Basic references

C* algebras that one can see

C*-algebras are operator algebras that form the conceptual foundation of noncommutative geometry. Since commutative C*-algebras yield categories anti-equivalent to categories of locally compact Hausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebras are viewed as function algebras on quantum spaces. Their study from this point of view is referred to as noncommutative topology. Here KK-theory and index theory are among prime tools leading to significant applications.

Frobenius manifolds

The aim of the course is to introduce the audience to the analytic theory of Dubrovin-Frobenius manifolds.

Linear ODEs in the complex domain and isomonodromy deformations

The aim of the course is to introduce the audience to linear systems of differential equations in the complex domain, their monodromy and essential monodromy data. Then, the theory of isomonodromy deformations will be studied, with applications to some examples. These notions play an important role in modern mathematical physics, for example in integrable systems.
  •     Existence and uniqueness theorems in the complex domain.
  •     Linear systems
  •     Singularities and monodromy

Integrable systems

Integrable systems are special  systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems, nonlinear wave equations   and probability. This  course covers the origins of the subject as well as modern topics in  nonlinear waves  and  integrable probability.

1.  The Korteweg de Vries equation (KdV) 

Self-adjoint operators in quantum mechanics

The course will give an overview on the operator-theoretical tools needed for a rigorous formalization and treatment of a quantum mechanical model with special emphasis on unbounded self-adjoint operators.

Topics:

Hamiltonian methods for integrable systems

The course is centered on the Hamiltonian aspects of integrable systems of Ordinary and, especially, Partial Differential Equations, with a focus on the geometrical side.

Integrability will mean

Existence of a “sufficient number” of conservation laws.

Contents & schedule

Pages

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