Mathematics

The course aims at introducing the notion of Random MAtrices and the analysis of their spectral statistical properties. We will study the classical Wigner ensemble with the proof of the celebrated Wigner semicircle law for the eigenvalues. We wil then move on to the definition of more general Unitary Ensembles (where the underlying symmetry is given by the Unitery group) and prove fundamental structural results of Dyson on how to relate their statistical properties to the study of orthogonal polynomials.

C*-algebras are operator algebras forming 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.

The course will present an introduction to the theory of determinantal point processes (DPP) and its use for the solution to the problem of the length of the longest increasing sub- sequence in a large random permutation. A celebrated result, belonging to what is now called ”integrable probability” and first proved by Baik-Deift-Johansson in 1999, asserts that the fluctuation of this length around its average is asymptotically distributed according to the Tracy-Widom distribution, similarly to the largest eigenvalue of a random Hermitian Gauss- ian matrix.

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