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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

Topics in complex geometry

The aim of the course is to give an introduction to some topics of current interest in the global geometry of compact Kähler manifols.

Part I – Basic notions. Hermitian and Kähler manifolds, harmonic theory, Kodaira Vanishing and Embedding, Kempf-Ness Theorem, notions of canonical Kähler metrics.

Some textbooks:

3-dimensional mirror symmetry and elliptic cohomology

This is an advanced course that will explore connections between recent developments in geom- etry, mathematical physics and homotopy theory. Its format will be that of a working seminar. Mirror symmetry has been at the center stage of geometry for the last thirty years. It is a sophisticated dictionary relating the symplectic geometry of a variety X and the algebraic ge- ometry of its mirror Y , and vice versa.

Algebraic geometry

 The course aims to introduce the student to the language of schemes, the central object of study in modern algebraic geometry.At the end of the course the student will master the basic dictionary, shared by all algebraic geometers,regarding the theory of schemes, affine and projective varieties, their morphisms and cohomology of coherent sheaves on them.All fundamental results will be presented with fully detailed proofs, and exercises will be proposed constantly during the course. 

Mathematical methods for solid state physics

The transport properties of topological quantum matter are of central importance in Mathematical Physics, both from a fundamental viewpoint and for technological applications.
While we are still far from a complete, fundamental theory of topological transport based on a microscopic dynamical law, the last few years witnessed important progresses in the understanding of several specific questions in this context.

Enumerative geometry and quasi-modular forms

Exapended title: A rapidly divergent tale about Hurwitz, Lambert, Frobenius, enumerative geometry and quasi modular forms.

The course starts in April. Dates of the course and course notes are available here

Indicative program:

Week 1: Introduction to Hurwitz theory, enumeration of graphs for fixed degree

Week 2: The Lambert Space, Enumeration of Hurwitz numbers for fixed degree

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.

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