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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

Standard and less standard asymptotic methods

  In every branch of mathematics, one is sometimes confronted with the problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of  formation of a sequence of numbers {A_n} of which one knows, for  instance, the first couple of hundred values.  The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much  less so, with lots of examples.   Here are three typical examples: 1.

Introduction to Topological Recursion theory and moduli spaces of curves

Topological Recursion (TR) can be thought as a universal procedure, or algorithm, to generate solutions of enumerative geometric problems related directly or indirectly to moduli spaces of curves. 
 - the input is the so-called spectral curve (think e.g. an algebraic curve with two particular meromorphic functions on it), 
 - the output is an infinite list of numbers (think e.g. Gromov-Witten invariants of some kind).

Introduction to Noncommutative Geometry

 These lectures focus 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.

Noncommutative Geometry: Banach and C∗-algebras + Selected topics

The first part of the course will supply some basic materials in functional analysis that are frequently used in noncommutative geometry. More details can be found in the attached syllabus.

The lectures will be streamed via Teams with the same link ( Tuesday and Thursday 11-13 weekly, the last one is the week of Oct 25):

Riemann surfaces and integrable systems

  1. Riemann surfaces: definitions and examples
  2. Holomorphic and meromorphic functions on Riemann surface
  3. Compact Riemann surface: genus, monodromy, homology
  4. Differentials on Riemann surface and integration
    • Riemann bilinear relation
    • Jacobi variety and Abel theorem
    • Divisors and Riemann-Roch theorem
  5. Jacobi inversion problem and theta functions
  6. Integrable systems: the Toda Lattice with periodic boundary conditions
    • Integrable systems with random initial data and connection with the theory of random ma

Point processes and random measures

The course will give an elementary introduction to random measures. The key example will be point processes of random matrix theory, especially determinantal and pfaffian point processes. Connections with ergodic theory, asymptotic combinatorics, harmonic analysis on infinite-dimensional homogeneous spaces will be emphasized.

Volterra series, shuffles and permutations

Volterra series are universal asymptotic expansions of solutions of well-posed evolution equations. These series have rich algebraic and combinatorial structures which we will focus on. The mentioned structures are all related to the combinatorial analysis of words, trees and permutations.

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