Mathematics

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):

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

The theory of point process is soon turning 500, yet very many questions remain unanswered.

The first part of the course will give a gentle introduction to point processes, with a special emphasis on determinantal point processes.

The second part will be devoted to corresponding random measures such as the Gaussian multiplicative chaos. We will start from the basics, no prerequisites assumed and will go on to topics of current research.

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.

In this course I will present a set of techniques which make it
possible to rigorously implement the renormalization group for several
euclidean Fermionic field theories and related models in equilibrium
statistical physics (such as planar Ising and dimer models).  This will
mainly be illustrated with the example of the planar Ising universality
class.

:This course aims to show that many PDEs studied in differential geometry can be considered as moment map equations for some infinite-dimensional group action. This point of view gives a unifying perspective for the study of various problems and can provide important insights by establishing a parallel with known phenomena in finite dimensions. We will be most interested in explaining a formal connection between the existence of solutions to PDEs of geometric interest and algebraic stability conditions, along the lines of the Kempf-Ness Theorem.

Smooth Ergodic Theory is the study of dynamical systems on smooth manifolds from a probabilistic and statistical perspective.

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