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2025

Numbers Theory

Francesco Pappalardi

Summary:

Elementary Numbers Theory: 

  • Quadratic reciprocity
  • Arithmetic fonctions 

Analytic Numbers Theory : 

  • Dirichlet Arithmetic progression
  • Prime Number Theorem
Sun. 20 Apr, 2025 → Wed. 30 Apr, 2025

Integrable systems

Maciej Dunajski

Summary: Integrable systems are nonlinear differential equations which ‘in principle’ can be solved analytically. This means that the solution can be reduced to a finite number of algebraic operations and integrations. Such systems are very rare - most nonlinear differential equations admit chaotic behaviour and no explicit solutions can be written down. Integrable systems nevertheless lead to a very interesting mathematics ranging from differential geometry and complex analysis to quantum field theory and fluid dynamics.

Sun. 20 Apr, 2025 → Sun. 27 Apr, 2025

Theories of Ordinary Differential Equations (ODEs): Special Topics

Abdulhakeem Yusuf

Summary: The following topics were covered during the period of stay 

1. GENERAL FIRST-ORDER EQUATION i. General Introduction of ODE ii. iii. Equivalence of an Initial Value Problem Existence & Uniqueness Theorem 

2. LINEAR SYSTEM OF FIRST-ORDER EQUATION i. Characrerisation of the Fundamental Matrix ii. iii. iv. v. vi. vii. Properties of Linear Systems Adjoint systems Homogeneous and non-homogeneous system Autonomous Differential equations LINEAR SYSTEMS WITH PERIODIC COEFFICIENTS THEORY OF OSCILLATION 

Sun. 13 Apr, 2025 → Thu. 24 Apr, 2025

Lattices and codes : arithmetic for communication systems

Frédérique Oggier

Summary: In this course, we taught show how lattices and codes, both independently and jointly, are used in the context of communication systems. The course was structured as follows : 

  • Introduction to lattices and geometry of numbers
  • Introduction to linear codes and lattices from codes
  • Introduction to number fields and lattices from number fields
  • Introduction to quaternion algebras and codes from quaternion algebras
  • Practical aspects : channel modeling
Mon. 10 Mar, 2025 → Sat. 15 Mar, 2025

Introduction to real dynamical systems with applications

Jorge Mozo Fernández

Summary: The idea of this course is to give an introduction to the theory of real dynamical systems from scratch, focusing on the main elements of qualitative theory: equilibria, periodic orbits, limit cycles, … An outline of the contents is as follows:

Tue. 11 Mar, 2025 → Sat. 22 Mar, 2025