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2025

Singularités, équations différentielles, transcendance

This Thematic Month aims to cover topics related to singularity theory of algebraic or analytic spaces, algebraic study of differential equations, and their applications to questions of transcendence.

Mon. 27 Jan, 2025 → Fri. 28 Feb, 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

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