2025

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.

Coordinator: Ramesh Gautam, Mathematical Biology Research Centre (MBRC), Nepal

Gourav Arora (India), Youcef Mammeri (France)

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

Francesco Pappalardi

Summary:

Elementary Numbers Theory: 

• Quadratic reciprocity
• Arithmetic fonctions 

Analytic Numbers Theory : 

• Dirichlet Arithmetic progression
• Prime Number Theorem

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.

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