Algebra, Geometry, and Calculus

This course explores various aspects of effective calculus in algebra (linear algebra over Euclidean rings, resultants, etc.) and introduces the fundamentals of algebraic geometry.

To master the “effective” aspects of the algebraic concepts studied in the first semester and to prepare students for the modeling option of the Mathematics Agrégation exam; the course unit includes computer lab sessions using SageMath (formal computation). 

Standard Fourier series in^L2(R/Z), endomorphism reduction, Jordan, the first-semester course “Algebra 1.”

Recommended prerequisites: Standard Fourier series in^L2(R/Z), reduction of endomorphisms, Jordan, the first-semester course “Algebra 1.”

1. Effective linear algebra over Euclidean rings: Hermitian and Smith normal forms, Frobenius reduction, invariant factors of a matrix. The adapted basis theorem and its applications.

2. Fast polynomial multiplication: computational complexity and standard multiplication, Karatsuba’s algorithm, the Fourier transform of a finite Abelian group, the discrete Fourier transform for polynomials, and the fast Fourier transform algorithm.

3. Resultant: Sylvester matrix, elimination methods, expression in terms of roots, discriminant.

4. Introduction to Algebraic Geometry: affine space, study of plane curves (e.g., quadric curves), the Nullstellensatz, Zariski topology.

5. Gröbner bases: monomial orders, Buchberger's algorithm, and elimination algorithms.

Hourly volumes:

            CM: 9:00 p.m.

            TD: 9:00 p.m.

            TP: 0

            Land: 0