Algebra, Geometry, and Calculus

This course develops various aspects of effective calculus in algebra (linear algebra on Euclidean rings, resultants, etc.) and introduces the first elements of algebraic geometry.

Mastering the "effective" aspects of the algebra concepts studied in the first semester, and preparing students for the modeling option of the mathematics competitive examination; the course includes computer labs in SageMath (formal calculation). 

Standard Fourier series in^L2(R/Z), reduction of endomorphisms, Jordan, 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 on Euclidean rings: Hermite normal forms, Smith normal forms, Frobenius reduction, invariant factors of a matrix. The adapted basis theorem and its applications.

2. Fast polynomial multiplication: complexity calculations and standard multiplication, Karatsuba algorithm, Fourier transform of a finite abelian group, discrete Fourier transform for polynomials, fast Fourier transform algorithm.

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

4. Introduction to algebraic geometry: affine space, studies of plane curves (e.g., quadrics), the nullstellensatz, Zariski topology.

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

Hourly volumes:

            CM: 9 p.m.

            TD: 9:00 p.m.

            TP: 0

            Land: 0