Since the 1960s, a new field of mathematics has been steadily growing in significance. Broadly describable as computational algebra, this field grew out of algorithms for computing Gröbner bases: that is, for finding explicit generators for ideals. Some notable successes have stimulated a wide interest in the area, from disciplines such as engineering and robotics through to algebraic geometry and cryptography.
Topics in abstract algebra will be approached from the perspective of what can be explicitly computed, with an emphasis on applications to algebraic geometry and practical skills in using computational tools. We use the term "computational" in the more formal sense to mean those things that can be explicitly calculated via algorithms, in contrast to much of your experience in pure mathematics to date, which has avoided the issue of direct computation.
Familiarity with the basics of algebra: rings, fields, etc.
(1) Polynomial rings and their ideals
Introduces commutative polynomials ring and their ideals, along with the key questions this course addresses:
- Are all ideals I \subset k[x_1,...,x_n] in a polynomial ring finitely generated?
- If I \subset k[x_1,...,x_n] is known to be finitely generated, can we write down a set of generators?
- Is this set of generators unique?
- Is there any systematic way of determining whether a polynomial f is in I?
- Answers for these questions for polynomials in one variable
(2) Taster on algebraic geometry
Introduces affine space A^n and projective space P^n, and their varieties V(I) where I is a (possibly homogeneous) polynomial ideals. Introduces the "inverse" process of moving from an affine variety to an ideal I(V), and asks the key question of when I and V truly are inverses of each other.
(3) The Nullstellensatz
Introduces the definition of a radical ideal \sqrt{I}, telling us exactly when V and I are inverses of each other.
(4) Primary decomposition
Introduces prime and primary ideals for the purpose of computing decompositions of algebraic varieties into easier pieces.
(5) Term orders and Gröbner bases
Introduces total and monomial orders, and the standard monomial orders: the lexicographic, reverse lexicographic, degree lexicographic and degree reverse lexicographic, and weight term orders. Defines the concept of a Gröbner basis for an ideal I in a polynomial ring, motivated as a way of addressing the failures of multivariate polynomial division, and as a way to solve the ideal membership problem.
(6) Computing Gröbner bases
Focuses on the practicalities of computing Gröbner bases: Defines the S-polynomial, and uses this to state and prove Buchberger’s Criterion, which tells us when a set of generators of I is a Gröbner basis. States and proves Buchberger’s Algorithm, allowing us to turn any set of generators into a Gröbner basis.
(7) Elimination Theory
Applies Gröbner theory to solve problems in algebraic geometry, mainly the computation of projections of varieties, and the computation of zero dimensional algebraic varieties. Our main results are the elimination and the extension theorem, which connect lexicographic Gröbner bases to the two problems.
(8) and (9) Free Resolutions
Introduces syzygies as a generalisation of S-polynomials, and introduces free resolutions. Introduces Betti tables, which allow us to distinguish algebraic varieties, and gives an algorithm to compute them. Introduces simplicial complexes, simplicial homology, and resolutions of monomial ideals
(10) Tropical Geometry (Bonus)
An introduction to an active area of research that employs tools from algebraic geometry and combinatorics