Let f be a polynomial in three variables and of degree d, for example f = x^d+y^d-z^d. How many singularities can f=0 have? This question is unsolved for d>6. Above, you see the current world record holders for d=4,5,6,7 of which only d=4,5,6 are proven to be maximal.

Computational Algebraic Geometry

Algebraic geometry classically studies solutions to polynomial equations, using abstract algebraic techniques to tackle geometric problems. With roots dating back to ancient Greece, it has since proven itself to be an indispensable tool in a variety of areas ranging pure mathematics such as number theory (Fermat's Last Theorem, Langlands Correspondence) and combinatorics (Rota's conjecture) to applied fields such as statistics (algebraic statistics). For a brief introduction to algebraic geometry, see this video by Aleph 0.

Computational algebraic geometry leverages the algebraic nature for concrete computations. For us, it will serve as an accessible entry point, providing a hands-on introduction to an otherwise highly abstract area.

The aim of the project is threefold:

1. learn the fundamentals of classical algebraic geometry (and commutative algebra),

2. gain foundational skills in modern software development,

3. produce something of value for the current research community.

The overarching theme of the project will be primary decomposition, the task of breaking down a large solution set into its minimal components. This concept is fundamentally important for many problems in algebraic geometry. For a single polynomial, this reduces to polynomial factorization. For example, the solution set of the equation xy - x - y + 1 = 0 consists of the solutions to x - 1 = 0 and y - 1 = 0, since the xy - x - y + 1 factors as (x - 1)(y - 1). For systems of multiple polynomials, a variety of strategies exist, each with its own strengths and limitations.

The project will begin with regular group meetings to build the necessary foundations in algebraic geometry. Afterward, each of you will select a topic that aligns with your interests. Some topics will lean more toward mathematical theory, others toward software development, but all will involve a combination of both.

Requirements

The project has no strict module requirements, but Algebra II (MATH2581) would be beneficiary. The project will involve coding, and experience in python or julia would be very useful.

Literature

• Cox, Little, O’Shea: Ideals, Varieties, and Algorithms (link)

• Greuel, Pfister: A Singular Introduction to Commutative Algebra (link)