Braid groups and configuration spaces
Luciana Basualdo Bonatto, The University of Oxford
Braid groups arise from the simple idea of strands crossing in space, yet exhibit remarkably rich algebraic and topological structure. They appear naturally as the fundamental groups of configuration spaces of n distinct points in the plane, linking geometric intuition with algebraic topology and group theory. In this course we will study braid groups and configuration spaces from both geometric and algebraic perspectives, exploring their connections with covering spaces, knot theory, representation theory, and several areas of current mathematical research.
Quadratic forms and K-theory
Oliver Gregory, University College London
Quadratic forms appear everywhere in mathematics, and have been studied by many of the great mathematicians of history. In this course we shall explore the "algebra" of quadratic forms. We will begin by constructing the Witt ring of a field together with its filtration by powers of the fundamental ideal. How deep a quadratic form lies in this filtration serves as an important invariant for classifying quadratic forms. To better understand this filtration, we shall introduce Milnor K-theory, learn how to compute it for fields of interest in number theory and derive some consequences. The techniques of the mini-course will provide you with the background to understand a more recent and arguably more fundamental construction called Milnor-Witt K-theory, which plays an important role in the modern topic of motivic homotopy theory. Milnor-Witt K-theory will be the topic of the project.
Group actions on buildings
Yusra Naqvi, University College London
Buildings provide a rich class of simplicial complexes with very nice local structure. They were originally invented by Jacques Tits as a way of studying linear algebraic groups over arbitrary fields, which act on these buildings in a natural way. Today, buildings play important roles in several areas of mathematics, including number theory, geometry, algebra, and combinatorics. In this course, we will study the construction of buildings, explore their combinatorial and geometric structure, and finally develop an understanding of the algebraic groups that act on them.
Problem Class Assistant: Yiannis Fam, LSGNT, Imperial College London
Coefficients of modular forms
Jenny Roberts, King's College London
Modular forms have emerged as a key tool in mathematics over the last century, from Ramanujan's fascination with their arithmetic properties to their use as a main ingredient in Wiles' proof of Fermat's last theorem. In this course, we will define modular forms and show that their nice symmetries allow us to write them as Fourier series expansions. Using some basic properties of modular forms and by studying their Fourier coefficients, we will prove some interesting arithmetic relations; for example, answering the question: can every integer be written as a sum of four squares?
Representation theory of finite groups - beyond C
James Taylor, The University of Padua
Representation theory, in its many forms, is ubiquitous throughout pure mathematics. Historically, the first instance of representation theory developed was the representation theory of finite groups over the complex numbers, as developed by Frobenius, Burnside and Schur. This beautiful theory is the bedrock of all further generalisations, and in the first part of this course we highlight the main results, accompanied by illustrative examples, with an emphasis on what properties of the ground field are actually being used. We then consider two natural extensions. The first is when the ground field is no longer assumed to be algebraically closed, but still has characteristic 0, which includes the case of R-linear representations. The second is when the ground field is not assumed to have characteristic 0, where all hell breaks loose.
Problem Class Assistant: Anna Bresciani, LSGNT, Imperial College London
Gröbner bases and convex polytopes: between algebra, geometry, and combinatorics
Sara Veneziale, Imperial College London
Gröbner bases are one of the most versatile tools in computational mathematics, underpinning algorithms across algebraic geometry, coding theory, cryptography, robotics, and statistics. But how many distinct Gröbner bases does an ideal have — and what geometric object organises them? This mini-course answers that question through the state polytope, then explores the rich interplay between polynomial algebra and convex geometry via toric ideals, with surprising connections to integer programming and triangulations. Interactive SageMath sessions accompany the theory throughout.