Talks

Geometric and Arithmetic Degrees on Unitary Shimura Varieties. 

Given for the Postgraduate Research Day 2026, Durham. 

Abstract: We discuss the problem of constructing generating series for the geometric and arithmetic degrees of special cycles on locally symmetric spaces associated to unitary groups. Via the machinery of the Siegel—Weil formula, these generating series are given as Eisenstein series for SL(2). We mention briefly the significance of such results in the broader landscape of the Kudla programme and arithmetic geometry. 

Fundamental Properties of the Funke—Millson Schwartz Form I. 

Given for the Workshop on Funke—Millson Theory, Bielefeld.

Lifting the Curtain: Theta Series in Arithmetic and Geometry.

Given for the GAndAlF (Geometry And Algebra Forum) postgraduate seminar, Durham. 

Abstract: Theta series (and their associated 'lifts') have long played a significant role in the theory of automorphic forms, with diverse applications from number theory to theoretical physics. In the 1980s, Kudla and Millson provided sweeping generalisations to prior work by Hirzebruch and Zagier, among others, studying the geometry of orthogonal and unitary Shimura varieties, constructing modular generating series for 'special cycles' valued in cohomology. In this talk, we begin with a famous classical example of theta series in arithmetic (Lagrange's 4-square theorem), before introducing the philosophy of indefinite theta series, with some applications in geometry.

Sums of Squares: Some Elementary Problems in Number Theory, from Fermat, to Lagrange, to the Modern Day. 

Given for the Van Mildert College Postgraduate Research Seminar, Durham. 

From Sums of Squares to Theta Series.

Given for the SiDur Symposium, Durham. 

Abstract: We discuss two well-known classical theorems in elementary number theory dating to the 17th and 18th centuries, and illustrate how these lead naturally to a first encounter with theta series, an important class of examples of 'modular forms', holomorphic functions on the complex upper half-plane satisfying a certain transformation condition. 

K₀ in Algebraic Geometry.

Given for the COGENT (Cohomology, Geometry, and Explicit Number Theory) seminar, Durham. 

Abstract: We outline the relevance of the zeroth K-group K₀ in algebraic geometry, with particular reference to its relation to the Chow ring CH*(X), and the Groethendieck—Riemann—Roch theorem. 

Pollyanna: An Introduction to the Symbology of Finite Multiple Polylogarithms.

Given for the Van Mildert College Postgraduate Research Seminar, Durham. 

Abstract: First defined by Leonard Euler in the early 1800s, polylogarithms have become an area of rich interdisciplinary research over the past several decades. This class of special functions has seen wide-ranging applications, in number theory, algebraic K-theory, hyperbolic geometry, and even high energy physics in recent years. In this talk, we aim to give a light introduction to polylogarithms, with a particular emphasis on their 'finite' variants.