Swinnerton-Dyer Celebratory Lectures

Dates:

• Friday 16 October 2026

Organisers:

• Adam Harper (Warwick)

Venue:

• Warwick Mathematics Institute, The University of Warwick

Summary:

For the last several years, Warwick has benefited from a donation in memory of Sir Peter Swinnerton-Dyer, which has funded several PhD studentships. This autumn we welcome our seventh and final Swinnerton-Dyer scholar. To mark the end of these scholarships, on Friday 16th October we will host an afternoon of talks commemorating Swinnerton-Dyer and his work.

Speakers:

The provisional timetable is

• 1.00pm-2.00pm, B3.03: Sean Eberhard (Warwick), "How fast can a group grow?"
• 2.00pm-3.00pm, Common Room: Tea/coffee/cake
• 3.00pm-4.00pm, B3.02: Rachel Newton (KCL), "Diophantine equations and local-global principles"
• 4.00pm-5.00pm, B3.02: Jack Thorne (Cambridge), "Swinnerton-Dyer and Arithmetic Statistics"
• 5.00pm-6.00pm, Common Room: Drinks reception

Abstracts:

• Sean Eberhard (Warwick), "How fast can a group grow?
  A finitely generated group can be explored by starting at the identity and repeatedly multiplying by elements from a fixed generating set. The number of elements reachable in n steps, often denoted v(n), is called the growth function of the group. Familiar examples have growth functions that grow polynomially (linearly, quadratically, ...) or exponentially. A founding theme in geometric group theory is that, surprisingly, such elementary counting data can reveal deep algebraic structure about the group.
  I will introduce the growth of groups through examples and explain some of the major dividing lines in the subject. A celebrated theorem of Gromov says that groups of polynomial growth are precisely those having a nilpotent subgroup of finite index, while a famous exotic example of Grigorchuk shows that the growth can be "intermediate", i.e., strictly between polynomial and exponential. These results lead naturally to a still-mysterious quantitative question: how slowly can a group grow without having rigid almost nilpotent structure?
  I will describe recent work on these questions, particularly for classes of groups with informative finite quotients.
• Rachel Newton (KCL), "Diophantine equations and local-global principles"

  Diophantine problems concern integer and rational solutions to polynomial equations with integer coefficients. These problems have been studied for thousands of years and remain a very active field of research. I will introduce local-global principles underpinning modern approaches to Diophantine problems, and the Brauer–Manin obstruction which can explain failures of these local-global principles. I will then describe work from two joint projects – one with Martin Bright and one with Emiliano Ambrosi and Margherita Pagano – answering a question of Swinnerton-Dyer concerning the primes that can play a role in the Brauer–Manin obstruction for certain families of polynomial equations, including those that define K3 surfaces.

• Jack Thorne (Cambridge), "Swinnerton-Dyer and Arithmetic Statistics"

  The story of how Birch and Swinnerton-Dyer used one of the first computers to calculate the rational points on many elliptic curves, leading to the formulation of their celebrated Conjecture, is famous. The calculation was based on a theoretical relation between rational points and binary quartic forms. This relation has reappeared more recently in the seminal work of Bhargava and Shankar, proving that the average rank of elliptic curves is bounded — one of many ways in which the themes of Swinnerton-Dyer’s work are reflected in some of the most recent exciting advances in “arithmetic statistics”. I will discuss some of these connections, including with some recent joint work with Jef Laga.