## Lent Term

**Stretching, Bending, Twisting, and Coiling: the Fluid-Mechanical Sewing Machine**

by Prof John Lister

*18:00–19:00, Wednesday 8 March*

*Boys Smith Room*

Idlers at breakfast watching a stream of honey falling from a knife may notice it buckle and coil as it reaches the toast. What happens if you move the toast (or the knife) steadily sideways? This talk will outline the mathematical description of the dynamics of a falling viscous thread, with possible diversions via chocolate fountains and Vienetta ice-cream.

## Centenary Events

*All talks will take place in the Main Lecture Theatre, Old Divinity School.*

### Saturday 18th February

**From Lichen to Lightning: Understanding Random Growth**

by Prof Amanda Turner, University of Leeds

*14:00–14:45*

Random growth arises in physical and industrial settings, from cancer to polymer creation. Despite considerable effort, mathematicians and physicists have been unable to answer fundamental questions such as “What do typical clusters look like?”. This talk will explore how combining probability and complex analysis can provide mathematical descriptions of random growth.

**Field Theories of Active Matter**

by Dr Rosalba Garcia-Millan, St John’s College, Cambridge

*14:45–15:30*Active matter, such as swimming bacteria and growing cell tissues, consists of entities that turn a local energy supply into mechanical action. Characterising such systems mathematically has become a major undertaking driven by the desire to describe biological phenomena quantitatively and develop a general framework of non-equilibrium physics. As active matter typically involves fluctuations, interactions, persistence and strong correlations, its description requires new tools. In this talk, I will outline how I use field theory to develop a new mathematical technique to study active matter.

###### Interval—Refreshments, 15:30–16:00

**Climate, Chaos, and Covid; how Mathematical Models Describe the Universe**

by Prof Chris Budd, University of Bath

*16:00–17:00*Mathematical models have been much in the news recently. They have been used to make predictions of the growth of Covid-19, to work out what is happening in climate change, and even to ‘predict’ A-level results. But how do these models work? How reliable are they, and can we trust them enough to advise government policy? In this talk, I will talk about how mathematical models are constructed and tested, how they can be applied and what their limitations are. The talk will focus on the application of mathematical models to Covid-19 and climate prediction, and will include a practical demonstration of chaos.

###### Interval—Refreshments, 17:00–17:30

###### Pre-dinner drinks, 19:00–19:30

#### Annual Dinner, 19:30

### Sunday 19th February

**Algebraic, Probabilistic and Statistical Aspects of Markov Chains**

by Dr Kweku Abraham, St John’s College, Cambridge

*09:30–10:15*

One beautiful property of Markov chains is how algebraic properties of matrix products couple with probabilistic intuition and arguments. I’ll talk about how one property of a matrix — the “spectral gap” – closely relates to the probabilistic notion of the “mixing time” of the chain, and in turn plays an essential role in statistical estimation of and with Markov chains.

**Fantastic Folds—the Mathematics of Origami**

by Phillipp Legner, Amplify and Mathigon

*10:15–11:00*

Origami not only looks beautiful, but also has important real-life applications—from solar panels in space to stents in your blood vessels. In this talk, we’ll explore the “Origami axioms”, impossible constructions, unsolved problems in polygon folding, as well as current research into folding algorithms and protein folding.

###### Interval—Refreshments, 11:00–11:30

**Learning in Biological Neural Networks: Hebb’s Rule and Beyond**

by Edward Young, St John’s College, Cambridge

*11:30–12:15*Artificial neural networks are typically trained using an algorithm known as back-propagation. Given the recent explosion of interest in such networks, most mathematicians are familiar with at least the basic ideas behind how back-propagation works. As powerful as it has been in training artificial networks, back-propagation remains an implausible rule for learning in biological neural networks. In this talk, we will examine some of the proposals theoretical neuroscientists have put forth for how learning may occur. We will start with Hebb’s rule, and from there examine some of the multitude of extensions and modifications scientists have put forward. Our discussion will highlight the diverse collection of tools used in theoretical neuroscience, covering ideas from statistics, dynamical systems, linear algebra, and information theory.

**Quasiconvexity, Microstructure and Dynamics**

by Prof Sir John Ball

Heriot–Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh

Hon. Fellow of St John’s College, Cambridge

*12:15–13:00*

Quasiconvexity is the central convexity condition of the multi-dimensional calculus of variations, and is roughly speaking necessary and sufficient for the existence of minimizers. But it is not known how to check whether or not a given function is quasiconvex. The free-energy function for elastic crystals is not quasiconvex, leading to the non-existence of energy minimizers and minimizing sequences that generate infinitely-fine microstructures. Yet quasiconvexity still plays a key role in understanding such microstructures, providing a direct link between experiments and fundamental unsolved questions of the calculus of variations. Still less understood is how to decide in a dynamical model whether solutions can generate infinitely-fine microstructures in the limit as time goes to infinity. The talk will give a panoramic survey of these issues.

## Michaelmas Term

**Numbers: Real, Complex and Super**

by Dr David Stuart

*18:00–19:00, 18th Oct 25th Oct, Tuesday*Castlereagh Room

Non-zero real numbers have strictly positive square while non-zero imaginary numbers have strictly negative square. Super numbers fill the gap: they have square zero (without themselves being zero!) Dr Stuart will try to explain their properties and how to do some simple geometry and calculus with them.

**Properties and Applications of the Gamma function**

by Vishal Gupta

*18:00–19:00, 1st Nov, Tuesday*Castlereagh Room

The gamma function is considered to be the least special of the special functions. This is because while it has a scary-looking definition, it has a lot of very nice properties and crops up in many areas such as number theory and quantum physics. Titi compared it to a large town with lots of little alleys and passages to get used to—this talk will be a guided tour around the landmarks of that town.

**Card Shuffles and Prime Numbers**

by Prof Jack Thorne

*18:00–19:00, 8th Nov, Tuesday*Boys Smith Room

How many prime factors does a natural number have, on average? What’s the probability that a randomly chosen prime ends with a three? We are used to thinking about probabilities when rolling dice and choosing balls from urns, but we can ask probabilistic questions about numbers too, provided we formulate them carefully. For this, we need to learn to ask big questions like ‘how big should it be’ and ‘how big can the error be’. Prof Thorne will introduce some of these ideas, starting with random card shuffles.

**When Is an Irrational Number Nearly a Rational Number?**

by Aled Walker

*18:00–19:00, 22th Nov, Tuesday*Castlereagh Room

This talk will be about an old area of number theory called ‘diophantine approximation’. Informally speaking, questions in this area ask about how closely one can approximate a fixed irrational number by a sequence of rational numbers. Undergraduates learn about how continued fractions can be used to do this, but this is just the start of a story that continues right up to the present day. In fact, a major conjecture in the area was recently resolved by Dimitris Koukoulopoulos and James Maynard: this was one of the results that won Maynard his Fields Medal in July. Though their proof was very technical, it had an unexpected relationship to very concrete questions in combinatorial number theory, involving greatest common divisors — some of which could have been asked on a Numbers & Sets examples sheet.