We’re grateful to have Prof. Kevin Buzzard as our speaker giving a talk on “Can a computer do your problem sheets?” at Nov 9th)  in Main Lecture Hall, Old Divinity School, St John’s College.

“We all know a computer can e.g. very quickly add up the first million prime numbers. But can a computer prove some random theorem on your vector spaces problem sheet or your algebraic geometry problem sheet? The standard programming languages which mathematicians and computer scientists are taught at school or uni can’t, because they are not expressive enough to state and prove theorems. Lean is a programming language which can. To see it in action, try https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/(link works on computers but not phones). I’ll talk about how I’m using Lean to do mathematics from undergraduate level to Fields Medal level and how you can get involved too (and even make money and get publications!). No background in programming is necessary. Suitable for 1st year (and higher year!) mathematicians.”

 

The annual football match between Adams Society and Hyperbolic Society happens on 6th November. Unfortunately, we lost the match as 1:2.

It’s been quite a long time since the last in-person talk. On 12th Oct, our Fellow – Professor Richard Samworth gave an inspiring talk about Stein’s Lemma.

Title: Stein’s paradox

Abstract: Stein’s paradox is one of the most striking results in Statistics. Although it appears to be a basic problem in mathematical statistics, it turns out to have profound implications for the analysis of modern, high-dimensional data. I will describe both the result and some of its consequences.

Fresher’s Squash 

Friday, 8th October 2021

A warm welcome was given to our new freshers, along with lots of Pizza and games.

The Adams Society garden party were held at the Fellow’s Garden, split into two sessions due to Covid restrictions of 30 people maximum.

 

Monday, 19th April, 19:00, 2021

The Tripos Experience Sharing Session is hold online due to Covid, led by Michael Ng, Nicholas Janisch, David Bai, Sophie McInerney, Jason Tang and Lennie Wells. The attendees peaked over 50.

Here is the link to the recording of the session
Here is the link to Michael’s slide

 

Lent Term

AGM

Friday, 26th March, 15:00

The Adam’s Society Annual General Meeting was held and a new committee elected for 2021-2022. Looking forward to more events, virtual or in person, in the coming year! The committee page will be updated shortly.

Truth by Prof. Michael Potter

Tuesday, 23rd February, 18:00

I am pleased to invite you to our last talk of this term. If any of you are tired of traditional mathematics by now, fear not, it is going to be something special. Our speaker is Prof. Michael Potter from the Faculty of Philosophy and the title of his talk is Truth. Please see the abstract below :

What does it mean to call a mathematical theorem (or indeed any other proposition) “true”? Frank Ramsey began while he was a maths undergraduate at Cambridge to propose a prima facie deflationary answer to this question. “There is no separate problem of truth,” he wrote a few years later, “but only a linguistic muddle.” But a few years after that Tarski proved a metamathematical result which seems to prove the opposite: so far from being eliminable, truth is indefinable. I shall explain both views and try to determine whether they are reconcilable.

How (not) to test your way out of a pandemic, using Part I maths, by Jordan Skitrall

Tuesday, 9th February, 18:00

The COVID-19 pandemic has made its presence felt in all of our lives, and has introduced the public to a well-known threshold phenomenon in disease dynamics: whether disease spread can be controlled or proceeds completely out of control depends exquisitely on small effects when the average number of people being infected by each case (R) sits around 1. One of the ways people have sought to affect R is by testing people, isolating infectious people from infecting others, and tracing the people they may already have infected. A simple discourse has arisen – sadly reminiscent of the early days of most screening programmes in history – that more testing is always better. But is it? I shall talk through some of my own work and show that a simple model of the effect of screening can be built using nothing more than early-undergraduate probability theory. I shall discuss the thinking required to construct such a model, and show how one goes back and forth between an understanding of the factors in a system and the underlying mathematics. Finally, we shall reach a working model of the effect of screening and show that some of the major factors affecting its success are those that are hardest for a mathematician to predict and control – those of human behaviour.

Morse Theory in Finite and Infinite Dimensions, by Dr Jonny Evans (Lancaster University)

Tuesday, 26th January, 20:00

If you put a drop of water on a sphere, there are exactly two points where it will stay put: the top, where it will sit if precisely balanced, and the bottom, where it will form a drip. If you try the same on a torus, you will find four such “critical points”. This difference is because the torus and the sphere are topologically distinct. Morse theory takes this observation further, and relates the topology of a space to the critical points of functions defined on it. We will think about some examples, some finite- and some infinite-dimensional.

Pizza and Game Night

Saturday, 23rd January, 19:00

An opportunity to talk to fellow Johnians, play games and eat a Pizza on us.

Michaelmas Term

PhD Talks: Poset Saturation Problems & Algebraic Geometry by Maria Ivan and Patrick Kennedy-Hunt

Tuesday, 24th November, 18:00

The final Adams Society Talk of the term is happening on Tuesday! We’ll have the chance to hear from two of our very own PhD Students: Maria Ivan and Patrick Kennedy-Hunt. They’ll both be giving short talks on their research areas, Poset Saturation Problems and Algebraic Geometry respectively, aimed at an undergraduate audience.

It promises to be a great chance to hear from mathematicians who were in your shoes not very long ago!

Title: “Poset Saturation Problems” – Maria Ivan

Abstract: “A poset is short for a partially ordered set. The most common example of a poset is the power set of [n] with the partial relation given by inclusion. Given a fixed poset P we say that a family F of subsets of [n] is P-free if there is no induced copy of P formed by elements of F. We further say that F is P-saturated if it is P-free and, for any other set X subset of [n], the family F\cup \{X\} contains an induced copy of P. The size of the smallest P-saturated family is called the induced saturation number of P, denoted by sat^*(n,P).The natural question is can we determine the saturation number for simple posets, or at least their order of magnitude? The posets that have been actively researched are the antichain, the diamond (one maximal element, one minimal element and two incomparable elements), and the butterfly (two maximal and two minimal elements). Until recently, the questions for all these three posets were open, despite substantial improvements. In 2020 the butterfly question was completely solved by two independent papers, the first of which I am the author of and proves the lower bound of n+1, with the second providing an example of size about 6n. These two papers combined give that the saturation number for the butterfly is of order n.In this talk I will introduce the audience to the field of poset saturation and focus on the butterfly poset, especially on the key ideas of my proof for the lower bound. There will be lots of diagrams which will hopefully make all the steps natural and easy to follow. I am looking forward to seeing you all there.

Title: “Algebraic Geometry” – Patrick Kennedy-Hunt

Abstract: Algebraic Geometry is an important area of modern pure maths in which we study the zero sets of polynomial equations. In this talk I will describe a visual construction in the relatively new tropical algebraic geometry. These `tropical curves’ are easier to think about than polynomials but remember lots of key information. We will use our construction and a result of Milkhalkin to sketch an unusual take on (which will not quite amount to a proof) Bezout’s theorem. Bezout’s theorem is one of the most famous results in algebraic geometry: it says two polynomials of degree d and e share d\times e common zeroes in projective space.

Algebra and Geometry in the Tropics, by Dr Dhruv Ranganathan

Tuesday, 10th November, 18:00

We are pleased to announce our second talk of the term, this time by Dr Dhruv Ranganathan. The title is “Algebra and Geometry in the Tropics”, abstract below:
There is a remarkable collection of ideas, with roots in high energy theoretical physics, that links the world of manifolds, polynomials, and string theory to the world of polygons, combinatorics, and graph theory. The bridge is ultimately based on the notion of an absolute value. While all of us understand the concept of absolute values of numbers, the ideas of tropical geometry give us an absolute value of geometry itself. We’ll explore what on earth this might mean, and stare at a large number of poorly drawn pictures to understand what makes this odd sounding concept a powerful new tool in modern mathematics.

On Conway’s Numbers and Games, by Dr Jessica Fintzen

Tuesday, 27th October, 18:00

Our first talk of the term will be by Dr Jessica Fintzen and hosted on zoom. If you would like to attend, please sign up for the facebook event here and we will send you further details closer to the date.

Fresher’s Squash, by the Adams Society

Tuesday, 13th October 2020, 16:00-17:30

Join as at Gazebo 3 (in Merton Court next to Cripps) for an opportunity for the new members to meet and greet the other mathmos in college.

As you’re all probably very bored of by now, there are various of restrictions which we need to abide by, so please through the following guidelines and stick to them:

• 1m distancing at all times
• Wear a mask
• Do not attend if you have symptoms of COVID-19

Easter Term

Quiz night, by the Adams Society

Monday, 4th May 2020, 7pm

A mathematical themed quiz on zoom. A chance to chat with your fellow mathematical friends at St John’s from home.

The Adams Society has had a quiet but successful year. The unavoidable impact of the pandemic meant that unfortunately we weren’t able to do many of our normal activities, such as the Garden Party or the Annual Dinner. However, the society managed to remain one of the most active subject societies in college by hosting regular talks and social events, albeit online.

Michaelmas term kicked off with the only in-person event of the year, as we welcomed the new Johnian mathematicians at the Freshers Squash meet & greet. The term also offered some interesting talks, including one in honour of the late Johnian, John Conway, given by Dr. Jessica Fintzen, and a talk from Dr. Dhruv Ranganathan about ‘Algebra and Geometry in the Tropics’. Lent term began with yet another lockdown, but nonetheless we kept spirits high as we hosted an online pizza and games evening. Later in the term we had talks from Dr. Jonny Evans on ‘Morse Theory’, Prof. Michael Potter on ‘Truth’, and a highly topical talk from Dr. Jordan Skittrall on the maths behind testing strategies for pandemic control, titled ‘How (not) to test your way out of a pandemic’.

A sincere thank you is due to the committee for their hard work and ingenuity in tackling the challenges that this year posed. We leave the society in capable hands and know that it will go from strength to strength, in what will hopefully be a year much closer to normality!

          Nic Janisch, President 2020-2021

You can view other Presidents’ reports here.

 

Lent

Ranks of elliptic curves, by Dr Tom Fisher

Wednesday 8th February, 6pm – Hall Signup

Can the sum of the first n square numbers itself be a square number? Are there any right-angled triangles with rational side lengths and area a given integer? Which integers can be written as the sum of two rational cubes? These are just some of the classical problems in number theory that lead to the study of elliptic curves. I will give an informal introduction to elliptic curves, and the group of rational points on an elliptic curve. I will then discuss what is known, what is conjectured, and what is still completely mysterious about the number of generators for this group.

At the dawn of a new era in astrophysics: Gravitational waves have arrived, by Dr Ulrich Sperhake

Wednesday 15th February, 6pm – Hall Signup

Dr Ulrich Sperhake, the amazing lecturer of Part II General Relativity, will be joining us for a talk on gravitational waves – here is the abstract:

On Sep 14 2015, gravitational waves were for the first time detected directly. This observation by the LIGO interferometric detectors marks the dawn of a new era in our observational study of the cosmos as a qualitatively new window to its exploration has been opened. This talk reviews some of the fundamental concepts of gravitational waves and the methodology employed for their observation. The first two events, dubbed GW150914 and GW151226, as well as the properties of their sources, as inferred from the observations, will be discussed. The talk concludes with a selected set of the most important topics where we expect gravitational-wave observations to deepen and either challenge or confirm our present understanding of the laws and the history of our universe.

Symmetries of manifolds, by Dr Oscar Randal-Williams

Wednesday 1st March, 6pm – Hall Signup

Dr Oscar Randal-Williams from DPMMS will join us for a very promising talk. Many of you will know Dr Randal-Williams for his excellent lectures for Part IB GRM. His research interests include algebraic and geometric topology. Here is the abstract:

Wednesday 15th March, 6pm – Hall Signup

Prof Adrian Kent from DAMTP will join us to give a talk about relativity, quantum information and cryptography. Abstract:

The goal of cryptography is to control access to information. For example, we may want a secret message to be readable by selected allies but not by adversaries, or an encrypted prediction to be unveiled only if we choose to supply a key. In recent decades, we have discovered how to use fundamental physical laws to guarantee cryptographic security. Quantum cryptography exploits the distinctive properties of information encoded in quantum systems, while relativistic cryptography uses the fact that information cannot be sent faster than light speed. I will show how some simple but perfectly secure cryptosystems can be built using these principles and describe the current state of the art of physics-based cryptography.

Michaelmas

Hunting for viral packaging signals, by Dr Julia Gog

Wednesday 2nd November, 6pm – Hall Signup

Dr Julia Gog from DAMTP (and “Mathematical Biology” fame) will join us for an exciting talk. Abstract:

Influenza has a genome split into several segments, and this complicates virus particle assembly as each particle must have one of each of the segments. This means that each of the RNA segments must contain some signal, and that this signal ought to be fairly conserved. Is this enough to go and hunt them down using mathematics? The answer turns out to be yes. However, this required some creativity in algorithm design, drawing inspiration from a number of apparently unrelated problems. This hack seems to work, but leaves some interesting mathematical problems.

I’ll also briefly talk about some of the other problems in influenza and infectious disease that interest me, and general joys and challenges of being a mathematician trying to research biology.

Do we really know what computers can do? – On the foundations of computational mathematics, by Dr Anders Hansen

Thursday 10th November, 6pm – Hall Signup

Dr Anders Hansen from the Cambridge Center for Analysis at DAMTP will join us for an exciting talk – note that this is on a *Thursday* rather than our usual Wednesday, and we also have talks on three consecutive weeks. Dr Hansen’s homepage can be found here. Abstract:

Questions regarding what a computer can do have fascinated mathematicians since the beginning of the 20th century. Hilbert initiated much of the research in the area by posing his “Entscheidungsproblem” (decision problem) in the late 1920s. Turing’s solution to this problem, as well as his intuitive definition of what a computer should be (the Turing Machine), made him one of the founding fathers of modern computer science. In the late 1940s von Neumann contributed substantially to the field, and this was the beginning of what we know as scientific computing, namely how to use computers to solve problems in the sciences. Over the next decades (50s until today) computer science and scientific computing, although rooted in the same questions, developed in parallel without much interaction. This discrepancy was pointed out by Smale in the 1980s when he asked basic questions on convergence of Newton’s method, one of the most basic algorithms in scientific computing, as well as questions on existence of algorithms for fundamental problems such as polynomial root finding. Several of these questions were answered by McMullen and the solutions were part of the justification of his Fields medal in 1998.

I will give an informal historical account of the developments in the field leading up to the current unsolved problems and demonstrate that, despite about 100 years of research and the enormous impact of computers, we still do not really know what a computer can actually do.

Loop Erased Random, Uniform Spanning Trees and Percolation, by Dr Sebastian Andres

Wednesday 16th November, 6pm – Hall Signup

In graph theory spanning trees have been investigated already since the 19th century. They appear for instance as objects in a number of algorithms. On the other hand, in modern probability theory certain random spanning trees, so called uniform spanning trees, have had a fruitful history. Most notably, around the turn of the millennium the study of these spanning trees led Oded Schramm to introduce the SLE process, work which has revolutionised the study of two dimensional models in statistical physics. One reason for the importance of uniform spanning trees is their intimate relation to another model, the loop-erased random walks.
In this talk we will introduce both models and explain their connection by means of Wilson’s algorithm. In the last part we will discuss some relations to percolation theory.

Computerized tomography and the X-ray transform, by Prof Gabriel Paternain

Wednesday 30th November, 6pm

Professor Gabriel Paternain will join us for an exciting talk. His department homepage can be found here. Abstract: I will describe some of the mathematics that underpins CT scans. The main inversion formula was discovered by J. Radon 99 years ago.