# Events

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Trinity Term
2017

Invariants Social

Come and join us for free snacks in the Maths Institute. This is going to be an opportunity for you to talk about you to us about the future of the Invariants while enjoying our Social Secretary’s finest choices of food and drink (There are rumours about pizza). Beer will be available. Members only. See More
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Invariants Puzzle Competition

Bring your friends or form a team of up to 4 people on the spot and solve some fun maths puzzles! See More
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Undergraduate Talks
Vlad Margarint

An evening "flight" over two modern topics in Mathematics; Random Fractal Curves and Rough Paths Theory. See More
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Undergraduate Talks
James Martin

None other than the one and only James Martin comes to the Invariants to give a talk. We are already very excited! See More
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End of term dinner

Pre-drinks will be hosted in the Larkin Room (St John's college) at 6.15pm. Formal hall afterwards. Dresscode: SMART.
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Hilary Term
2017

Learn to code with Morgan Stanley

Writing code is very common nowadays, whether it's a cool new app or some scientific script to automate calculations. Have you ever wanted to learn how to do it, but think it sounds complicated? It doesn't have to be! See More
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Invariants Puzzle Competition

This week it's time for the Invariants Puzzle Competition next week and it promises to be great! It will be in teams of 4 people (or 3 people), problems will be timed and prizes very appealing: the best team will get Invariants jumpers and a prize in money! Yay! Drinks and snacks provided as well for the whole evening! Hence bring your friends and join us for an evening with the best Maths Puzzles :) See More
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Invariants Movie Night: The Man Who Knew Infinity

Invariants is watching a movie this week! We will be seeing The Man Who Knew Infinity next Tuesday - the story of the life and academic career of the pioneer Indian mathematician, Srinivasa Ramanujan, and his friendship with his mentor, Professor G.H. Hardy. See More
Facebook event

The Ultimate Limits of Privacy
Artur Ekert, University of Oxford

Among those who make their living out of science of secrecy, worrying and paranoia are just a sign of professionalism. Can we protect our secrets against those who wield superior technological powers? Can we trust those who provide us with tools for protection? Finally, can we trust ourselves, our freedom of choice, our free will?
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Valentine's Day Puzzle Competition

Owing to the success of the last puzzle competition, the puzzle masters of the Invariants decided that it needs a sequel. So, get hyped, because next Tuesday, on Valentine's Day, your brain will face new riddles in the quest for the winning LOVEly prizes!
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Cracking the Myths of Big Data Forecasting
Xuxin Mao, UCL

Amid recent economic and political uncertainties, it is the worst of times and it is also the best of times for forecasters. While few have foreseen economic recessions or political populism, robust predictions have become high sought-after treasures that everyone is determined to possess.
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Invariants Committee Elections

This Tuesday is Pizza and Elections! The old committee is changing and we need new amazing people to run the society next year!
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Rational points on elliptic curves
Andrew Wiles, University of Oxford

Andrew will talk about the problem of describing rational points onelliptic curves and recent ideas on the problem.
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Michaelmas Term
2016

Caught Between Dimensions
Richard Earl, University of Oxford

Examples of fractals like the Mandelbrot set and Julia sets are easy to define in terms of sequences of complex numbers. No doubt you will have seen many weird and wonderful images of these fractals - but what is it to be a fractal? This is an introductory talk about some ideas of dimension theory and how it is that a set can have a dimension that isn’t a whole number. See More
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TBA
Nikolay Nikolov, University of Oxford

One of the first results in Mathematics studied at university is that every vector space has a basis. In this talk I will explain how this basic result helped to solve Hilbert's third problem: Can a cube be assembled from a regular tetrahedron of the same volume by cutting them into finitely many tetrahedral pieces?
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Invariants Social: Maths, Drinks and Pizza

It’s already week 3 so congratulations to all our freshers for surviving the first few weeks in Oxford! To celebrate this huge achivement we want to organise next week Maths’ Drinks. We will have pizza and drinks, board games and lots of fun.
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Squirrels, Cancer and Animal Coat Markings
Philip Maini, University of Oxford

Mathematical models are being more and more widely used in the biological and medical sciences. In this talk, I will show how mathematical models initially used to understand ecological interactions (for example, the invasion of the UK by the grey squirrel) are now being used to understand cancer invasion.
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Dr Robin Knight

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General Meeting

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Trinity Term
2016

Brainlabs: Pizza and Coding

Join Brainlabs for a night of coding at the Mathematical Institute. Whether you’re an experienced coder, just starting out, or anywhere in between, we’ll provide problems, puzzles and pizza for you to get your teeth stuck into. So bring your laptop, your brain and an appetite and see you on 26th April! See More
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Chameleon Tongues and Differential Equations
Derek Moulton

Seashells, ballistic tongues, and explosive plants (and differential equations!) Until recent decades, mathematics and biology were primarily seen as completely separate, non-overlapping fields. In this talk, I will show through several examples how mathematics can provide the perfect language to comprehend the marvels of the biological world.
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Undergraduate Talks
Chan Bae and Eliza Casapopol

This event gives our members the chance to shine by presenting on a topic themselves. The time is usually divided into a number of short lectures by members of the society. All members are welcome to give a paper! See More
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Invariants Social

Join us in the common room for pizza, beer and lots of fun! See More
Facebook event

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Hilary Term
2016

Combinatorics and the Fourier Transform
Tom Sanders

Hilary Term 2016 Social

Join in us in the common room at the Mathematical Institute for free pizza, snacks and drinks. This event is open to members only (and lifetime membership will be available on the door for £15 as usual). See More
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How to solve a problem by making it harder
Vicky Neale

Surprisingly, sometimes the way to make progress on a mathematical problem is to tackle a harder problem instead. I'll describe an example of such a problem from number theory that has led to some beautiful mathematics, involving ideas that at first sight seem to have nothing to do with the integers. We'll also see an application of the important fact that 1 is the smallest positive integer. See More
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Curves - An Introduction to Algebraic Geometry and Topological String Theory
Richard Thomas

For centuries mathematicians have generalised statements like "there is a unique line through any 2 points", but with increasing technical difficulties. It was not until the late 1990s that new ideas from symplectic geometry and string theory allowed rigorous definitions to be made of these "curve counting problems". See More

Smashing the racket: Detecting match-fixing in tennis via in-play betting irregularities
ATASS Sports

This talk summarises the results of a match-fixing study spanning over 5,000 in-play tennis betting markets. After developing a natural point-by-point probabilistic model, featuring novel mechanisms for selecting parameters robustly from the data, we demonstrate that the observed market trajectories correlate extremely closely with this model. See More

80th Anniversary Dinner

We will be following our 80th Anniversary lecture by a formal dinner at Balliol College. Over dinner, Prof. Robin Wilson will be giving a brief talk on ‘80 years of the Invariant Society’. The dress code for this event is smart. The dinner is open to everyone. The cost of dinner is £35 per person. See More
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80th Anniversary Lecture
Robin Wilson

To celebrate the 80th Anniversary of the Invariants society, we are hosting a special lecture and dinner with Prof. Robin Wilson. The lecture will be free for everyone, both members and non members. Lifetime membership will still be available on the door for £15 for those who wish to join the society. See More
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Tropical Geometry
Diane Maclagan

Tropical geometry is a combinatorial shadow of algebraic geometry. It is geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. See More

How Fourier series inspired transfinite induction
Charles Batty

Transfinite induction is a process which extends ordinary induction beyond the natural numbers, for certain types of inductive hypothesis. It was first formulated by Cantor, inspired by a question about Fourier series put to him in 1869. See More

Annual General Meeting 2016

The AGM is a chance for members to make their mark on the society by raising important issues or running for committee positions. We are holding elections for every committee position, so come along and get involved. Our Annual General Meeting in the Common Room upstairs. See More

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Michaelmas Term
2015

Rational Points on Curves
Prof. Victor Flynn, University of Oxford

I shall consider the problem, given a curve with rational coefficients, of how one tries to find all of its rational points. See More

Challenging Mathematics: The UK Mathematics Trust and its work
Dr Peter Neumann, University of Oxford

Planning for the United Kingdom Mathematics Trust (UKMT), now nineteen years old, began just twenty years ago. There were some difficulties in the early years, but now it is going from strength to strength under the leadership of Frances Kirwan, Tutorial Fellow of Balliol and Professor in the Oxford Mathematics Department. I propose to speak about its past, present and future (so far as I know them), and show examples of its work. See More

Michaelmas 2015 Puzzle Off

The annual Invariants Society Puzzle Off, where teams compete to solve Mathematical Puzzles for great prizes. See More

Is mathematics invented or discovered?
Prof. Minhyong Kim, University of Oxford

I somewhat disapprove of this question. In this lecture, I will try to explain why. See More

Linear Algebra and the SVD
Prof. Gilbert Strang, MIT

The polynomial method
Dr Miguel Walsh, University of Oxford

In broad terms, the polynomial method is the idea of understanding a set of points, for example in euclidean space, by studying the properties of polynomials vanishing on that set. See More

Mathematics of the Faraday cage, or why is it so hard to see into your microwave oven?
Prof. Nick Trefethen, Mathematical Institute and Balliol College

Since Faraday in 1836, wire meshes have been used to shield electrostatic and electromagnetic fields. The mathematics starts from the Laplace equation, and it must be well understood and in lots of textbooks, right? See More

Michaelmas 2015 Social: Ice Skating

We're holding our main social for the term this week. In keeping with the season, the plan is to go ice skating around 6pm, followed by snacks, pizza and drinks at the Maths Institute. See More
Facebook event

Michaelmas 2015 Social: After Party

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Trinity Term
2015

Maths, Music, AND Maiolica: Alfred Pringsheim (1850-1941)
Keith Hannabuss

Every mathematics student is likely to encounter some of Pringsheim’s ideas, even if they do not recognise his name. This is the story of the mathematician who took part in a duel - and survived; who inherited a large fortune, and lost it; a fine pianist who knew and championed Wagner, publishing arrangements of Wagner’s music for smaller ensembles; who inspired characters in two novels by a Nobel prizewinner; and who, as an octogenarian, found his life turned upside down by the Nazi regime in Germany. See More
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Finding a rational point on the elliptic curve \(y^2 = x^3 + 7823\)
Jennifer S. Balakrishnan

An elliptic curve E can be thought of as a smooth curve of the form $$ y^2 = x^3 + Ax + B , \quad (*) $$ together with a special point \(O\) 'at infinity'. Suppose we fix integers \(A,B\). The set of solutions in rational numbers to \((*)\), together with \(O\), has the structure of an abelian group, and understanding this group \(E(Q)\) has led to many interesting developments in number theory. See More
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Manifolds
Ciprian Manolescu

Manifolds (spaces that look locally like \(\mathbb{R}^n\)) are the basic objects of study in topology. In this talk I will describe what is known about their classification. I will mention the different versions of the Poincare conjecture, and also strange phenomena that appear in high dimensions: exotic smooth structures on the same manifold, and manifolds that cannot be triangulated. See More
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The Mathematics of Lending
Mark Harrison

Crowd Funding is a boom area, and one that broadly splits into three sections - projects that ask for donations in return for some kind of reward (T-shirts are popular), startups that ask for funding in exchange for a claim on future profits (share issues), and companies that borrow money for interest (loan funding.) See More
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Trinity 2015 Social

We're running a social at the usual time on Tuesday in the Mathematical Institute to give everyone a chance to relax and have fun before, during or after exams (for the lucky few who are already done!). We'll be in the common room so come along for snacks and pizza. Entrance is free for members or £3 for non members. And don't forget that lifetime memberships can be bought on the door for just £15! See More
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Annual General Meeting (Constitutional Reform)

We will be holding a vote on our new constitution. Most of the changes are purely cosmetic, so that the constitution follows the guidelines set out by the proctors. We understand that many of you are busy with exams and revision so we'll try to keep this meeting short and functional. That said, it is important that we bring our constitution in line with the new guidelines so we'd really appreciate it if you could come along briefly to vote. The latest draft of the new constitution, as well as the old one for comparison, are below. See More

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Hilary Term
2015

Random Trees
Christina Goldschmidt

Tree structures are ubiquitous in nature, and it is natural to want to model trees which are somehow randomly generated. In mathematics, we think of trees as connected graphs (or networks) with no cycles; these are a fundamental object of study in combinatorics. See More

Mathematics underpins the world we live in
Philip Maini

What do the grey squirrel invasion of England, and the growth of cancer, have in common? How are animal coat markings and chemical reactions related? What links the life cycle of the slime mold and heart attacks? Answer: Mathematics! Come and find out how. See More

Why gravity isn't so attractive after all
James Binney

We experience gravity as an attractive force but that's just because our experience of it is very limited. In our experience a gravitational field is generated by mass-energy. But Lorentz Covariance requires that it is also generated by pressure. See More

Hilary 2015 Social

The "dual" way of doing geometry
Tomasz Miller

The talk will be about some weird ways of doing geometry and physics. See More

Progress in Prime Number Theory
Roger Heath-Brown FRS

This will be a low-brow talk about some of the classical problems in the theory of prime numbers, and the progress that has been made in the recent (and not-so-recent) past. See More

The Tutte Polynomial and Computational Complexity
Leslie Goldberg

The Tutte polynomial is an interesting two-variable graph polynomial defined by a deletion-contraction recurrence. I will first define a specialisation, the chromatic polynomial, which was introduced earlier by Birkhoff as an approach to the 4-colour problem. See More

Annual General Meeting

The AGM is a chance for members to make their mark on the society by raising important issues or running for committee positions. We are holding elections for every committee position, so come along and get involved. Our Annual General Meeting in the Common Room upstairs. See More

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Michaelmas Term
2014

An unqualified success in promoting Mathematics!
Johnny Ball

Johnny was writing comedy for TV in the late 60's. When asked what subject he might write his own series on, he said, "Maths" and produced Think of a Number - which won BAFTA in its first year - and subsequently 20 TV series on Maths and Science. Amazingly Johnny had left school aged 16 with just 2 O'levels in Maths (of course - 100%) and Geography. But he was clearly gifted and made Recreational Maths his hobby while in the RAF. See More

Decision Problems for Linear Recurrence Sequences
Joel Ouaknine

Linear recurrence sequences (LRS), such as the Fibonacci numbers, permeate vast areas of mathematics and computer science. In this talk, we consider three natural decision problems for LRS, namely the Skolem Problem (does a given LRS have a zero?), the Positivity Problem (are all terms of a given LRS positive?), and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). See More

Invariants Puzzle Drive 2014

Join us for a competitive evening of puzzle solving. Compete in teams to complete a variety of entertaining problems set by our puzzle master. Prizes will be awarded to the winners and other successful teams and there will be free snacks. First prize this year is a huge £200, courtesy of Oxford Asset Management. This event is free! See More

Points and lines
Ben Green

Suppose you have n points in the plane, not all on a line. A famous result called the Sylvester-Gallai theorem states that there must be at least one "ordinary line", by which we mean a line through precisely two points of the set. See More

It’s a talk of two halves: A crash course in football and tennis forecasting
Dr Tim Paulden, ATASS Sports

Forecasting the outcomes of sports events is vital to the sports betting industry, sports clubs and - like me - those with fantasy football teams. Dr Tim Paulden will be presenting models to predict football and tennis. The talk will provide an insight into the work that ATASS Sports carry out and after the talk there will be an opportunity to talk to Dr Tim Paulden and Rich Hill about career opportunities with firm. See More

Gravity and the Arrow of Time
Julian Barbour

Since the time of Boltzmann, one of the great mysteries in physics has been the apparent mismatch between the time-reversal symmetry of the laws of nature and the irrerversible growth of entropy expressed by the second law of thermodynamics. It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behaviour of any kind that defines an 'arrow of time'. In a recent paper in Physical Review Letters (arXiv:1409.0917), collaborators and I have shown this is not so. See More

Why I am not a Platonist: quantum information, life and the universe
Vlatko Vedral

I will start by introducing the concept of information due to Shannon and will then argue that it actually only starts to make sense within the framework of quantum physics. Here, the key will be the intrinsic randomness existing within all elementary quantum phenomena and which allows for information to be created ex nihilo (out of no prior information). This counterintuitive feature will be seen to be linked with the bizarre quantum effects such as being in two places at the same time as well as the famous “spooky action at the distance” that Einstein alleged quantum physics to permit. See More

Early computers, and how I met my wife - Number theory on the EDSAC 55 years ago
Bryan Birch

I shall reminisce about the early days of computers, and work I did with Peter Swinnerton-Dyer on Wround about 1958-61. See More

Profinite group theory
Dan Segal

A suitably coherent family of finite groups can be ‘stuck together’ to make a compact (infinite) group, by forming an ‘inverse limit’. The resulting object is called a profinite group. For example, the family (Z/p^nZ) (n>0) corresponds to the ring of p-adic integers Z_p. All sorts of interesting questions about infinite groups, or about infinite families of finite groups, can be approached by studying the associated profinite group. I will give various illustrations. See More

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Trinity Term
2014

The 27 Lines and Other Stories
Nigel Hitchin

In the 1860s it was discovered that a general cubic surface defined over the complex numbers has exactly 27 lines on it, and the configurations of these attracted a lot of attention. It was followed by similar statements relating to two families of algebraic curves. Why were these so interesting? See More

Selling Category Theory to the Masses: A Tale of Food, Spiders and Google
Bob Coecke

We will demonstrate the following. Category theory, usually conceived as some very abstract form of metamathematics, is present everywhere around us. Explicitly, we show how it provides a kindergarten version of quantum theory as well as a new process-based foundation of it, how it helps to automate quantum reasoning, and how it will help Google to understand sentences given the meaning of their words. See More

Machine Learning and Sentiment Analysis, or How to find Happiness
Alex Davies

Machine learning is a hot area at the moment - DeepMind gets bought by Google for 500 million, IBM's Watson beats the world champion at Jeopardy and Siri is occasionally useful. This talk will give an introduction to machine learning through the example of sentiment analysis, the art of getting computers to understand human emotions in written text. See More

Trinity 2014 Social

The Dark Matter Mystery and the Large Hadron Collider
Ben Allanach

I shall describe the dark matter mystery, and we shall go on a speculative journey to solve it. Going back to the beginning of time, we shall witness the birth of a proton, following it through to the present day, where it ends up in the Large Hadron Collider at CERN. See More

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Michaelmas Term
2013

The Simpsons and Their Mathematical Secrets
Simon Singh

Simon Singh, author of Fermat's Last Theorem and Big Bang, talks about his latest book, which explores mathematical themes hidden in The Simpsons. Everyone knows that The Simpsons is the most successful show in television history, but very few people realise that its team of mathematically gifted writers have used the show to explore everything from calculus to geometry, from pi to game theory, and from infinitesimals to infinity. Singh will also discuss how writers of Futurama have similarly made it their missions to smuggle deep mathematical ideas into the series. See More

Oxford Figures
Prof. Robert Wilson, Pembroke College and Open University

This illustrated talk traces the eventful history of Oxford mathematics over the past 800 years, from the founding of the University to the opening of the new Mathematical Institute earlier this month. Included among the expected names are such ‘unlikely’ figures as Geoffrey Chaucer, Christopher Wren, Lewis Carroll and Florence Nightingale. This talk also provides a guided tour of the mathematical sights to be seen around Oxford. See More

Using Mathematics to Discover New Biology
Prof. Philip Maini, St John’s College, University of Oxford

In this talk, we will show a number of examples from over the years of how mathematical models have been used to gain new insights into aspects of biology. Examples will include understanding what controls cell movement in early development, how tumour vasculature affects tumour growth, and how we can predict the growth and dynamics of AIDS in India. See More

Puzzle Hunt!

Van der Waerden's Theorem
Prof. Imre Leader, Trinity College, Cambridge

Suppose that we have a long line of beads, each of which is red or blue. Can we always find three beads of the same colour, equally spaced? So for example if the 4th, 9th and 14th beads were red then this would count. See More

The Lavrentiev phenomenon
Prof. Sir John Ball, Queen's College, FRS

Innocent-looking problems of the calculus of variations can have different minimizers in different spaces of functions. This can lead, for example, to finite-element methods finding the `wrong’ minimizer. Examples of this and related phenomena will be given, and their philosophical implications for models of nature discussed. See More

The random graph and its relations
Prof. Peter J Cameron, Queen Mary, University of London

In 1963, Erdős and Rényi proved the astonishing result that, if a countably infinite graph is chosen at random by selecting edges independently, there is one particular graph which will almost surely arise: this is the celebrated “random graph”. This graph is met with in many different parts of mathematics, including logic, group theory, Ramsey theory, and topological dynamics. I will talk about it and some of its properties, and some related structures. See More

Christmas Party

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Trinity Term
2013

Trinity 2013 Social and Election

The first week of the most work heavy term heralds the newest Invariants Society Social, a chance for everyone to come and have fun while discussing holidays and playing games and eating copious quantities of jaffa cakes and drinking tea, but most importantly, applying for one of the remaining committee positions! See More

A Low-Bit Rate Introduction to Information Theory
Dr Richard Earl, Worcester College, University of Oxford

How much information is there in being told the roll of a die or the colour of a person's eyes? A mathematical theory of information dates back to Shannon's seminal 1948 paper. "Shannon entropy" measures how much information is being conveyed (e.g. by a person speaking) and can be shown to be a lower bound for just how quickly that information can be encoded and transmitted. The talk also discusses some of the main issues of coding - optimal codes, instantaneous codes, error-correcting codes, and a little on cryptography. See More

Members' Papers

This annual event gives our members the chance to shine by presenting their own papers. The usual time is divided into a number of short lectures by members of the Invariants society. All members are welcome to give a paper! See More

Trinity 2013 Social

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Hilary Term
2013

The Secret Mathematicians
Marcus du Sautoy

Artists are constantly on the hunt for interesting new structures to frame their creative process. From composers to painters, writers to choreographers, the mathematician's palette of shapes, patterns and numbers has proved a powerful inspiration. Often subconsciously artists are drawn to the same structures that fascinate mathematicians. See More

Genes and Geography
Peter Donnelly

Genes and Geography: using genetic data and mathematical models to understand the history of the people of the British Isles. See More

Hilary Term 2013 Social

The Reality Game
Doyne Farmer

We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g. heads or tails in a coin toss) depend on the amount wagered on those outcomes. See More

Surfaces and Strings
Frances Kirwan

This talk is about surfaces: their geometry and topology, and how they are related to string theory in mathematical physics. It is intended to be accessible to first-year undergraduate mathematicians, but has the (no doubt over-optimistic) aim of providing a small taste of some recent research in geometry. See More

The Gap of Ignorance: How Uncountable are the Reals?
Robin Knight

After Cantor provided a method for measuring the sizes of infinite sets, the following question arose: are there any sets of real numbers that are uncountable, so having more elements than the set of rationals, but having fewer elements than the entire set of reals? See More

Invariants on the Curve
Alain Goriely

In this talk, Alain explores properties of curves in three dimensions. In particular, he shows how to attach topological and geometric descriptors to curves and show how they are related to each other through a fundamental identity. See More

Annual General Meeting 2013

The AGM is a chance for members to make their mark on the society by raising important issues or running for committee positions. We are holding elections for every committee position, so come along and get involved. See More

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Michaelmas Term
2012

Lewis Carroll in Wonderland
Prof. Robin Wilson, Open University & Pembroke College, University of Oxford

Charles Dodgson is best known for his ‘Alice’ books, 'Alice's Adventures in Wonderland' and 'Through the Looking-Glass', written under his pen-name of Lewis Carroll. If he hadn’t written them, he'd be mainly remembered as a pioneering photographer, one of the first to consider photography as an art rather than as simply a means of recording images. But if Dodgson had not written the Alice books or been a photographer, he might be remembered as a mathematician, the career he held as a lecturer at Christ Church in Oxford University. But what mathematics did he do? See More

Desert Island Maths
Dr David Acheson, Jesus College, University of Oxford

If you were marooned on a desert island, which 8 pieces of mathematics would you want with you, to help keep your spirits up? David presents his own personal choice, drawn from both pure and applied mathematics, with reasons. See More

Michaelmas 2012 Social: Poker Night

Quite self-explanatory; come along to the Mathematical Institute for a night of poker and generally good fun! See More

Did Galois deserve to be shot?
Dr Peter Neumann, Queen's College, University of Oxford

Évariste Galois died aged 20 in 1832, shot in a mysterious early-morning duel. His ideas, after they were published fourteen years later, changed the direction of algebra and have had a huge influence on mathematics. See More

Members' Papers
Various Speakers

This annual event gives our members the chance to shine by presenting their own papers. The usual time is divided into a number of short lectures by members of the Invariants society. All members are welcome to give a paper! See More

Mathematical Potpourri on Airplane Boarding
Anne Henke, Jesus College, University of Oxford

When passengers board an airplane, they queue in arbitrary order. For simplicity, we assume that passengers are arbitrarily fast to get to their seat row, they are arbitrarily broad and arbitrarily thin. Each passenger will need precisely one minute to store his luagge away and take his seat. During this time, he blocks the way for passengers in rows further back in the plane. See More

Attempting to Model the Mathematical Mind
Prof. Sir Roger Penrose, Wadham College, University of Oxford

Alan Turing’s ground-breaking 1937 paper introduced his concept of Universal Turing machine, which underlies the modern general-purpose computer. In 1939, he proposed generalizations based on ordinal logic and oracle machines, these being apparently motivated by attempts to model the mathematical mind in ways that evade the apparent limitations presented by Gödel’s incompleteness theorems. See More

Christmas Party

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Trinity Term
2012

Paradoxes of Mathematical Space
Dr Richard Earl, Worcester College, University of Oxford

Through the 19th century and into the 20th, mathematicians began increasingly to uncover surprises (some nice, some nasty) amongst their notions of geometry and space. Seemingly axiomatic ideas like the parallel postulate, crucial notions such dimension and volume were, all of a sudden, much less certain than had been previously been the case. See More

Members' Papers
Various Speakers

This annual event gives our members the chance to shine by presenting their own papers. The usual time is divided into a number of short lectures by members of the Invariants society. All members are welcome to give a paper! See More

Blowing up a Balloon
Matthew Saxton

When you try to blow up a balloon, the hardest part is near the beginning when it takes a lot of effort to make it expand more than a little. After blowing hard enough, the balloon expands quite suddenly past this stage and then the process becomes much easier. See More

Fuss about Fusion
David Craven

Conjugacy is a fundamental notion in mathematics, and fusion - which makes non-conjugate things conjugate - has been central to modern group theory. In this talk I will take a look at conjugacy and why it is so important, discuss fusion, and give a potted history of the events surrounding its rise to the interface of algebra and topology. See More

Fractional Calculus: differentiation and integration of non-integer order
Marcus Webb

In 1695, Leibniz and L'Hôpital were discussing the newly developed differential calculus by letter. Considering the notation \(d^n/dx^n\), L'Hôpital asked '... and what if \(n\) be \(1/2\)?'. Leibniz replied: 'It will lead to a paradox, from which one day useful consequences will be drawn.' This was the birth of the fractional calculus. See More

Hilary Term 2012 Social

Building Bombs using Quantum Mechanics
Daniel Hoek

Some physicists will tell you that Quantum Mechanics (QM) is a perfectly clear, sensible and well-defined theory that has rigorous mathematical underpinnings (except for the bits that don't). Those physicists either have no idea what they're talking about, or they're lying. The truth is that QM is insane, and my talk will tell the truth. See More

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Hilary Term
2012

How the fish got its spots.
Dr Thomas Woolley, St John's College, University of Oxford

2012 marks the centenary of a true mathematical genius and war hero, who's short life met a tragic end. Alan Turing is probably best known for his huge advances in computational logic and breaking the enigma code. However, very few people know about his groundbreaking work in biology. This evening's talk is a celebration of his counter intuitive ideas that captures the breathtaking beauty of the natural world with amazing simplicity. See More

A model of learning under uncertainty
Prof. David Leslie, Lancaster University

Reinforcement learning is a very popular model of machine, human, and animal learning in which the value of taking actions in different states is estimated online from observations. We will observe that convergence of the learning relies only on the fact that averages converge to expected values. However these simple convergence results only hold when an individual receives unambiguous information as to which state the world is in. See More

Complex Numbers, Quaternions and Beyond
Sam Evington, Undergraduate Mathematics Seminar

In 1835 Hamilton gave a rigorous definition of a complex number as simply an order pair of real numbers \((a,b)\). He defined addition coordinatewise and multiplication by \((a,b)(c,d) = (ac-bd,ad+bc)\). He then went on to show that the expected properties follow from these definitions and nothing else. There’s no need to pluck a square root of \(-1\) out of thin air! See More

Members' Papers
Various Members

DNA, Cells, Organisms and Populations - A Tour of Mathematical Biology
Annekatherin Meiburg, Undergraduate Mathematics Seminar

I specialise in Mathematical Biology. – Oh, so you do Genetics and Statistics and stuff? – No actually I am really bad at Statistics. – Then what do you do? See More

Games of Persuit and Evasion
Prof. Imre Leader, University of Cambridge

A scorpion wants to catch a beetle, a porter wants to catch a student, and a lion wants to catch a man. The beetle, student and man do not want to be caught. What tactics should they adopt? See More

Berry Phases
Jonathan Sykes, Undergraduate Mathematics Seminar

A discussion of the Gauss-Bonnet theorem's application to Berry phases. The lecture will briefly introduce the Gauss-Bonnet theorem for simple closed curves. It will then move to derive the Berry phase for a laser light in a helix and for magnetic field of a monopole. Conclusion will mention a brief discussion of measurement and further applications. See More

Infinitesimals
Kobi Kremnitzer

After the work of Cauchy and Weierstrass on the foundations of analysis it seemed like infinitesimals are unnecessary incoherent entities. Infinitesimals have survived and flourished in algebra, algebraic geometry and model theory, even after the brutal attack on them by analysts. I will discuss synthetic differential geometry. See More

Beyond the Möbius Strip
Will Perry, Undergraduate Mathematics Seminar

Consider the phenomenon of a vector space attached to each point of a shape, much as the Möbius strip can be formed by attaching a line to each point of a circle with some global "twisting". See More

The Return on Research
Michael Peyton Jones

Rubik's Magic Cube in Oxford
Dr Peter Neumann, Queen's College, University of Oxford

Rubik's Cube first came to Oxford in September 1978, brought by Professor Roger Penrose from the International Mathematical Congress that had been held in Budapest that summer. It has been enjoyed, analysed, solved, written about ever since. See More

The Story of Arithmetic Descent Part 1
Simon Myerson, Undergraduate Mathematics Seminar

In 1659 Pierre de Fermat claimed to have "astonished the greatest experts" with his methods of ascent and descent. They were the first attempt at a systematic approach to solving polynomial equations in whole numbers (Diophantine equations). See More

Cambridge Problems Drive

Trapping a line with a curve
Prof. David Epstein, University of Warwick

I will talk about a problem that I have used in the past to motivate undergraduate courses on topology or on metric spaces. In fact I would usually present 6 different problems at the beginning of each course, but I'm only planning to talk to the Oxford Invariants about one of them. See More

The Story of Arithmetic Descent Part 2
Simon Myerson, Undergraduate Mathematics Seminar

This is the second of two seminars on this topic, but it is completely independent and no prior knowledge is required. See More

Annual General Meeting 2012

The AGM is a chance for members to make their mark on the society by raising important issues or running for committee positions. We are holding elections for every committee position, so come along and get involved. See More

Graph Theory: an Introduction to Cycle Spectra
Lauren Kutler, Undergraduate Mathematics Seminar

One attribute of graphs that can be investigated is cycle length. Roughly, a graph has a cycle of length n if you can pick some vertex, then travel along n edges back to that initial vertex without going through the same vertex twice (excepting, of course, the initial vertex). See More

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Michaelmas Term
2011

Problems, Pizza and the Electric Guitar
Dr David Acheson, Jesus College, University of Oxford

What is it, exactly, that makes for a good maths problem? Is there a place for pizza in serious mathematics? And what has all this got to do with playing the guitar? These are some of the questions we will consider in an off-beat look at some of the highlights of mathematics. See More

Great Mathematicians
Dr Raymond Flood (formerly of Kellogg College) and Prof. Robin Wilson (Pembroke College and the Open University)

Dirichlet's Theorem on Arithmetic Progressions
Phil Tootill, Undergraduate Maths Seminar

One of the best known proofs in mathematics is Euclid's proof that there are infinitely any prime numbers. Arguments similar to Euclid's can be used to show that infinitely many primes lie in certain progressions, but fail to show the result in full generality. Using the methods of analysis, Dirichlet's theorem shows the result for any progression \((a+cn)\), with \(a\) and \(c\) coprime. See More

Can computers deal with infinite objects in finite time?
Martin Escardo, University of Birmingham

For example, can they calculate with real numbers represented as infinite sequences of digits, so that all digits printed at any given time will be correct, and so that they print digit after digit in a never ending fashion? Can they check an infinite number of possibilities and answer yes or no after calculating for a finite amount of time? See More

Careers Talk
IMA

Someone from the IMA will talk to us about careers at 4.00. See More

Quantum Cryptography
Matt Williams, Undergraduate Maths Seminar

Throughout history the desire to securely share secret messages has driven research into cryptography; from the simple Caesar cipher to the Public Key Cryptography systems that protect our credit cards every time we buy textbooks on Amazon. The discovery of quantum mechanics at the start of the last century fundamentally changed the way we think about the world around us. See More

Maths In and Out of the Zoo
Prof. Chris Budd, University of Bath

Is maths a tame animal, confined to the classroom or university, or is it a wild animal, free to roam in the jungles of the worlds problems. My talk will try to answer this question by taking maths on a tour around the zoo. See More

Contact Line Dynamics of an Evaporating Droplet
Matt Saxton, Undergraduate Maths Seminar

The dynamic contact angle is the angle between a moving liquid/vapour interface and a solid surface, measured within the liquid at the contact line where the three phases (solid, liquid, gas) meet. There is empirical evidence that the contact angle is related to the velocity of the contact line by a so-called contact line law. See More

The Mathematics of Ageing
Dr David Steinsaltz

Why do humans and many (but not all!) other organisms build up elaborate bodies over the course of months or years or even decades, but are then incapable of maintaining what they have built up, so that they gradually (or in some organisms very rapidly) become weaker and more susceptible to disease and death as they age? See More

D-modules and Automated Proof
Chris Campbell, Undergraduate Maths Seminar

The theory of D-modules is an rich and deep algebraic theory that describes differential equations that, since it's inception in the 60's, has left few areas of mathematics untouched. The algebraic structure allows for the use of algorithms designed for attacking systems of polynomial equations to be used for differential equations. See More

The Friendship Theorem
Martin Liebeck

This theorem that tells you that if you go to a party where any two people have exactly one mutual acquaintance, then there is someone who knows everybody. Apart from its obvious fundamental applications to the theory of going to parties, it also has a wonderful proof which uses a completely different and unexpected part of maths. The talk should be comprehensible to all undergraduates, including those who have just started. See More

A Logic with Computable Infinite Expressions, for Foundational Study
Catrin Campbell-Moore, Undergraduate Maths Seminar

It would be good to find a logic which can characterise the natural numbers. In order to give a logic that can both express the natural numbers and have certain other desirable properties we need to allow infinite conjunctions. See More

Hypercomputation: exploring the extreme theoretical limits of computing
Dr Toby Ord, Balliol College, University of Oxford

In computer science, there are many theoretical machines in a hierarchy of increasing power. The most powerful is the Turing machine, which is roughly like a desktop PC with unlimited time and memory. See More

Knot theory and the Jones polynomial
Jakob Blaavand, Undergraduate Maths Seminar

In this talk we will introduce the basic notions in knot theory. We will start by defining what we mean by a knot and a link. A mathematical knot is almost the same as knots in the real world. If you tie a knot on a rope, you have to glue the ends of the rope back together again, so that your knot is on a circle. A link is just several knots, which might be linked together. We will also define, what it means that two knots are isomorphic. See More

Christmas Party!

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Trinity Term
2011

Proof by Chocolate, and other Goodies
Dr David Acheson, Jesus College, University of Oxford

What do we mean, exactly, when we say that a mathematical proof is elegant, or even beautiful? We will explore many examples from both pure and applied maths, ranging from the elementary and well-known to rather more exotic methods...including proof by chocolate. See More

Members' Papers
Various Speakers

Who's afraid of the universal set?
Dr Brian King, Junior Research Fellow, Worcester College, University of Oxford

It is a straight-forward theorem of standard set-theory (ZF) that there is no set of all the sets, the so-called universal set. But why? See More

Interactive Proofs
Jakub Závodný, Computing Laboratory

In mathematics, a proof is a monologue: a coherent and self-contained argument, conclusively convincing any reader about the truth of some proposition. Computer scientists, however, turned the proof into a dialogue between a prover and a verifier. See More

Trinity 2011 Punting and Social

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Hilary Term
2011

Noncommutative Geometry & Physics
Dr Keith Hannabuss, Balliol College, University of Oxford

Keith Hannabuss' research includes Quantum Theory, Noncommutative Geometry and Operator Algebras. In the talk, we will see how some of these interact with each other, stressing the importance of geometry in the study of physics. See More

75th Anniversary Talk: Euler – 300 years on
Prof. Leonhard Euler, Imperial Academy of Arts and Sciences, St Petersburg

The Invariant Society celebrates its 75th anniversary in style with a man who made large contributions to a great variety of fields, including the famous 'Seven Bridges of Königsberg' problem. Prof. Euler will reminisce about his life and mathematical works. See More

Information is Beautiful (with OU Scientific Society)
Mr David McCandless

Data visualisations and information design is a rising trend across many disciplines: science, design, and journalism. Some interesting and unexpected things can happen when you visualise data, knowledge and ideas. See More

Cosmology
Prof. Sir Roger Penrose, Wadham College, University of Oxford

Renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology, Sir Roger Penrose has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking. He will speak to us about the wonders of the cosmos and some exciting recent advances. See More

Members' Papers
Various Speakers

A Variety of Flags
Dr Jan Grabowski, Keble College, University of Oxford

To a mathematician, a flag is a collection of subspaces of a vector space. The collection of all flags has a fascinating geometrical and algebraic structure; we will explore some examples and a few related topics. See More

Emmy Noether
Prof. Harvey Brown, Wolfson College, University of Oxford

Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. David Hilbert, Albert Einstein and others described her as the most important woman in the history of mathematics. We will look at her work in physics. See More

Mythical beasts in algebra
Dr James Cranch, University of Leicester

For decades, mathematicians have motivated much work by referring to several deeply interesting algebraic objects which sadly fail to exist under the normal definitions. We will see several of these. See More

Annual General Meeting

The AGM is a chance for members to make their mark on the society by raising important issues or running for committee positions. We are holding elections for every committee position, so come along and get involved. See More

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Michaelmas Term
2010

All you ever wanted to know about pi
Prof. Robin Wilson, Open University & Pembroke College, University of Oxford

The circle number pi has fascinated people for thousands of years?. Who first called it pi? Who had it engraved on his tombstone? Who tried to pass a law declaring it to be 3.2? See More

How to Be as Small as Possible
Dr David Acheson, Jesus College, University of Oxford

Some of the most attractive and thought-provoking problems in mathematics involve minimisation. We will take an informal look at some examples, ranging from elementary geometry and packing problems to soap bubbles and catastrophe theory. Practical demonstrations will be included. See More

The Mathematics of Smallpox
Prof. Tom Körner, Trinity Hall, University of Cambridge

Smallpox is a dreadful disease. In the 18th century a method for preventing it was introduced into Europe - but the method itself was dangerous. Daniel Bernoulli's discussion of the resulting dilemma remains relevant. See More

Challenging Mathematics
Dr Peter Neumann, Queen's College, University of Oxford

Peter was the chairman of the UKMT for over 7 years which is best known for the Junior, Intermediate, Senior, and Team Maths Challenges. The talk will be an account of the UKMT and its work featuring some of the more challenging problems, both mathematical and non-mathematical. See More

The Secret Mathematicians
Prof. Marcus du Sautoy, New College, University of Oxford

Artists are constantly on the hunt for interesting new structures to frame their creative process. From composers to painters, writers to choreographers, the mathematician's palette of shapes, patterns and numbers has proved a powerful inspiration. Often subconsciously artists are drawn to the same structures that fascinate mathematicians. See More

Indra's Pearls: A mathematical adventure
Prof. Caroline Series, Warwick University

This talk is based on the book of the same title by David Mumford, Caroline Series and David Wright. It tells the story of the authors' computer explorations of the geometry of repeated Moebius maps, whose interactions produce intricate fractals. See More

Adding Prime Numbers
Prof. Ben Green, Trinity College, University of Cambridge

Ben Green is the Herchel Smith Professor of Pure Mathematics at the University of Cambridge and Fellow of Trinity College. He has also held positions at Princeton, the Rényi Institute, PIMS, MIT and Harvard. See More

Christmas Party