maths academy 2013-2014 homepage
The Maths Academy 2013-2014

The Maths Academy is running two projects in the 2013-2014 academic year; an Interactive Lecture Series and a University Mathematics Project. You can find details of lecture dates, times, and locations in the descriptions below, and registration and contact details at the bottom of the page.

Interactive lecture series

This is an interactive lecture series aimed at eager and interested year 12 and year 13 students. The lectures will be highly interactive and so the session content may differ from the titles and descriptions below. There is a restricted number of places and priority will be given to year 12 students. All lectures and tutorial will take place in Hicks Building (see the map below).

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Semester 1

The Semester 1 sessions were held in Lecture Theatre 5 (Floor E) of the Hicks Building at the University of Sheffield. Session titles, times, dates, and further reading are given below.

What is pi? by Dr. Fionntan Roukema

You probably met π in primary school, but what is it? In this talk we will think about π and realise that real numbers depend on a limiting process. This limiting process was not "properly" understood by Newton! In this talk we may recover a powerful and beautiful 19th century construction that will give you a glimpse of a university topic called mathematical analysis.

Location/time: 07/10/2013, 16:00-17:15, LT5 Hicks Building.

Remarks and links following the session:
  • Everyone did very well and we were all extremely impressed by how you managed to rediscover the beginnings of one of the most important mathematical stories ever told! Very well done to you all!
  • We were supposed to see some great questions, and hear a great song but unfortunately we were let down by some audio visual problems...
  • We uncovered a powerful and fundamental concept; the idea of taking a limit. This limiting process underlies all of the calculus and most of the physics that you will do at A-level. Here's a neat video that illustrates this limiting process and points out a couple of incredible facts about pi.
  • We didn't quite get there, but we were on route to constructing the real numbers. If you're interested, you can have a look at the real number wiki page and ask questions at the next lecture.
What can you do with a straightedge and compass? by Dr. Fionntan Roukema

I learnt how to draw a circle in primary school. What I did not learn was how to use a compass and a straightedge (a ruler with no markings) to do nice geometric constructions. More remarkably, I did not learn a stunning result about straightedge and compass constructions until my final year of undergraduate mathematics! In this talk we may talk about a duel to the death and touch on some deep university algebra!

Location/time: 21/10/2013, 16:00-17:15, LT5 Hicks Building.

Remarks and links following the session:

Elliptical orbits of the planets by Dr. Sam Dolan

Planets move on ellipses around the Sun. Today, this may not sound like a scandalous claim -- but back in 1633 it was enough to get Galileo in trouble! We will learn about the ellipse, and how elliptical orbits are a simple consequence of Newton's laws of motion and of universal gravitation.

Location/time: 04/11/2013, 16:00-17:15, LT5 Hicks Building.

Remarks and links following the session:
  • Another great session; well done everyone!
  • You can find the lecture's presentation notes here and you can find the lecture's worksheet here. Remember; you've been challenged to provide a solution to the difficult exercise 3. The first correct submission will be posted here and we're still waiting; good luck!
  • We saw that ellipses are found at different scales throughout our universe. To get a sense of scale in our universe have a look here.

Aperiodic tilings: kites, darts and the golden ratio by Dr. Sarah Whitehouse

A collection of tiles of various shapes is called aperiodic if it tiles the whole plane (without gaps or overlaps) with no repeating pattern. It is surprising that this is possible, but in this talk you will see several examples, including the famous Penrose kite and dart tiles. You'll learn how to make kite and dart tilings and how we can figure out the ratio of kites to darts.

Location/time: 18/11/2013, 16:00-17:15, LT5 Hicks Building.

Remarks and links following the session:
  • We continue to be very impressed by how well everyone is engaging with the advanced material we're presenting. Keep up the great work!
  • Tiles are found in kitchen and in our bathrooms. In this talk we started by think about what a tiling of the real plain should be with a little help from these pictures. In the end decided that a tiling of the plain should cover the plane with tiles without gaps or overlaps. We then went on to speak about "periodic" tilings like (1), (2), (5), (8) here.
  • The ratio of tiles in a periodic tiling with two types of tiles must be rational. We looked at kites and darts tiles (see (5) and (7) here) and introduced some gluing conditions to form "Penrose tilings".
  • We looked at how to inflate and deflate tilings.
  • We summed up by seeing a nice illustration of how interconnected mathematics can be; in the limit (see "What is Pi?") the ratio of kites to tiles was the golden ratio (see "What can you do with a ruler and compass") meaning that a Penrose tiling is aperiodic!
  • For more reading on aperiodic tilings you can have a look at this essay.

How big is infinity? by Sarah Browne, Thomas Cottrell and Gemma Halliwell

A set either has finitely many objects or doesn't, and that's all there is to say, right? Wrong, incredibly, there is a lot to say! Cantor shook the world of mathematics in the late nineteenth century with a wonderful result that will change the way you think about infinity! Come to this talk and learn what all the fuss was about.

Location/time: 02/12/2013, 16:00-17:15, LT5 Hicks Building.

Remarks and links following the session:
  • The mysteries of infinity were in no way intimidating to our visitors!
  • Hilbert could accommodate extra guests at his his hotel even if it was full, but this was nothing compared to what the odd things we discovered once we agreed that two (possibly infinite) sets should be the "same size" when there is a bijection between them.
  • We then looked at odd examples and decided that there were as many even positive integers as there were positive integers, as many integers as there were positive integers, and incredibly that there were as many integers as there were rational numbers!
  • Over mince pies and casual mathematical discussions, some of us were shown how Cantor found a beautiful way to show that some infinities are bigger than others!

Links to further reading will be posted after the sessions above so that you can independently research beyond what you saw during the lecture.

Semester 2

All Semester 2 sessions will be held in Lecture Theatre 10 (Floor H) of the Hicks Building at the University of Sheffield. Session titles, times and dates are given below.

Do knots exist? by Dr. Fionntan Roukema

We tie and untie our shoelaces everyday (or maybe you use velcro in which case you'll have to pretend). So, is it possible to tie a knot that can't be undone? And, if there is such a knot, can we somehow convince ourselves beyond reasonable doubt (prove) that our knot can never be undone? This talk will be about topology (a world of geometry in which distance is irrelevant), and with any luck we might get to a piece of 20th century Fields medal winning mathematics.

Location/time: 10/02/2014, 16:00-17:15, LT10 Hicks Building.
Remarks and links following the session:
  • It was quickly noticed by a few different people that you can never tie up your Christmas lights in such a way that they can't be undone. The reason being that you can always "rewind" the knotting process.
  • We then thought about whether we could make a knot if we were allowed to seal the ends together to form a closed piece of string like those found in Celtic artwork (have a look at some Celtic knots). In this case a knot would be something that couldn't be deformed into a circle.
  • We were certain that not all closed bits of string could be deformed into a circle, but then we saw some videos which made us think otherwise (see this and this).
  • We then turned our statement about being able to deform knots into a circle into a question whether a knot diagram could be changed into a circle via the Reideimeister moves. This gave us a mathematical framework to prove that knots exist by constructing the tricolourability invariant of knots.
  • Tricolourability could only show us that at least one knot existed (we didn't develop a way of telling whether two knots that could both be coloured with three colours could be different). If you want to learn about a very powerful and important knot invariant you can read about the Jones polynomial.
  • As we were thinking about things being the "same" up to some notion of movement, we saw an incredible video of a sphere turning itself inside out! You can see a longer more detailed video giving an account of the process here.

How can maths help when playing games? by Dr. Tim Heaton

This is was an introduction to game theory based on the first lecture of a game theory course taught at Yale University. You can see the course webpage here.

Location/time: 24/02/2014, 16:00-17:15, LT10 Hicks Building.
Remarks and links following the session:
  • Owing to the session coinciding with the half term break, the session was not well attended. This meant that those who attended got a great student-staff ratio, and a very healthy serving of drinks and snacks!
  • You can watch all 24 lectures of the Yale University Game Theory course here. So, if you missed this session, have a look at lecture 1 to see the sort of thing that we spoke about.

Probability, Monopoly and Google by Dr. Jonathan Jordan

Lots of things evolve in time in a way which looks random. Examples can include asset prices in finance, biological populations and membership of social networks. I'll introduce Markov chains, which are a type of probabilistic model with a lot of theory which can be applied in many of these situations, and illustrate it by showing you how to win at Monopoly (well, possibly!) and some of the ideas behind Google's search algorithms.

Location/time: 10/03/2014, 16:00-17:15, LT10 Hicks Building.
Remarks and links following the session:
  • We started thinking about Monopoly, and realised that probabilities starting from Go looking something like this. In the same way, we understood that the probabilities starting from Go To Jail looked like this, and from a Chance square looked like this.
  • Of course, we could play the same game from any square, and it turned out that the whole network looked like this.
  • This all seemed a bit tough to get our heads around, so we started thinking about post-apocalyptic Monopoly where the board only has four squares; Go, Jail, Free Parking, and Go To Jail. Here the picture was much clearer and we found that in the long run 2/17-th of the squares visted are Go, 6/17-th are Jail, 4/17-th are Free Parking, and 5/17-th are Go To Jail. This is an example of a Markov Chain tending to an equilibrium behaviour.
  • We justified the convergence to equilibrium by thinking about the Kruskal count card trick.
  • We then saw the corresponding equilibrium distribution for (an approximation to) real Monopoly, showing the orange and red properties tend to be most landed on.

Humans vs computers: problem solving with the Raspberry Pi by Dr. Sam Dolan

Are you cleverer than your mobile phone? Let's find out! Some mathematical problems can be quickly solved on a computer; others require more subtle and creative reasoning. In this session we will learn some simple programming skills, and test our newfound knowledge against pen-and-paper techniques.

Location/time: 24/03/2014, 16:00-17:15, LT10 Hicks Building.
Remarks and links following the session:
  • This was a hugely popular session. The class was broken into 6 teams of students working with graduate students and lecturers. Each group was aiming to work out their best solutions to these problems. Three teams consisted of students and graduate students/lecturers working with pen, paper and their brains, two teams used Python and a Rasberry Pi and a third team wrote Psuedocodes.
  • After a frantic hour of problem solving the scores were taken and the Humans beat the Machines with a score of 9.5 to 7. Well done Humans!

The students' choice by Dr. Fionntan Roukema

If you've attended the interactive lecture series you will get to vote on what the final topic will be. This talk will be an absolute wild card!

Location/time: 7/04/2014, 16:00-17:15, LT10 Hicks Building.
Remarks and links following the session:
  • Erdős and "The Book" was the students choice taking almost 35% of the votes.
  • We started with an introduction to the great mathematician's idiosyncratic language (all of which can found in his biography The man who loved only numbers). However, we all agreed that his own introduction was much better (see six minutes into this BBC documentary about his life).
  • To Erdős, "The Book", belonged to God and contained the most elegant proofs for every mathematical theorem. Of the hundreds of known proofs of Pythagoras' theorem, we thought that proof 7 (presented in plainer English) from this list of a 101 proofs must be in "The Book".
  • Erdős proved that there were infinitely many primes by showing that the sum of the reciprocals of the primes diverged. This was the content of one of Erdős' research articles (so beyond undergraduate level). We very quickly ran through the proof (so the most enormous Kudos to you if you got any of his argument). During snacks we spoke about some of the things that his told us about the distribution of primes.

Links to further reading will be posted after the sessions above so that you can independently research beyond what you saw during the lecture.

University Mathematics Project

This project invites year 13 students to attend the first year undergraduate module MAS114 Numbers and Groups. The module will be novel and differ from anything you will have seen at school. However, the material will be accessible to anyone who enjoys mathematics. You are invited to attend the first semester of Numbers and groups. There is a restricted number of places and priority will be given to students who attended the interactive lecture series last year.

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