maths academy 2014-2015 homepage
The Maths Academy 2014-2015

The Maths Academy is running two projects in the 2014-2015 academic year; 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.

Contact, support and registration:

To register to attend the Interactive Lecture Series or the University Mathematics Project or if you have any questions, please email Dr. Fionntan Roukema (f.roukema(at)

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. The semester two (February and March 2015) sessions will take place in Hicks Building, Lecture Theatre 6 (see the map below).

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The interactive lectures will take place in Lecture Theatre 6 (Floor E) of the Hicks Building at the University of Sheffield. The lectures will take place at 16:30-17:45 on Thursday the 26th of February, and the 12th of March student presentations session is likely to take place at King Edward VII school (exact location/times for the student presentation to follow).

Last Minute Cowboy Mathematics by Dr. Fionntan Roukema

The initial plan was to hear about Probability, Monopoly and Google, but our speaker was sick and then John Wayne came to the rescue!

Location/time: Thursday 26/02/2015, 16:30-17:45, LT6 (Floor E) Hicks Building.

Remarks and links following the session:
How to Count Without Counting by Magdalini Flari

Although counting is innate, it can very easily get out of hand! We will see how we can answer questions of the type "How many..?" quickly, using techniques that combine more than one area of mathematics.

Location/time: Thursday 12/03/2015, 16:30-17:45, LT6 (Floor E) Hicks Building.

Remarks and links following the session:
  • To follow
Student Team Presentations by representatives from our participating schools

At the end of the Maths Academy we ask students to form teams and tell the maths academy team about some area of mathematics that they have independently researched and found interested. In keeping with our philosophy of no pressure or stress, participation is optional (but encouraged). Talks should be in teams with greater or equal to one person, and the talks should be about 10-15 minutes long. Friends and family are warmly invited with the presenters permission.

Location/time: To be announced.

Remarks and links following the session:
  • To follow.

Links to further reading will be posted after the sessions above so that you can independently research beyond what you saw during the lecture. You can have a look at details of the sessions from semester one below.

Do you take the blue pill or the red pill? by Dr. Fionntan Roukema

This will be a talk about infinity; take the red pill and see how far the rabbit hole goes! In this lecture we'll think about what infinity is, and rediscover a stunning result that shook the world of nineteenth century mathematics.

Location/time: 20/10/2014, 16:30-17:45, LT9 (Floor H) Hicks Building.

Remarks and links following the session:
  • The Maths Academy organisers were EXTREMELY impressed with everyone who attended! Together, as a team, we rediscovered advanced ideas and managed to see that there should be different sizes of infinity.
  • While we didn't get to speak about nDonald's Theorem, we started by asking ourselves whether there was a "biggest number". From there, we realised that there existed sets that don't contain finitely many elements and agreed to say that the size of such a set was infinite.
  • We considered some different sets with infinitely many elements (such as the even natural numbers, the integers, and the rational numbers) but kept finding that we could partner all of these elements up with them up in a "bijective correspondence with the elements of \(\{0,1,2,3,\dots\}\). This boiled down to understanding how to squeeze people into a full hotel with infinitely many rooms. So, all the evidence was pointing to one thing; every set with infinitely many elements should be the same size. That's when we took the red pill.
  • We recovered Cantor's diagonalisation argument and concluded that there are different sizes of infinity! If you would like to see this again, then you can look at this video which covers the argument and leaves you on the trail of some amazing and unbelievable mathematical statements to chase. Enjoy!
Codes and Cryptography by Dr. Ines Henriques

Way beyond the days were a padlock was enough to lock away precious belongings, we will talk about a modern version of secret telling and keeping, using Mathematics.

Location/time: 3/11/2014, 16:30-17:45, LT9 (Floor H) Hicks Building.

Remarks and links following the session:
  • It was fantastic to share my apreciation for Alan Turing's work, and for modern cryptography with you.
  • I hope this talk opened your appetite to go learn more about this amazing piece of British legacy, and that the previews made you want to go watch The Imitation Game at the cinema.
  • You may also be interested in this exhibition.
Topology and the Euler characteristic by Sarah Browne

Leonard Euler observed a formula regarding relating a number to a polyhedron in 1751. In fact it gives a way of distinguishing between different spaces. We will learn in particular what this means and see how to calculate the Euler Characteristic.

Location/time: 17/11/2014, 16:30-17:45, LT9 (Floor H) Hicks Building.

Remarks and links following the session:
  • You all impressed us with your understanding of the concept of topology.
  • We learnt that topology is the mathematical study of spaces with respect to the properties preserved by deformations, twisting and stretching of spaces, and this in no way could include tearing our spaces.
  • We saw that a coffee mug can be deformed in to a torus (sugar ring doughnut), to see this again following this link:
  • We looked at two ways of deciphering between spaces: The Euler Characteristic, a number that represents any polyhedron with no holes and the alterations for a space with holes, but also how we can view spaces by the circles/loops they have on them.
  • We saw that a circle can be represented by the integers when we think about its loops, since you can go around multiple times, and also go clockwise and anti-clockwise.
Big numbers by Dr. James Cranch

At the beginning of the term, Fionntan spoke about infinity. How high we can make our numbers, without hitting the ceiling of infinity? Well, as high as you want, obviously! We'll talk about how to make big numbers efficiently.

Location/time: 1/12/2014, 16:30-17:45, LT9 (Floor H) Hicks Building.

Remarks and links following the session:
  • Thanks for coming! I hope everyone enjoyed it!
  • Here is the first of a series of youtube videos on the subject. They cover the same ground a bit differently (and go much, much further than I could in my talk):
  • For those who have access to a good library or a good bookshop, J E Littlewood's book "Littlewood's Miscellany" is a classic read. It has a short chapter on the uses of large numbers, covering some of the same ground as my talk (but more real-life large numbers).
Aperiodic tilings: kites, darts and the golden ratio by Prof. 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: 15/12/2014, 16:30-17:45, LT9 (Floor H) 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".

University Mathematics Project

This project invites year bright and interested year 13 students to attend a series of lectures based on the first semester's material from a first year undergraduate module (MAS114 Numbers and Groups) taught at the University of Sheffield. 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. The course will be about the fundamental properties of numbers. There is a restricted number of places and priority will be given to students who attended the interactive lecture series last year.

  • Lecturers: The people who will lecturer this module are Dennis Almeida, James Cranch and Fionntan Roukema.

  • Material: Relevant material for this module will be posted on this site.

  • Lecture locations and times: Lectures will take place on Tuesdays and Thursdays afternoons from the week starting 8/10/2014 until the week ending 17/12/2014. All lectures will take place in the Hicks Builing (see the map below) at the University of Sheffield. Times, dates, and rooms can be found at:

    • Tuesday lectures, 7th of October - 14th of December, will take place at 16:30-17:30 in F41 of Hicks Building.

    • Thursday lectures, 7th of October - 20th of November, 4th of December - 16th of December, will take place at 16:30-17:30 in LT11 of Hicks Building.

    • Thursday the 9th of October, will take place at 16:30-17:30 in F35 of Hicks Building.

    • Thursday the 27th of November, will take place at 17:00-18:00 in F30 of Hicks Building.

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