An A-Level to University guide on all things Maths

Have you looked at maths degree modules and had no idea what any of it means? Compared to A Levels, university modules can sound terrifyingly vague. What in the world will you study in Methods? But fear not, this guide will break down how the A Level Maths topics you enjoy link to the modules you can study at UCL, so you can tailor your UCL degree to what you like learning. Taking a look at the mandatory modules I studied in first year:

If you like… well all of A Level Maths, you’d enjoy Mathematical Methods!

Mathematical Methods (or the shortened name ‘Methods’) is a continuation of almost all your favourite A Level Maths and Further Maths topics like vectors, polar coordinates, integration, differentiation and more! By balancing new ideas with revision of familiar concepts, UCL structures this course to make studying undergraduate maths friendly and never monotonous. 

If you like sequences, series and proofs, you’d enjoy Analysis!

Analysis 1 and 2 dissect the reasoning behind classic maths topics such as calculus and functions. Building on A Level proof types like contradiction and induction, new notation is introduced to prove fundamental ideas around limits and differentiation. These concepts are standard in any undergraduate maths degree but UCL’s well-structured, supportive framework means that you’ll understand the reasoning every step of the way. This module is a favourite amongst most students as it uses clever reasoning to provide dimension to ideas you already know, making the content feel relatable. So if you’ve always asked why in class (why is the limit of 1/x = 0 as x tends to infinity? Why are we able to differentiate this curve?) and love the certainty of maths, Analysis at UCL is perfect for you!

If you like matrices, functions and set theory, you’d enjoy Algebra!

If Analysis focuses on the fine print, Algebra’s about the big picture. Algebra 1 and 2

discusses what links mathematical objects (such as matrices and vectors) together – what can we say about the way these groups behave? The first year centres around the underlying complexity of different linear equations like functions and matrices. This unlocks greater depth to the ‘simple’ A Level topics you know to create cohesion between Algebra and other mandatory modules. Algebra 1 and 2 are the modules I enjoy the most because its straightforward concepts are easy to conceptualise. This makes the ideas grounded and logical, enabling them to feel accessible and engaging.

UCL’s mandatory modules build a strong mathematical base which allows you to confidently explore any area you choose, inviting you to shape your own education so you study what you’re interested in. This personalisation is provided through the wide range of UCL maths degree programmes and modules you can pick. So, looking at the modules on offer:

If you like modelling, differential equations and conservation of energy, choose Applied Maths!

This mandatory module for Mathematics BSc students provides a great foundation on how maths is utilised in real life. Using ideas like second order differential equations and conservation of energy, this module applies pure maths concepts to build different population and oscillation models.

If you prefer statistics, why not pursue Mathematics and Statistical Science BSc?

This combined-studies degree allows you to take three Statistics modules in First Year:

•   Introduction to Probability and Statistics
•   Introduction to Practical Statistics
•   Further Probability and Statistics

With an emphasis on probability, distribution and coding in R, these modules incorporate data from a range of fields such as astronomy, medicine and finance to show theory in action!

And if you prefer less maths-focused topics, check out the other degree programmes you can take at UCL:

• Mathematics with Mathematical Physics BSc
• Mathematics with Economics BSc
• Mathematics with Modern Languages BSc
• Mathematics with Management Studies BSc
•  Mathematics and Physics BSc

With the cohesive structure and the flexibility to individualise, UCL is the perfect place for you to study maths depending on what you enjoyed at A Levels!

You can visit our module information page for more details on the syllabi of each module.

You can find more details on different degree programme structuresere. on the UCL Mathematics website.

Written by: Teren Lee, UCL Mathematics

Christmas, a beloved tradition for many families around the world, which has held a special place in the hearts of Christian families since the 9^th Century, has now become a universal tradition, allowing different cultures to adopt the holiday and shape it as per their wishes. However, with the advent of the industrial revolution and its associated technologies, it has evolved into an increasingly non-denominational event with a decidedly commercial nature, thanks to advances in manufacturing, shipping logistics, and marketing, all brought by technological change.

The mass production of consumer goods has forever changed the tradition of gift-giving on Christmas (and other holidays or festive occasions such as birthdays, anniversaries, Bat Mitzvahs, etc.).

Gone are the hand-crafted trinkets and ornaments of the past – the modern Christmas relies on a steady flow of commodities from manufacturer to consumer, making products that were once expensive and rare cheaper and more widely available than ever before. For instance, in the past, there was a practice among the wealthy to give citrus fruits on the holidays. It was a symbol of status – citrus fruits are out of season in the winter, and must be imported to a rainy, cool place like Britain. It was a sign that the giver of such a gift had the means to afford such a luxury. However, when considered in our modern context, this practice seems quaint. In fact, it is quite easy to picture a child’s face, full of disappointment while receiving a citrus fruit when all they really wanted was a PlayStation.

Furthermore, the wide availability of these commodities is facilitated by the vast, somewhat reliable network of modern shipping and logistics. Planes, trains, trucks, and powered boats have made access to merchandise much quicker and less expensive. These advances in manufacturing and distribution enabled by technology have led to cheaper, more plentiful consumer goods, and modern Christmas is changed because of it. In some sense, it is a positive change – revolutions in these sectors have made it easier and cheaper to bring joy to a loved one, awarding it with inclusivity. However, the other side of the coin is the negative effects these technologies bring – the rapid increase of pollution and global warming due to the methods in which mass production of goods is executed, amongst other undesirable effects.

The current state of Christmas is inextricable from the technology it is built upon – it would look foreign to us without the mass production of consumer goods, the vehicles and manpower used to distribute them, and the platforms on which they are sold to us. Even the inability to track Santa Claus’ journey across the globe seems rather antiquated. This demonstrates the inevitable social shaping of technology and how technology has reshaped Christmas (along with perhaps every other tradition known to the human being).

Written by: Andréa Lekare, UCL Science & Technology Studies

Are you stuck trying to find a gift for the mathematicians in your life? As much as we love t-shirts with cheesy mathematical puns, it’s great to get something more intellectually satisfying for any mathematician you know this holiday season. Whether you’re passionate about numbers or know someone who loves solving problems, here are some of my favourite gifts that will make any prospective mathematician happy!

1) Maths Books

The Indisputable Existence of Santa Claus: The Mathematics of Christmas

These are great presents for anyone who wants to build their mathematical knowledge and venture into new areas of mathematics. Ever wanted to know how to use game theory to figure out who you should buy Christmas gifts for? Well, ‘The Indisputable Existence of Santa Claus’ written by UCL’s very own Dr Hannah Fry and Dr Thomas Oleron-Evans solves that exact question as well as all other mathematical Christmas conundrums (including the metabolic rate of Santa Claus!).

If these popular science topics pique your interest, definitely check out the rest of Dr Hannah Fry’s novels such as ‘The Mathematics of Love’ which finds patterns in romance and ‘Hello World: How to be Human in the Age of the Machine’ which tackles the good, bad and ugly of algorithms. If novels are too long for your liking, why not gift issues of Chalkdust magazine, a bi-annual magazine tackling all things mathematical? Founded by UCL Alumni Rafael Prieto Curie and published in UCL, each issue uses different areas of maths to solve real life problems so there’ll be something interesting for anyone who’s mathematically curious. Click here to check out some of their articles and order a copy!

2) Puzzles

Mathematicians are driven by a love of solving problems so gifting any sort of puzzle will definitely be fun. From chess sets and sudoku books, to mathematical board games like Decrypto and mechanical puzzles, the possibilities are endless. I’d particularly recommend puzzle books by Raymond Smullyman as they blend recreational maths problems with different fields of logic and set theory, the perfect brain teaser for on-the-go or a relaxed afternoon.

3) Klein Bottles

Klein bottles are key shapes in topology since the neck of the bottle twists into itself such that there’s no outside or inside to the shape. This topological surface isn’t actually possible to make in three dimensions, but 3D approximations are fun gifts as bottle openers, water bottles, lamps and decorative places!

4) Maths Pun utensils

No mathematician can deny the fun of mathematical puns, especially when they are in a practical application. Think ‘proof is in the pudding’ pudding bowls where mathematical proofs are written at the bottom, or a π pie plate. This will definitely put a smile on their face.

I hope this short list gave you some inspiration for the types of presents you can get to develop someone’s passion for maths. Regardless of what you give, what we value most is time with loved ones and a well-deserved break so I have no doubt that anything mathematical will be appreciated.

Happy Holidays!

Written by: Teren Lee, UCL Mathematics

In the run up to Halloween, UCL Science & Technology Studies student Andrea explains the Day of the Dead, its significance in Mexican culture and how technology helps to connect with traditions.

Do you have math phobia? Well, you’re not alone! To celebrate the spooky season, UCL Mathematics student Teren asks UCL students to share their tricks to treat the fear of maths. Happy Halloween!

If you’re following UCL Mathematical & Physical Sciences on social media, or on this blog, be prepared to meet our Digital Student Ambassadors!  Our Ambassadors are current MAPS students who are happy to share their experiences studying at UCL and give you a real taste of student life here at UCL, by creating heaps of exciting content including videos, blogs, podcasts and social media takeovers!

To celebrate the second Black Mathematician Month, Chalkdust magazine and UCL maths department organised a maths workshop day in Tottenham for 60 Year 9/10 students from 6 local schools. The aim of the day was to highlight the minimal progress of diversity in our field and to improve it in the best way possible. After the success of last year, we were very excited to showcase our passion in Tottenham. We hope that it will encourage young black students to engage more with maths, enjoy it and view it as an important skill for employment.

To kick start the day, the MP for Tottenham, David Lammy gave a speech about the importance of believing that you can achieve anything, regardless of what your upbringing may have been like. Lammy spoke of WhatsApp founder Jan Koum, who had to migrate to USA as a child with his parents, who both passed away soon after. Lammy also spoke about the recent Brexit proposal and how his team of researchers were needed to check it and discuss how it would impact the public’s finances. Each of these anecdotes served the purpose of highlighting the significance of maths in life’s difficult situations. Ultimately, the technological and economic climate is changing quickly, and Lammy hopes that maths students – such as the ones who were sat before him – will be well-placed to lead the way.

Keen to remind us that maths isn’t all just formulas, Dr Nira Chamberlain then arrived with an enthusiastic, interactive and humorous presentation about how maths can be applied to the real world. All the students thoroughly enjoyed these talks and could not wait for the workshops.

These workshops included fun and intriguing maths ranging from Number Theory, Modelling to Frieze patterns and Topology. They were run by UCL PhD students and Dr Naz Miheisi from Kings college. In the topology class, we made several types of Möbius loops, combined them and understood their properties. A discussion was also held about the Klein bottle and Projective plane. The students were all trying to predict what the different shapes would turn out as, but were completely astonished by the results. Some did say they would make great Christmas decorations!

In the afternoon, UCL undergrad volunteers formed a Q&A panel to answer numerous, interesting and funny questions from the pupils, some including about turning up to lectures! On a serious note, the students did find this very helpful and informative, further establishing whether maths is for them.

To round off the day, there was a question relay between schools where the winning team got their very own Chalkdust T-shirt. The pupils did get competitive against each other, albeit for the love of maths!

For me personally, I loved this whole day and the idea of inspiring the next generation felt very rewarding and I will be hoping to do more outreach events like this in the future.

To conclude, thank you to London Academy of Excellence for hosting the event. The feedback from the schools shows that they thoroughly enjoyed the day and that it will really influence their pupils’ future decisions. After all, that’s what this Month is all about!

Written by Amin Sabir

This July, University College London (UCL) hosted the first ever Science and Technology Facilities Council (STFC) summer school in Artificial Intelligence (AI) and Machine Learning.
120 PhD students from universities across the country were brought together to spend a week and a half learning about the forefront of AI technology. With lectures from several industry leaders; including Intel, NVidea, ASI, and Dell, the summer school aimed to give a solid grounding in the basics of machine learning before providing a glimpse into some of the latest technologies now being applied.

This included being granted access to some of the best interactive platforms currently available; and allowed students to follow along with demonstrations during the lectures. To start with, ASI’s ‘SherlockML’ enabled students to familiarise themselves with the most common tools used in machine learning. By making use of the Intel ‘DevCloud’, students were then shown how to optimize Neural Networks in TensorFlow and Caffee, before learning about computer vision and making use of the Movidius stick to be able to turn a webcam into a tool to classify everyday objects. NVidea also demonstrated how to make the most out of currently available computing clusters, using Graphics Processing Units (GPUs) to parallel process tasks, and students were taught how to build a convolutional neural network in ‘Digits’ which could then classify whale faces.

On top of the extensive technical training there were additional guest lectures highlighting how machine learning can be applied to a variety of different fields, from high energy physics and astrophysics to in industry. It was also made evident how machine learning techniques will be vital in the future of astronomy with new methods such as gravitational waves and new observatories like the Square Kilometer Array (SKA) which will produce petabytes of data every day.
Over the weekend a further 100 sixth form pupils were invited to take part in a hackathon-style event lead by the visiting PhD students, where during the afternoon the students demonstrated how machine learning could be applied in a real research scenario at the Large Hadron Collider (LHC).

One of the best things about assembling people from such a wide range of disciplines, with a common interest in machine learning and AI, was the incredible diversity of the work presented. Over the course of three poster sessions, we had the opportunity to discover what was being researched; from classifying solar winds and galaxies to using neural networks for novel detection methods in the LHC. And for one evening, this was combined with a chance to meet and talk with professionals from various CDT industry partners with companies like ASOS, TFL, and NCC Group.

As well as networking with the industry partners, a key aim of any summer school is to develop links between the students themselves. Although never an issue when helped along with some free food and drinks, it was especially encouraging to see everyone come together for the quiz night, with each team setting a round of obscure questions, and the school ended in spectacular fashion making use of the record-breaking summer with a river cruise BBQ on the Thames.

The Summer School was directed by Jonathan Tennyson, and the LOC/SOC included Tim Scanlon & Jason McEwen (CDT Research Directors), Jaini Shah (CDT Manager),  and Nikos Konstantinidis & Ofer Lahav (CDT co-Directors).

-Written by Ben Henghes (Research Student, UCL Physics & Astronomy)

In 1999 the physics community was stunned when it was discovered that far from slowing down under the pull of its own gravity, the universe is actually expanding at an accelerating rate. This is the cosmic acceleration problem and it poses one of the largest challenges in physics today. Here, at UCL’s Mullard Space Science Laboratory (MSSL) we are leading the effort to find a solution.

Parts for the Euclid Satellite, which will conduct a massive astronomical survey, are being built on site. Meanwhile we prepare for the arrival of the groundbreaking Euclid data by developing and testing techniques to distinguish between theories of cosmic acceleration.

We take advantage of an effect of Einstein’s theory of general relativity called gravitational lensing. As light from distant sources travels towards Earth, it is deflected by the gravitational attraction of intervening mass slightly distorting the shape of the observed image. This effect is normally very small, but by measuring the shape of billions of galaxies, amazingly we can statistically differentiate between theories.  This type of analysis has been conducted before, but to get the most out of next generation experiments we must hone our methods. When comparing the two leading statistical techniques, we found both have serious drawbacks. Nevertheless we have shown that the two methods are just special cases of a more general theory: narrowing our search for a better approach.

Irrespective of the statistical technique that is ultimately used, we must always accurately predict the lensing signal to compare to the real universe. An intermediate step involves determining the power spectrum that describes the ‘clumpiness’ of matter on different length scales, at different times. We are working on a publicly available code that will help researches who generate these spectra determine whether their output is accurate enough to meet the needs of upcoming lensing experiments like Euclid.

Gravitational lensing is very a delicate business. Large collaborations must identify, understand and mitigate annoying instrumental and astrophysical effects that contaminate the lensing signal. UCL is leading the charge in preparation for Euclid’s launch in the early 2020s.

Image credit to ESA

Many students send emails requesting summer internships at UCL Physics & Astronomy, but one particularly caught Dr Anasuya Aruliah’s eye. Vishal Ray was a 2nd year Aerospace Engineering undergraduate at the prestigious Indian Institute of Technology Bombay, India (IIT). He belonged to a student team building their own miniature satellite. Vishal was just the student that Dr Aruliah needed for her new direction of research: satellite drag. After a successful application to the International Students Dean’s Summer Student Scholarships, he was awarded a 2 month internship in summer 2015.
Dr Aruliah’s group, the Atmospheric Physics Laboratory (APL), is a subgroup of the Astrophysics Group. It has a long history of researching the upper atmosphere using a global circulation atmospheric model. They also operate a network of Fabry-Perot Interferometers (FPIs) in Arctic Scandinavia to observe the aurora. The Earth’s atmosphere is like an onion skin, with the troposphere (the domain of weather forecasters), stratosphere and mesosphere as layers on top of each other. The thermosphere is the final layer of the Earth’s atmosphere, and is the altitude region between 90-400km. Low Earth Orbit (LEO) satellites occupy the top of the thermosphere, and rely on upper atmospheric models to predict their orbits.

Recently the APL group found a discrepancy between measurements of thermospheric winds calculated from Doppler shifts of airglow photons, and winds determined from atmospheric drag on the Challenging Minisatellite Payload (CHAMP) satellite. This is an important puzzle to solve because satellite drag measurements are put into atmospheric models to bring them as close to reality as possible. If the ground and satellite measurements do not agree, then which is correct?

The IIT miniature satellite, commonly called a cubesat, is composed of a single cube, only 30 cm in length, width and breadth, and weighing only 10 kg, as much as a few bags of sugar. Their cubesat is called Pratham. This fits perfectly with UCL’s involvement in the European Union FP7-funded QB50 project, in which fifty cubesats carrying miniaturised sensors will be launched nearly simultaneously. This is an international collaboration involving many universities, academic institutes and the space industry. It is an unprecedented science operation, with potential for future Space Weather monitoring campaigns. The QB50 cubesats will be carried by rocket into the upper thermosphere, and fall to Earth in decaying orbits while sampling regions of the thermosphere and ionosphere that were previously poorly understood owing to the lack of detailed measurements.

“The simplicity and low cost of cubesats has spurred much excitement and creativity amongst young (and old) engineers and scientists over the last few years. There are new frontiers being opened by this miniaturised space technology,” said Dr Aruliah.

The UCL Mullard Space Science Laboratory (MSSL) designed and built one of the three key sensors: the Ion Neutral Mass Spectrometer, which will be carried on several of the cubesats, as well as their own cubesat called UCLSat. The QB50 cubesats are scheduled for launch in three batches over the winter period of 2016-2017. Two batches from a Ukrainian-Russian Dnepr rocket, and a third from the International Space Station. During Vishal’s internship at UCL he met with the MSSL cubesat and sensor team, led by Mr Dhiren Kataria and Dr Rob Wicks; and with Dr Stuart Grey in the UCL Department of Civil, Environmental & Geomatic Engineering.

Dhiren Kataria holding the UCLSat designed and built at MSSL

Vishal used his experience to write several sophisticated computer programs to calculate drag coefficients from simulations of a cubesat orbiting in our 3-dimensional atmospheric model called CMAT2. This work was subsequently built upon by Dr Aruliah’s 4th year project student, Jennifer Hall. Jennifer wrote her own programs to derive and compare satellite drag coefficients from CMAT2 simulations and EISCAT radar measurements. Jennifer’s project won the UCL Physics & Astronomy Tessela prize for best use of computer technology in a 4th year project.

Pratham team at the Indian Institute of Technology Bombay. Vishal Ray is 2nd from the right in the top row.

One year on, after a busy 3rd year of studies, Vishal has written up his summer project as a journal paper, and “Pratham” was scheduled for launch at 0530 UTC on the 26 September 2016. The IIT Bombay student team installed their cubesat on the launch vehicle PSLV C-35 on the remote island of Sriharikota in South India. Vishal said that he “…had goosebumps when we actually placed the satellite on the launch vehicle module and completed the testing for one last time!”. “Pratham” was successfully launched and will measure the total electron count from 800 km altitude in a Sun Sychronous Orbit. MSSL were the first to receive Pratham’s beacon signal, which the students were incredibly excited to hear. You can hear the cubesat from 4:20 onwards as it passes within range of the detector at MSSL. The signal is decipered as “Pratham IIT Bombay Student Satellite”. The accompanying image is of Theo Brochant De Viliers (MSSL) beside the MSSL receiver.

The prospect of finally being launched is very exciting, with both projects having been nearly 10 years in the making. Once launched, the missions will change from the technical challenges of the innovation of miniature sensor devices to the scientific challenges of collecting, analysing and interpreting the measurements. The rewards will be great: from the new technologies surrounding cubesats; to the training of future space scientists and engineers, and to the Space Weather community.