Contours 2016-17

uoemaths

Stories from the School of Mathematics. Undergraduate students interview researchers to find out what the life of a mathematician is like.

Contours

Stories from the School of Mathematics


The team

IMAGE CREDITS

P7: ©ISTOCK.COM/DANIL MELEKHIN

P8: ©ISTOCK.COM/ALEKSANDARVELASEVIC

P13: ©ISTOCK.COM/EDWARDSAMUELCORNWALL

P14: ©ISTOCK.COM/IGOR ZHURAVLOV

2 Contours


Contents

4

6

7

8

10

12

14

16

18

20

22

23

Contours 3


Projects: Knot as hard as they seem

Fourth Year Project


4 Contours


Summer Vacation Project


◄ The output from Imogen's Matlab

program, showing a 12x24 Celtic

plait, with each link component

of the plait coloured differently.

Find out more about options for the

summer vacation scholarships online:

www.wiki.ed.ac.uk/display/SciEngSc

hol/Scholarships+Overview

Contours 5


MathPALs

All first‐year students can take part in a weekly MathPALs session

led by two higher‐year students who have been trained in Peer

Assisted Learning.

The sessions are informal, giving students a chance to ask

questions and meet other students on the course – as well as

benefiting from the experience of the higher‐year students.

“the best way to

learn and improve is

to ask questions,

and to learn from

other peoples’

experiences”


6 Contours


The Alan Turing Institute


Why Alan Turing?

There is no doubt that Alan Turing is

a giant in the world of computer and

data science. During the Second

World War, Turing played a pivotal

role in decoding intercepted German

messages – work which is thought to

have shortened the length of the war

by two to four years. But Turing was

also a pioneer in computer science,

taking steps towards artificial intelligence

many years before it became a

mainstream idea. It was partly thanks

to his ability to combine techniques

from mathematics, computing and

statistics that Turing was so successful,

and so he truly represents the

spirit of modern data science research.

Naming the institute after

Alan Turing seeks to to give people a

visionary understanding of its aims.

Contours 7


From molecules to Big Data

8 Contours



Thermostat methods

Say we want to find out the distribution

of certain particles in a system.

The movement of these particles can

be simply modelled using Newton’s

equations. In reality, this model is

fairly poor, because it doesn’t take

into account friction, and other

forces, amongst the particles. Taking

these forces into account, we obtain

a modelling equation with two extra

terms.

But when we simulate the system using

the equations, we find that the

temperature of the simulated system

changes over time, when this

doesn’t actually happen in real life.

To solve the problem, mathematicians

like Ben and his PhD student

Xiaocheng Shang introduce what’s

called a ‘thermostat’ into the system.

Like a thermostat that keeps a

house a constant temperature, a

thermostat in this context is an extra

term in the modelling equation that

regulates the temperature of the

model. Similarly, when studying

large sets of data, introducing a

thermostat is essential to remove

the ‘noise’ that appears when

sampling.

Various thermostat methods exist,

but the traditional ones are not appropriate

when looking at such large

sets as those studied in Big Data research,

being either too slow or too

inaccurate. The method produced by

Ben and Xiaocheng has performed

very well in tests, converging much

faster than former methods, and

having a high accuracy. Similar

methods are also used by Google, in

machine learning – the study of pattern

recognition. It’s this that allows

Google to optimize its search results

so that you can find what you’re

looking for faster.


Contours

9


Hunting for Data

“You get to derive

cool results and then

on top of that you

can play with

numbers”

10 Contours


XKCD.COM/1132

▲ This is one of Ruth's favourite XKCD comics. Next year, Ruth will be teaching

a postgraduate course called Bayesian Theory, as part of the newly

established MSc in Statistics with Data Science.


Statistics is very important in Big

Data research because statistical

modelling allows you to deal with

missing data in formal ways. The

techniques Ruth has worked on can

be really useful when your data set

is imperfect, for example, if you

don’t have full access to people’s

salaries because of privacy.

Ruth is on the programme committee

for the Alan Turing Institute (see

p.7), providing advice regarding the

scientific and innovation programmes

for the institute, and is

also a part of the recruitment committee

for the institute.

Contours 11


The Stochastic Nature of Life

“Research is not

always easy. It can be

a lengthy procedure,

and sometimes

frustrating as things

often don’t work out

the way you intend

them to. As such, it’s

imperative to really

enjoy the process, not

just the end result.”

12 Contours



José is one of the founding members

of the Edinburgh Mathematical

Physics Group. Founded in 1999, it

is a joint research institute made up

of the Mathematics departments of

the University of Edinbugh and Heriot‐Watt

University. The group covers

many areas of research, including

Gravitational physics, noncommutative

geometry and field theory

amongst many others.

To find out more on the group visit

https://empg.maths.ed.ac.uk/

Contours 13


Great Waves

“...you have to go

further than anyone

else to make

progress. You’re way

more likely to do

that if you’re

passionate about

what you’re doing.”

14 Contours



Maths in Action

This year, Lyuba has been teaching a

brand new course which is closely

related to her research interests.

Mathematics in Action is aimed at

students in years 4 and 5, and focuses

on using mathematical techniques to

analyse real‐world data – usually

building on mathematics that appears

in earlier courses. The course is very

hands‐on, giving practical experience

of analysing data in Matlab – ranging

from comparisons of different imageprocessing

methods, to analysing Met

Office data on the mean temperature

in Scotland.

Contours 15


Analysing a Hiro

“His office is full of

academic books, and

when asked if he has

any hobbies, Hiro

gestures towards the

books and says: ‘I

think of mathematics

as a hobby’”

16 Contours



Contours 17


Travels in mathematics

“This could have

been a safe position

for a settled life…

I decided to try

something new”

18 Contours


▲ Vanya's Brompton bike has been with him around the world. These are just a few of the images he has posted on his

website – you can see the full collection at www.maths.ed.ac.uk/cheltsov/brompton


Contours 19


Spreading Inspiration

20 Contours



▼ As an undergraduate, David made a series of stained glass windows to illuminate mathematical concepts he found

inspiring. The common tile motif at top and bottom is a ‘proof by pictures’ of the Pythagorean theorem.

Perhaps you can try and figure out why!

1 The Klein bottle, and its ‘fundamental

group’.

2 A proof that the same number, pi, is the ratio

of a circle's circumference to its diameter,

and of its area to the square of its

radius.

3 Designed by David's brother, Alex Jordan,

this depicts Tartaglia's solution of the cubic

equation.

4 The golden spiral, and its relationship to the

Fibonacci numbers and the golden ratio.

Contours 21


Why Mathematicians Should Knit


22 Contours


Puzzles

Stuck? Solutions are on the website:

www.maths.ed.ac.uk/contours

Sangaku problems were hung on tablets in shrines and temples during

the sakoku period (1639‐1853) in Japan. Since the country was secluded

in those years, Japanese mathematicians used a distinct kind of mathematics

known as wasan to solve the problems.

In the School, Professor José Figueroa‐O'Farrill (p. 12) offers a 4th year

project which involves studying some sangaku, and their connections

with Western approaches to similar problems.

ρ

ρ

The radius of each

circle is written at

its centre.

Show that R=5r.

1 2

3

4

Maths

Cryptic

No 2

r .

R

Contours 23

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