Applied Mathematics at Oxford

Christian YatesCentre for Mathematical BiologyMathematical Institute

Who am I?

‣ Completed my B.A. (Mathematics) and M.Sc. (Mathematical Modelling and Scientific Computing) at the Mathematical Institute as a member of Somerville College.

‣ Currently completing my D.Phil. (Mathematical Biology) in the Centre for Mathematical Biology as a member of Worcester and St. Catherine’s colleges.

‣ Next year – Junior Research Fellow at Christ Church college.

‣ Research in cell migration, bacterial motion and locust motion.

‣ Supervising Masters students.

‣ Lecturer at Somerville College

‣ Teaching 1st and 2nd year tutorials in college.

Outline of this talk

‣ The principles of applied mathematics

‣ A practical example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves on strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves on strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Calculus of variations

‣ Mathematical Biology

‣ Reasons to study mathematics

Principles of applied mathematics

‣ Start from a physical or “real world” system

‣ Use physical principles to describe it using mathematics

‣ For example, Newton’s Laws

‣ Derive the appropriate mathematical terminology

‣ For example, calculus

‣ Use empirical laws to turn it into a solvable mathematical problem

‣ For example, Law of Mass Action, Hooke’s Law

‣ Solve the mathematical model

‣ Develop mathematical techniques to do this

‣ For example, solutions of differential equations

‣ Use the mathematical results to make predictions about the real world system

Simple harmonic motion

‣ Newton’s second law

‣ Force = mass x acceleration

‣ Hooke’s Law

‣ Tension = spring const. x extension

‣ Resulting differential equation

simple harmonic motion

‣ Re-write in terms of the displacement from equilibrium

which is the description of simple harmonic motion

‣ The solution is

with constants determined by the initial displacement and velocity

‣ The period of oscillations is

Putting maths to the test: Prediction

‣ At equilibrium (using Hooke’s law T=ke):

‣ Therefore:

‣ So the period should be:

Experiment

Equipment:

‣ Stopwatch

‣ Mass

‣ Spring

‣ Clampstand

‣ 1 willing volunteer

‣ Not bad but not perfect

‣ Why not?

‣ Air resistance

‣ Errors in measurement etc

‣ Old Spring

‣ Hooke’s law isn’t perfect etc

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves on strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

Celestial mechanics

‣ Newton’s 2nd Law

‣ Newton’s Law of Gravitation

‣ The position vector satisfies the differential equation

Solution of this equation confirms Kepler’s Laws

How long is a year?

‣ M=2x1030 Kg

‣ G=6.67x10-10 m3kg-1s-2

‣ R=1.5x1011m

‣ Not bad for a 400 year old piece of maths.

Kepler

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves on strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

Waves on a string

‣ Apply Newton’s Law’s to each small interval of string...

‣ The vertical displacement satisfies the partial differential equation

‣ Known as the wave equation

‣ Wave speed:

Understanding music

‣ Why don’t all waves sound like this?

‣ Because we can superpose waves on each other

=

‣ By adding waves of different amplitudes and frequencies we can come up with any shape we want:

‣ The maths behind how to find the correct signs and amplitudes is called Fourier series analysis.

Fourier series

More complicated wave forms

‣ Saw-tooth wave:

‣ Square wave:

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves of strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

Fluid mechanics

‣ Theory of flight - what causes the lift on an aerofoil?

‣ What happens as you cross the sound barrier?

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves of strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

Classical mechanics

‣ Can we predict the motion of a double pendulum?

‣ In principle yes.

‣ In practice, chaos takes over.

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves of strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Mathematical Biology

‣ Reasons to study mathematics

How we do mathematical biology?

‣ Find out as much as we can about the biology

‣ Think about which bits of our knowledge are important

‣ Try to describe things mathematically

‣ Use our mathematical knowledge to predict what we think will happen in the biological system

‣ Put our understanding to good use

Mathematical biology

Locusts

Switching behaviour

‣ Locusts switch direction periodically

‣ The length of time between switches depends on the density of the group

30 Locusts 60 Locusts

Explanation - Cannibalism

Outline of this talk

‣ The principles of applied mathematics

‣ A simple example

‣ Mods applied mathematics (first year)

‣ Celestial mechanics

‣ Waves on strings

‣ Applied mathematics options (second and third year)

‣ Fluid mechanics

‣ Classical mechanics

‣ Calculus of variations

‣ Mathematical Biology

‣ Reasons to study mathematics

Why mathematics?

‣ Flexibility - opens many doors

‣ Importance - underpins science

‣ Ability to address fundamental questions about the universe

‣ Relevance to the “real world” combined with the beauty of abstract theory

‣ Excitement - finding out how things work

‣ Huge variety of possible careers

‣ Opportunity to pass on knowledge to others

Me on Bang goes the theory

I’m off to watch Man City in the FA cup final

Further information

‣ Studying mathematics and joint schools at Oxford

‣ http://www.maths.ox.ac.uk

‣ David Acheson’s page on dynamics

‣ http://home.jesus.ox.ac.uk/~dacheson/mechanics.html

‣ Centre for Mathematical Biology

‣ http://www.maths.ox.ac.uk/groups/mathematical-biology/

‣ My web page

‣ http://people.maths.ox.ac.uk/yatesc/