Abstract: We will talk about percolation, loops and the interplay in-between the two. A number of open questions will be introduced as well as potential routes of attacking them.

Samuel Forbes | MathSys 3rd Year | Wealth Distribution Models

Abstract: I will introduce the wealth distribution, discuss empirical data for the UK and summarise some wealth distribution models.

Finally, the SPAAM poem puzzle, we state,
Just a bit of fun - for you to procrastinate!
They’ll sometimes be easy, sometimes fascinate,
But we do hope you’ll try and investigate,
For answering correctly will surely elate!
Don’t worry if you fail, you can always wait,
With a new week comes the next iterate.
And come along for the answer – don’t be late!

Ten coins are in a two-by-five grid,
Exactly four of them are to be slid,
Behold! Five straight lines of four coins now,
But the question is: can you tell me how?

Jaromir Sant | MASDOC 3rd Year
Mathematical Genetics - A Song of Wright & Fisher

Abstract:

Have you ever wondered why your eyes are blue,
And not some other beautiful hue?
Or perhaps you’d like to know
Why it is that your dad’s got such a prominent brow.
Or why your brother has a chin
That could easily fit an industrial bin.
Then worry not for I shall tell a story
About genetics in all it’s shining glory.
From Darwin to Fisher, from Mendel to Wright,
Their theories started to shed some light.
Then came Kimura claiming selection was dead,
And the issue has been all but put to bed.
This and some more I shall present,
To those who choose the SPAAM seminar to attend,
On Tuesday afternoon when the clock strikes three,
Just an hour before we serve coffee and tea.

Hanson Bharth | MASDOC 3rd Year | Rolls of Castamere

Abstract:

For this talk I shall
Introduce a mathematical model to
Simulate a cylinder rolling along an elastic
Half-space.
By taking a Fourier transform we
Obtain a general solution but require a
Wiener-Hopf technique to
Learn more about the solution.

SPAAM poem puzzle #2:

“I’ve found a really nice puzzle”, said Flo,

“Oh that's great!” exclaimed Mo,
“Come on, let’s have a go!”

“C is a circle with centre O”,

“This is exciting!” shouted Mo
“I like geometry - didn’t you know?”

“Corners chosen uniformly from C,
form a random triangle - let’s call it T”

“Probability!? This I did not foresee,
And is almost definitely not for me!”

“The probability that O is in T,
That’s exactly one half - don't you agree?”

“No!” replied Mo, “on the contrary,
It’s closer to 1 over 3!”

“You are all wrong!” interrupts Lee,
"But the actual answer alludes me,
Can you solve the mystery?"
"And how about a bonus round?"
"Can the expected area of T be found?"

And as the question began to hound
From the 3 came no more sound.

Mattia Sanna | 3rd Year PhD | Computation on Galois representations

Abstract: Galois representations have a special role in algebraic number theory. They arise from different theories for example arithmetic geometry and particluar holomorphic functions called modular forms. It is quite natural then to ask whether two such Galois representations are isomorphic, and therefore provide a connection among several areas of study. One of the most famous result in this direction is the proof of Fermat's last theorem due to Andrew Wiles in which he proves that all the Galois representations attached to elliptic curves defined over comes from the ones associated to particular modular forms. We would like to extend Wile's result in the most general setting but unfortunately it is a very hard task. However, there exist some explicit methods to check whether two given Galois representations are isomorphic. In this talk we will present a brief introduction on the subject and the ideas behind the methods.

Emma Southall | 3rd Year MathSys
Early Warning Signals of Disease Elimination

Abstract: TBC.

SPAAM Poem Puzzle 3:

A conspiracy theorist named Lee,
Frank, a risk mitigation analyst was he,
Jane, an inventor with robots she had three,

The 3 friends and 3 robots on a riverbank,
All stood still and looking blank,
”How would we cross the river?” thought Frank,

A small boat and an oar is all they had to hand,
The opposite riverbank, they wanted to land,
Luckily, the robots could operate the boat - this was well planned,

Unluckily, the boat was small and could only carry two people/robots at a time,
This line is only here so that I can rhyme,
Let’s go back to the story of maritime,

The conspiracy theorist didn’t trust the robots at all,
He was convinced that they would cause a brawl,

The risk analyst more sceptical now,
Thought "I’m sure we can sort something - but how?"

Lee insisted that no one should,
Ever be outnumbered by robots - that would be good!
The inventor wondered why her robots were so misunderstood,
But carried on anyway and quickly solved it for good.

Can you find a solution?
How many times did Jane have to cross the river?

Michael Luya | MathSys
The Mathematics behind Rotation Invariant Basis Representations for Machine Learning Models within Computational Chemistry

Abstract: How much mathematics is present in chemistry? In this presentation, Michael hopes to answer this question from a background of mathematics from the perspective of his PhD project, which is based in the design of machine learning infrastructures in computational chemistry. We will be looking at a variety of mathematical models that provide us with symmetry invariant parameters, that specifically satisfy rotation invariance, deriving them from quantum mechanical principles. Quantum mechanics itself provides us with a high level of accuracy, which we are currently interfacing with machine learning methods, which will allow us to make a variety of calculations. This talk however, will focus on the mathematics that is integral in getting us to such a stage.

Connah Johnson | MathSys
Spatial inhomogeneities in biochemical reaction networks

Abstract: Living systems may be thought of as perturbations in a chemical space. Of the set of possible chemical combinations a select few occur within biology, held within specific ranges of environments. These environments themselves being host to maelstrom of chemical interactions which may be modeled as a reaction network. Some ways in which these physical and chemical aspects combine and subsequent emergent phenomena will be the subject of this talk.

SPAAM Poem Puzzle 4:

I spent a £5 note on 2 ice creams,
They were Chocolate and Caramel Supremes,
At £1.20 each, this was certainly the dream,
The number of coins in change I got was 19,
At least one 2p, 5p, 20p, 50p and £1 coins could be seen,
And exactly one 1p and one 10p I got,
How many 2p coins could be in the lot?

Nicolò Paviato | MASDOC 2nd Year | Convergence to Brownian motion in a Deterministic System

Abstract: In this talk we will discuss statistical properties of dynamical systems, i.e. an application of probability theory to a deterministic case. If a system displays a "random behaviour", then both the Central Limit Theorem and Donsker's Invariance Principle are valid for regular observables; these follow from an intrinsic martingale property of the dynamic. Finally we will provide a gist on how to improve rates of convergence for a deterministic functional CLT and to find some multidimensional estimates.

SPAAM Poem Puzzle 5:

I used to play noughts and crosses,
But all I got was a number of losses,
So all the rules I decided to change,
And you might find this strange,
When players play uniformly at random,
"No skill!" you cry and cannot fathom,
But see, the skill is in choosing whether to go second or first,
Can you figure out which one is best and which one is worst?

Diogo Caetano | MASDOC 2nd Year
The Cahn-Hilliard equation on an evolving surface

Abstract: In this talk, we develop some tools to pose PDEs on evolving hypersurfaces and derive the Cahn-Hilliard equation with a logarithmic potential on a 2-dimensional moving surface. We then compare this problem with the corresponding case of a fixed domain, and conclude by discussing some results regarding existence and uniqueness of (weak) solutions.

Mohammad Noorbakhsh | MathSys
Causal network discovery from climate time series

Abstract: TBC.

SPAAM Poem Puzzle #6:

There once was a frog called Etienne,
Who used to sit on a riverbank in the Glen,
Lilly pads on water there was one shy off N,
Formed a straight line just in front of Etienne,
He chooses a number uniformly from 1 to N,
And jumps to this numbered lillypad then,
But if he chooses N, he crosses the river in the Glen.
And if he lands on a lillypad, the process repeats again,
Until on the other side of the riverbank is our dear Etienne

Given he jumps forwards on every jump, what is the expected number of jumps for Etienne to cross the river?

Letizia Angeli | MASDOC | Interacting Particle Systems Approximations of Feynman-Kac Formulae in continuous time

Abstract: In this talk, I will present a class of numerical algorithms - based on the evolution of interacting particle systems - for estimating large deviation quantities of additive path functionals of stochastic processes. Adapting already established results from the literature of particle filters and sequential Monte Carlo methods, we can study the convergence of the algorithms and provide a rigorous framework to evaluate and improve their efficiency.