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The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgraduates to present mathematics, either from their research or just a topic of interest. If you would like to give a talk or have any comments or suggestions as to the organisation of the seminars please contact Matthew Taylor or Nic Clarke. Every week, a reminder will be sent to all pure postgraduates. If you are not a pure postgraduate and would also like to be sent a reminder then please e-mail us to be added to the list.

The seminars are held in Frank Adams 1 in the Alan Turing Building, on Fridays from 4pm to 5pm. We will have tea, coffee and biscuits before the seminar at 3:45pm on the Atrium bridge. Afterwards we usually go to Sandbar.

You are currently looking at the Spring 2014 schedule. For the Autumn 2013 seminar timetable, please click here.

## Upcoming seminars

• 17th January 2014
The Weyl Algebra
Andrew Davies
Abstract (click to view)

Some past attendees may recall previous seminars in which Sian has spoken about the quantum plane, a simple enough looking ring that still has many interesting properties. In this seminar I will show that the Weyl algebra (another easy-to-define ring) also displays some very curious behaviour. Among other things, I will speak about its relations to Lie theory, which illustrate some useful techniques when working with noncommutative algebras in general.

• 24th January 2014
Local cohomology
Nic Clarke
Abstract (click to view)

In the early 1960's Grothendieck, in a series of seminars, introduced the idea of local cohomology. Given a ring R, an ideal I and an R-module M, I will show how to construct the Cech complex and compute the local cohomology of M with support in I. There will be plenty of examples. From the definition it will be apparent that the local cohomology inherits the R-module structure. However, the behaviour of local cohomology modules remains quite mysterious and is an area of active research. I will highlight some of the important structure theorems and will describe applications to finding the minimal number of defining equations of an affine/projective variety. If I have time at the end I will explain how the algebraic perspective ties in with Grothendieck's original definition and try to give some intuition as to why local cohomology is 'local'.

• 31st January 2014
An intro to the theory of monads
Sam Dean
Abstract (click to view)

Throughout the ages, various interpretations of the words "algebraic theory" have arisen. These have given rise to a field of mathematical logic called universal algebra, which studies these different interpretations and how they are related. In this talk, I will discuss just two of these. First, I will discuss how model theorists might interpret the words "algebraic theory", and then go on to what a category theorist is likely to mean by "algebraic theory": a monad. I’ll discuss how these approaches compare, in particular showing that the model theoretic method fits into the theory of monads. I’ll also use the theory of monads to show how you can PROVE that topology is not algebra.

• 7th February 2014
Transfer Operators
Anthony Chiu
Abstract (click to view)

An iterated function system (IFS) is a fixed set of maps that are applied at random in a sensible manner. How can we study the typical behaviour of an IFS? A powerful approach is to use a transfer operator, a tool that encodes the IFS in a way that is easier to deal with. Although some of the definitions and theorems may look like messy functional analysis, I will use a simple diagram to explain the details in a much easier way. I will also give an idea of how different kinds of transfer operators can be used together to prove certain properties of the IFS.

• 14th February 2014
Linearizing certain Boolean algebras
Amit Kuber
Abstract (click to view)

Boolean algebras are quintessential objects in almost all areas of mathematics. A (freely generated) Boolean algebra can be analyzed using a family of integer valued valuations (i.e., finitely additive measures). I will describe the construction of such valuations using localization and discuss techniques combining (semi-)lattice theory and simplicial homology. By the end of the talk, you will certainly see the hidden geometry in these algebraic objects!

• 21st February 2014
The Ping-Pong Lemma
Jamie Phillips
Abstract (click to view)

“Freedom is like drink. If you take any at all, you might as well take enough to make you happy for a while.” – Finley Peter Dunne.

Free objects in a category (whatever they may be) are the most basic objects in mathematics. A paradigm is the theory of free groups. They arose naturally through the study of the geometry of hyperbolic groups but their fundamental role in group theory was recognised by Nielson (who named them), Dehn, and others.

We'll begin with a crash course in free group theory before proving The Ping-Pong Lemma, a statement which ensures that several elements in a group acting on a set freely generate a free subgroup of that group. We’ll see examples of it in action and reformulations of the result in other areas of pure mathematics. Time permitting, I’ll also discuss the role the lemma plays in the proof of Tits Alternative, an important theorem about the structure of finitely generated linear groups.

The group theoretic ideas involved are fairly elementary so the talk should be approachable to all.

• 28th February 2014
Making Clocks with Maths
Tom Withers
Abstract (click to view)

Sundials are usually a simple object where pointing a rod south produces a shadow which we can draw a clock around and use to tell the time. They might look pretty in the garden, but they are very difficult to use practically. I'm going to construct an argument showing the existence of a sundial which can arbitrarily accurately give the time using a digital clock face, the only moving part being the sun. We will need ideas from dimension theory and fractal geometry.

There hasn't been a dimension theory talk yet this year, so, I'll start from scratch and look at the basics and why we bother with it. Then I'll do some easy examples of constructing fractal sets using iterated function systems and the properties of projections of these fractal sets; eventually arguing for the existence of our clock, hopefully just before the sun goes down.

• 7th March 2014
Introduction to amalgamating structures
Alex Antao
Abstract (click to view)

Amalgamation is a way of gluing together overlapping structures to form a structure which is similar to all the orginal ones. A little more precisely, let $\mathscr{K}$ be a class of structures. Then $\mathscr{K}$ is said to have the amalgamation property (AP) if any $\mathcal{B}, \mathscr{C} \in \mathscr{K}$ with a common substructure $\mathcal{A}$ (not necessarily in $\mathscr{K}$) can be embedded in some structure $\mathcal{D} \in \mathscr{K}$, such that the embeddings agree when restricted to $\mathcal{A}$. This notion of amalgamation was introduced by Roland Fraïssé in the 1950s.

Subsequently variants of amalgamation have been formulated, some of which will be considered in the talk, given enough time. Such results are vital tools in model theory for constructing new structures with desirable properties or classifying existing structures by looking at how they are "built up" around substructures.

The talk will contain mathematical examples (and maybe non-mathematical pseudo-examples) of amalgamation in action. Unlike Sandbar's finest whiskey, it should also be fairly light on proof.

Note: There will be some model theory, but the necessary "evils" will either be casually defined or described intuitively in the talk ad hoc. In other words, everyone is welcome!*

* Though if you think this all looks as dour as Gordon Brown at a funeral, probably stay away.

• 14th March 2014
The Maths Behind Bitcoin
Matthew Taylor
Abstract (click to view)

As something of a computer geek, I've been asked several times in the past to explain what Bitcoin is. There's no easy answer, but you can think of Bitcoin as a digital currency. You can buy things with it and sell things for it just like with real money; Bitcoin has "value" because enough people believe it's worth something. That, however, is for an economics talk. We're not economists.

When your currency is entirely digital, a number of problems arise. How do you regulate the creation of new Bitcoins? How do you verify who's sending what to where? And how do you protect people's money? The answer to all of these is public-key cryptography, a mathematical endeavour which underpins a surprising amount of the entire system. In this talk, I'll be giving a brief overview of what a "difficult" problem is in cryptographic terms - and how the Bitcoin system works - followed by a look at hash functions, Bitcoin transactions and what "mining" actually is.

• 21st March 2014
Introduction to supermanifolds
Matthew Peddie
Abstract (click to view)

Supermathematics became useful in physical theories when supersymmetric models were used to relate two types of elementary particle, the boson and the fermion, which have a different quantum nature. These supersymmetric models provide a link between classical and quantum physics and provide a much simpler way to analyse a quantum field theory that possesses supersymmetry. Despite mounting arguments and a lack of evidence, suggesting that supersymmetry may just be fundamentally wrong, the maths is still there!

In this we introduce the supermanifold, the space where our theories are set, and with time permitting, look at how we might integrate over this new space with the Berezin integral. This should be widely accessible with absolutely no understanding of physics needed.

• 28th March 2014
Simple Lie algebras and their maximal subalgebras
Tom Purslow
Abstract (click to view)

In 1894 Cartan produced a classification of finite dimensional simple Lie algebras over the complex numbers, sadly it has been 120 years and the classification of simple Lie algebras over fields of positive characteristic is still incomplete. But that’s not a problem we will be solving this Friday!

Instead we will be looking at some of the tools Dynkin used in his classification of simple Lie algebras. Then we’ll ask the question of whether this helps us in characteristic p and see some of the differences that happen and if there is time we will look at maximal subalgebras in certain exceptional Lie algebras in fields of positive characteristic.

Like David Moyes we will haplessly attempt to reuse some of the building blocks laid out for us by former legends, but unlike David Moyes we will have some success!

• 2nd May 2014
Hilbert's 17th Problem
Laura Phillips
Abstract (click to view)

Hilbert's 17th problem asks whether a polynomial in $n$ variables over $\mathbb{R}$ that is non-negative on all of $\mathbb{R}^n$ can be written as the sum of squares of rational functions. In this seminar I'll present a (positive) solution to the problem and describe the wider context that it sits in. If there's time I'll talk about some of the more quantitative aspects of the problem and some variants.

• 9th May 2014
Metric spaces aren't torsors
David Wilding
Abstract (click to view)

But (you might reasonably ask) was there any danger of them being torsors in the first place? Actually, you are probably just wondering what a 'torsor' is. A torsor is essentially a group, except we have "forgotten" which element is the identity element. Given a torsor for the additive group of real numbers, we can define a distance function on the torsor that satisfies all but one of the metric space axioms. This suggests that torsors generalise metric spaces, but it turns out that a torsor can never be a metric space. Instead, metric spaces and torsors have a common generalisation, which I will describe.

• 16th May 2014
Vector fields and Lie algebras
Matthew Peddie
Abstract (click to view)

The structure of a Lie algebra can be described by a weight +1 homological vector field on the space of shifted parity considered as a supermanifold. We give some background and then show how this is done.

• 23rd May 2014
Fermat's Last Theorem, 20th Anniversary
Goran Malic
Abstract (click to view)

This year will mark the 20th anniversary of Andrew Wiles' proof of Fermat's Last Theorem (FLT). As it is very well known within the maths community, in June 1993 Wiles presented what he thought was a proof of FLT. However, in August of the same year, during the review process, Nick Katz alerted Wiles of a flaw in his argument. But all was not lost; in September 1994 Wiles corrected the proof and in October 1994 submitted two papers to the Princeton Annals of Mathematics (which were published in May 1995), and the rest is, as they say, history.

• 30th May 2014
TBD
Lyndsey Clark
Abstract (click to view)

An abstract.

• 6th June 2014
TBD
Abstract (click to view)

An abstract.

• 13th June 2014
TBD
Office 2.121
Abstract (click to view)

An abstract.

• 20th June 2014
TBD
Simon Baker
Abstract (click to view)

An abstract.