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# Pure Postgraduate Seminars

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 Andrew Davies or David Ward. 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 the Alan Turing Building, room G107, 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 a pub.

## Autumn seminars

- 21
^{st}September 2012**The World's First Fractal**

Dave NaughtonAbstract (click to view)Well, that pretty much sums it up. It's probably true anyway. Actually I have no solid basis on which I am making this claim but I needed a good title! To start the seminars off once again, I'm going to give a bit of an overview of Hyperbolic Geometry, one of the main examples of a geometry where not all of Euclid's postulates hold. Hyperbolic geometry has some particularly interesting uses, for example in tiling the plane. In euclidean geometry we can tile the plane just 3 ways using regular polygon, squares, triangles and like a Blockbusters board. But what about in Hyperbolic geometry? I will then look at some seemingly simple maths dating back to the ancient Greeks (they were just like the modern Greeks but without things like microwaves and nectar points) and how this led them to unwittingly discover what was probably the first fractal. I don't know this for sure, but it sounds cooler if I say that. Although these have been around for a while (since ancient times in fact!), very little is in fact known about them. I'm just going to talk about some of these examples and how this links back to Hyperbolic Geometry. It will all be very simple, with no hard maths (Friday afternoon ain't meant for hard maths) and loads of cool pictures.

- 28
^{th}September 2012**Mathematics Research Students' Conference (MRSC) 2012**Abstract (click to view)An annual conference for mathematics research students from all disciplines to meet and hear short presentations about some of the research that is undertaken in the school. For more information, visit the conference website at the link above.

- 5
^{th}October 2012**Makeshift last minute seminar**

Sian Fryer, Simon Baker and Dave NaughtonAbstract (click to view)Due to illness we had to find speakers at the last minute. Luckily Simon, Sian and Dave put something together to save the day. Simon spoke about Pisot numbers, while Sian spoke about the ergodic theorem and composition algebras. Dave ended by speaking about dynamical systems related to decimal expansions.

- 12
^{th}October 2012**Joint Pure/Applied seminar - MRSC Puzzles**

Katie Steckles and Paul TaylorAbstract (click to view)Last month, you may have attended the MRSC, the world's greatest postgraduate conference. The puzzle sheet which was given out at lunch was put together by Paul Taylor and Katie Steckles, two former Manchester PhD students, who are returning in blaze of glory to deliver a seminar which not only gives the solutions to some or all of these puzzles, but also includes a non-zero amount of actual maths.

- 19
^{th}October 2012**Alon's Combinatorial Nullstellensatz and Applications**

Amit KuberAbstract (click to view)P. Komjath asked a question: How many affine hyperplanes in R

^{n}are necessary to cover all points in the unit cube {0,1}^{n}but one? The answer is of course n. We will see a short proof of this theorem via Combinatorial Nullstellensatz. I will explain the connections between well known Hilbert's Nullstellensatz and this combinatorial version due to Noga Alon. I will give a short proof of this theorem and discuss a couple of applications in Number Theory and Algebra. - 26
^{th}October 2012**Non-standard Snakes and Computers**

Sian FryerAbstract (click to view)We all know how to add numbers together, but do we really think about what we're doing when we do it? Could you explain it in terms a computer would understand? I will explain what exactly your computer is doing when you type 1+1 into it, and demonstrate how we can use this to build adders (things that add things together) from almost anything... including toppling rows of dominoes! If there's time I'll also talk about some other fun computer constructions and ways of counting. There will be no ergodic theory.

- 2
^{nd}November 2012**Undecidability in Number Theory**

Laura PhillipsAbstract (click to view)Hilbert's 10th Problem asks for an algorithm which on input of a diophantine equation determines whether it has a solution in the integers. I will briefly discuss the background and (negative) solution to this problem before moving on to variants of Hilbert's 10th Problem over rings of number theoretic interest.

- 9
^{th}November 2012**Kähler Differentials (rescheduled from 5th October)**

Nikesh SolankiAbstract (click to view)There exists deep in the bowels of the belly of mathematics, derivatives from which all other derivatives come (or should I say

*derive*) from. They go by the name universal derivatives. More precisely, given a commutative ring with unit R_{0}and R a R_{0}-algebra all R_{0}-derivatives of R (I will explain what this means in the talk) are `linear transformations' of this universal derivative. Furthermore, these universal derivatives give rise to a module called the module of Kähler differentials of R over R_{0}which holds a lot of algebraic information about R_{0}-algebra structure of R. In this talk I explain precisely what I mean by all this jargon I have spewed here and why it is so. - 16
^{th}November 2012**The Shadowing Lemma**

Rafael Alcaraz BarreraAbstract (click to view)The shadowing lemma (or shadowing property in certain cases) is a basic result (property) in dynamical systems theory. Nonetheless it has important implications. Roughly speaking, the theorem states that a numerically computed trajectory with rounding errors on every step stays uniformly close to some true trajectory (yes, this comes from Wikipedia). During this talk we will prove the shadowing theorem in a certain context, and this will allow us to give a nice (at least for me) introduction to Hyperbolic Dynamics.

- 23
^{rd}November 2012**Hercules, Hydras and Goodstein**

Tahel RonelAbstract (click to view)A hydra is a finite single-rooted tree with outer nodes known as heads and the following regeneration property: At stage n a head is chopped off node x. Go one node down from x, and grow n copies of the remaining subtree coming out of this node (so we have n new heads), unless node x is the root. In a battle between Hercules and a hydra, who wins? In this talk I will answer this question, draw a connection to Goodstein sequences, and explain why the results of this battle are significant to logic.

- 30
^{th}November 2012**Iterated fun with iterated function systems**

Anthony ChiuAbstract (click to view)Most people have heard of the "chaos game", even if they don't know what it is. The basic idea relies on an iterated function system (IFS), a set of maps with associated probabilities. Unlike a dynamical system, where we would normally apply the same map repeatedly, an iteration of an IFS involves choosing one of the maps at random (according to the probabilities) and applying it. After a large number of iterations, in some cases a pretty fractal will start to appear and the reason why can be explained by an ergodic theorem. We will see some pretty pictures with the help of a computer and an IFS analogue of the shadowing lemma, which I'm sure you all remember from Rafa's talk.

- 7
^{th}December 2012**Speed-dating seminar**

David Wilding, Rafa Alcaraz Barrera, Matthew Taylor, Inga Schwabrow, Alex Antao, David WardAbstract (click to view)Inspired by the makeshift seminar ealier this term we are having a `speed-dating' style seminar for the last seminar. The extra twist is that the speakers do not choose their topics, but were assigned to them randomly following a spectacularly bad drawing ceremony. We will get to hear short (8 minute) talks by the following speakers:

1. Dave Wilding - Little/Great Picard's Theorem

2. Rafa Alcaraz Barrera - Biography of a mathematician (born or died on 7th December!)

3. Matthew Taylor - p-adic numbers

4. Inga Schwabrow - van Kampen's Theorem

5. Alex Antao - Sturm sequences

6. David Ward - Smale's Paradox

## Spring seminars

- 25
^{th}January 2013**Come with us now on a journey through time and space...**

Simon BakerAbstract (click to view)In this talk I will discuss some of the techniques and theory used in dimension theory/fractal geometry. There will also me an emphasis on the history of the subject along with many examples and pretty pictures.

- 1
^{st}February 2013**An afternoon with homology**

Alex LongdonAbstract (click to view)The holy grail of algebraic topology is full knowledge of the homotopy groups, which are algebraic gadgets that take a topological space and spit out groups containing lots of information about said space. Unfortunately holy grails tend to be elusive and difficult to obtain, and the homotopy groups are no exception. When it comes to making down and dirty computations, topologists instead like to use something called homology, which preserves less information about the topological space but is far more tractable. In this seminar we'll meet homology, find out what it can do, find out what it can't do, and then see how simply dualising everything yields a more powerful algebraic invariant: cohomology.

- 8
^{th}February 2013**The Power of Complex Analysis**

Goran MalicAbstract (click to view)Ever since the 800's people have been wondering about the nature of the solutions to polynomial equations - a whole bunch of historic figures attempted to tackle the problem and it was finally resolved in 1806 by Jean-Robert Argand. Today we know it as the Fundamental Theorem of Algebra. Turns out it's neither fundamental nor does it involve much algebra, but rather - complex analysis.

Another problem concerning polynomials occupied the minds of Argand's contemporaries: can solutions to polynomial equations of any degree be represented in terms of radicals? P. Ruffini, N.H. Abel and E. Galois independently proved that the answer is no if the degree is greater-than or equal to 5. This result is known as the Abel-Ruffini theorem, but history will most likely remember it as the birthplace of Galois theory.

In this seminar I shall present a proof of the Abel-Ruffini theorem - but don't worry if you don't know any of the full-on awesome Galois theory, because I won't use any! As the title suggests, I will prove it using the power of complex analysis. - 15
^{th}February 2013**TQFTs and the Cobordism Hypothesis**

Alastair DarbyAbstract (click to view)Topological Quantum Field Theories, inspired by theoretical physics, provide invariants for topological spaces. This is an exciting area of mathematics with four Fields Medals being awarded so far for work relating to TQFTs. We will define these and state a version of the Baez-Dolan Cobordism Hypothesis, recently proven by Jacob Lurie, which is a classification of TQFTs.

- 22
^{nd}February 2013**The illustrated zoo of order preserving functions**

David WildingAbstract (click to view)Posets (partially ordered sets) underlie much of mathematics, but we often don't give them a second thought. I will describe some of the interesting posets and, more importantly, order preserving functions between posets that turn up in various branches of mathematics. The order preserving functions on show will range all the way from the very basic (monotone functions) to the most powerful (residuated functions and order isomorphisms).

- 1
^{st}March 2013**Galois groups and elliptic curves**

Adam BiggsAbstract (click to view)I will give a brief introduction into Galois actions on elliptic curves, an incredible relation that shaped a lot of important mathematics in the second half of the twentieth century.

- 8
^{th}March 2013**Fusion in Finite Groups**

David WardAbstract (click to view)Let G be a finite group and let p be a prime divisor of the order of G. We may form the partially ordered set, consisting of all p-subgroups of G, where the partial ordering is containment. We may associate to this partially ordered set a graph. However, when is this graph connected? It sounds an easy enough problem to consider, but trying to determine the answer can be extremely difficult.

In this talk we shall define this graph, and illustrate how its connectedness is linked to the fusion of finite groups and strongly p-embedded subgroups. All of these notions will be explained and examples will be given. We will conclude by showing that in a few specific cases, the above notions can be used to determine the connectedness of the given graph. - 15
^{th}March 2013**A Crash Course in Tropical Maths**

Matthew TaylorAbstract (click to view)Tropical mathematics is a comparatively new area of research that's exploded in popularity over the last decade. With links to algebraic geometry, metric geometry, cryptography and more scientific applications than you can shake a stick at, it's attracting a variety of academics from seemingly unrelated fields.

I'll be introducing tropical maths from a semigroup-theoretic point of view, covering all of the basics and going on to explain some of the work that's been done in Manchester. Depending on time and interest, I may go on to talk briefly about some of the related topics. - 22
^{nd}March 2013**The McKay Correspondence**

Andrew DaviesAbstract (click to view)The McKay correspondence relates finite subgroups of the special linear group, resolution of singularities and quivers. I will speak about the dimension 2 case in which the correspondence is very strong; in higher dimensions there are only partial results, which are in the setting of derived categories.

- 19
^{th}April 2013**Transitivity and specification in symmetric subshifts**

Rafael Alcaraz BarreraAbstract (click to view)During this talk I will introduce the notion of a symmetric subshift and some of its dynamical properties will be studied, namely transitivity and specification. If there is time left I will explain the measure theoretic consequences arisen by these dynamical properties.

- 26
^{th}April 2013**Hyperreals, Great and Small**

Malte KliessAbstract (click to view)When Leibniz and Newton did their work on analysis they used 'infinitesimals', which can be thought of as non-zero quantities that are infinitely small. These got replaced by the well-known epsilon-delta formulation in modern calculus, as some people thought of them as 'naive' and 'meaningless'. In 1960 with the help of formal logic, Abraham Robinson showed that the notion of an infinitesimal can indeed be a clear and meaningful one.

I will show the construction of the hyperreals, an extension of the real numbers by infinitesimals, and talk about methods used in this non-standard analysis. You will probably not learn any new facts about calculus, but you will see ways of obtaining them that are more intuitive, and in some cases much easier than the standard methods used to obtain them. - 3
^{rd}May 2013**Singularities of Weierstrass-type functions**

Dave NaughtonAbstract (click to view)I'm gonna talk about how graphs of continuous functions arise as invariant sets of dynamical systems and how certain (surprisingly weak) restrictions can be put on the dynamics to ensure fractal behaviour. In particular, nowhere differentiability.

Along with dimension theoretic results, much interest over the past 100 years has involved singularities of such functions. This theory comes from classic analysis, with useful results due to Weierstrass, Lebesgue, Boltzman and Hardy.

I will use these results, along with notions of recurrence and hyperbolicity in dynamical systems, to consider a specific type of singularity of these functions, and how we can show the existence of a large set of such points.

There is very little machinery required, so all will be able to follow, and I will give a nice sketch of how the result is proven. - 10
^{th}May 2013**An introduction to group cohomology**

Nic ClarkeAbstract (click to view)Given a group we can associate to it a graded commutative ring: its cohomology ring. I will introduce the notion of a classifying space and use this to define the cohomology ring of a group. I will then show how we can relate some properties of this ring with properties of the group.

- 17
^{th}May 2013**Ctrl+X Ctrl+V**

Amit KuberAbstract (click to view)This talk will take you back to high-school days and bring lost memories of some ``elementary'' geometry.

It all began when Gauss asked if there is any ``elementary'' proof of the formula for the volume of a pyramid. Much later, Hilbert's third (and perhaps the easiest) problem asked if two polyhedra of same volume can be cut-paste into each other in only finitely many steps. I will discuss some issues around this problem and the solution provided by Max Dehn.

Later on I will discuss polytope complexes - a gadget useful to formulate this problem in an algebraic K-theory setting. - 24
^{th}May 2013**The Matroid**

Goran MalicAbstract (click to view)This is your last chance. After this, there is no turning back. You take the blue pill - the story ends, you go home or to the pub and believe whatever you want to believe. You take the red pill - you stay in the Pure Postgraduate Seminar and I show you how deep the rabbit-hole goes.

A matroid is a ``combinatorial'' structure invented by the influential topologist Hassler Whitney (a.k.a. The Architect) in order to study the abstract properties of (linear) independence. It generalizes many ideas that arise from the study of matrices, graphs, finite geometries etc. such as linear independence, rank, bases and so on. Matroid theory is a richer theory than the aforementioned in the sense that to all matrices and graphs one can assign a matroid but there are matroids which can not be studied by linear algebra or graph theory. That's why Neo isn't The One (who's going to talk about it).

What's that you say? The level of abstractness is not large enough for you? Fear not 'cos matroids themselves are generalized by symplectic and Coxeter matroids and the former are used to study graphs cellularly embedded on surfaces. If time permits, I will describe a special case of symplectic matroids and show how to peel off the skin of a compact connected and orientable surface in one move.

...Why oh why didn't I take the BLUE pill?... - 31
^{th}May 2013**The Representation Theory of the Symmetric Groups**

Inga SchwabrowAbstract (click to view)The representation theory of finite groups has been a major topic in pure mathematics over the last 100 years. In my talk, I will give a short introduction to the theory for a particular nice class of groups, which we hopefully all know and love: the symmetric groups. Over the complex numbers, we have a good understanding of what's going on and we have an explicit way of constructing the irreducible representations, which form the basic building blocks for all other representations. In particular, there is a very intimate connection between these representations and Young tableaux, which will give this talk a lovely combinatorial flavour.

- 7
^{th}June 2013**Fourier analysis and its use in pure mathematics**

Lyndsey ClarkAbstract (click to view)Fourier analysis is one of those bits of mathematics that everyone comes across as an undergrad, and then the vast majority do their best to forget about it and never use it again. Everyone "knows" that it's useful for physics and engineering, but as pure mathematicians physics and engineering are not very fun, so what is Fourier analysis actually good for aside from this? Working from the assumption that any knowledge of Fourier series has long since been forgotten, I am going to introduce the basics of Fourier analysis and give some examples of how it can be used to prove what would otherwise be tricky results in pure maths.

- 14
^{th}June 2013**Groups, Graphs and Monsters**

Jamie PhillipsAbstract (click to view)For a group G and a subset X of G, the commuting graph of G on X is the graph whose vertex set is X with vertices joined by an edge if and only if they commute within the group G. If the elements of X are all involutions, then the graph is called a commuting involution graph.

In this talk I will define and construct commuting involution graphs for various finite groups, tell you why you should care about them, how they're used, and discuss the progress made in determining the graphs for more interesting/problematic cases. In particular, I will discuss the (not yet complete) commuting involution graphs associated with the Baby Monster group and show you some of the computational and theoretical wizardry that has been necessary to get as far as we currently have.

For those not well versed in the mysterious arcane ways of group theory, rest assured that this talk will still be approachable with plenty of illustrative pictures. - 21
^{th}June 2013**Quantum Algebras and the Stratification Theorem**

Siân FryerAbstract (click to view)Commutative algebraists love figuring out how to describe the prime ideals of a ring, often citing algebraic geometry or representation theory as a motivation but mostly just because they're there. Non-commutative algebraists don't have that excuse (NC algebraic geometry is defined via category theory) but they still like their prime ideals... at least once they've figured out what the right definition of prime should be in a noncommutative ring. I will talk about how to define noncommutative versions of some popular algebras and a powerful result called the Stratification Theorem which describes their prime and primitive ideals. If there's time I'll also talk about a (commutative) Poisson version of this theorem, and conjectured links between the commutative and noncommutative ideals.

- 5
^{th}July 2013**Some 'Elementary' Model Theory**

Alexander AntaoAbstract (click to view)In this talk I will give an introduction to some of the fundamentals of model theory, with attention paid to intuition. Chiefly I will be talking about the hierarchy of functions and equivalence relations between first-order structures, including a notion of homomorphism that generalizes those of groups, rings etc., as well as the concepts of elementary equivalence and elementary embedding. Associated with these notions are some interesting preservation theorems that I will also present.

Most of the mathematics is due to Tarski, Vaught, Löwenheim, Skolem, Maltsev and Loś.

## Previous Seminars

**
List of 2012/2013 seminars (Andrew Davies/David Ward)
List of 2011/2012 seminars (Simon Baker/Dave Naughton)
List of 2010/2011 seminars (Philip Bridge)
List of 2009/2010 seminars (Richard Harland)
List of 2008/2009 seminars (Ali Everett)
List of 2007/2008 seminars (Jacob George)
List of 2006/2007 seminars (Stephen Clegg)
List of 2005/2006 seminars (Marianne Johnson)
List of 2004/2005 seminars (Matt Horsham)
List of 2003 seminars (Matthew Craven/Sara Santos)
List of 2002 seminars (Sarah Perkins/Sara Santos)
**