# Pure Postgraduate Seminars

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The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. If you would like to give a talk or have any comments or suggestions as to the organisation of the seminars please contact Philip Bridge. Every week, a reminder will be sent to all Pure Postgrads. If you are not a pure postgrad and would also like to be sent a reminder then please e-mail me.

**The seminars are held in the Alan Turing Building, Frank Adams Room 2 (1.212), Fridays from 4pm to 5pm.** We will have tea, coffee and biscuits before the seminar at 3:45pm on the Atrium bridge. In the evening, we often go to a pub.

## Autumn Semester 2010

- 1
^{st}Oct 2010**Groups acting on polynomial rings**

Rob McKemeyAbstract (click to view)Take a polynomial ring and let a group act on it linearly. For each n, consider the vector space of the homogeneous polynomials in degree n. These vector spaces are related as they are made using the multiplication of the ring. Starting with an easy example, we will explore some aspects of these relations using a result from Galois Theory.

- 8
^{th}Oct 2010**The Quadratic Reciprocity Theorem**

Nicholas GreerAbstract (click to view)It is quite easy to determine whether solutions to a linear equation in modular arithmetic exist, and to find the solutions if they do. With quadratic equations, things are a little harder. It turns out that the existence of solutions can be determined by checking whether a particular element is both a square and a unit mod n for some n, and that this can be calculated simply by hand thanks to a theorem of Gauss, which he described as "the gem of higher arithmetic". A basic proof of this theorem will be presented.

- 15
^{th}Oct 2010**Truth vs. Proof: An Introduction to First Order Logic**

Nikesh SolankiAbstract (click to view)This talk is for all those that have always wanted to know what we logicy people are talking about or what a formal proof really is. I shall be covering several fundamental notions in logic. In brief, it shall be broken up into three parts- formulas and theories, semantic truth and syntactic proof. I shall explain what terms, formulas, sentences, theories etc. are, what truth in a mathematical structure is and what a formal proof is. This talk will be basic for all have done any kind of logic but feel free to come along anyway as I hope the talk will have plenty of open informal discussion.

- 22
^{nd}Oct 2010**The alternating group A**_{5}has many cunning disguises...

John BallantyneAbstract (click to view)Algebraic structures often have 'natural' actions on certain objects: for example, a matrix group acting on a vector space. However, there are often less obvious actions which we can study, and these can shed light on deeper properties of the structure which might not be otherwise apparent. In this talk I'll illustrate this via a nice group theoretic example.

- 29
^{th}Oct 2010**The Mathematics of Martin Gardner**

Katie StecklesAbstract (click to view)Martin Gardner was an American mathematics and science writer, who specialised in recreational mathematics, but whose interests encompassed magic, literature, philosophy, scientific skepticism, and religion. He published over 70 books, and wrote a regular column in Scientific American from 1956 to 1981 called Mathematical Games, which featured many interesting mathematical facts and puzzles, some with deep mathematical concepts behind them. I will give an overview of his life, and follow it with a few examples of his Mathematical Games to amuse and intrigue.

- 5
^{th}Nov 2010**Reading Week**

No Seminars Scheduled - 12
^{th}Nov 2010**Cobordism**

Alastair DarbyAbstract (click to view)The classification of smooth closed manifolds is really hard, so hard it's practically impossible. Cobordism is an equivalence relation that unintuitively leaves a lot of information and allows us to classify these manifolds up to it, providing many applications. Not only does it do that, but it forms a generalised homology theory, the one that Poincare failed to define when first trying to define homology purely in terms of manifolds in 1895.

- 19
^{th}Nov 2010**Coxeter Groups and Dynkin Diagrams**

Tim CrinionAbstract (click to view)In this seminar we will look at Coxeter Groups and things called Dynkin Diagrams. It turns out there is a one-to-one correspondence between these two. Also between Dynkin Diagrams and Root Systems, and Dynkin Diagrams and Lie Algebras, though we won't go into the latter. Dynkin Diagrams are very very very important in Group theory!

- 26
^{th}Nov 2010**Topological Data Analysis**

Beverley O'NeillAbstract (click to view)Topological Data Analysis applies tools from algebraic topology to extract coarse qualitative information on various kinds of data that allows for noise and statistical variations yet captures the higher-order features beyond clustering. In this seminar I will illustrate the techniques of persistent homology and give several applications of it....yeah you heard me....applications.... but ssshhhh, don't tell them next door!

- 3
^{rd}Dec 2010**Something, Something, Something, Cohomology**

Stephen MillerAbstract (click to view)Anyone who's been awake around a topologist knows that there is a covariant functor, called homology, from the category of topological spaces and continuous maps to the category of graded abelian groups. What's more, there appears to be a contravariant functor, called cohomology, doing a similar thing. In fact oridinary homology and cohomology are just specific cases of a family of functors called generalised (co)homology theories, which satisfy certain axioms (like excision and long exact sequences). I'll introduce these axioms, give some examples, and explain what this all has to do with stable homotopy theory and some very natural objects in topology called spectra.

- 10
^{th}Dec 2010**Graphs, Schottky surfaces and zeta functions**

Aoife McMonagleAbstract (click to view)By a well known result of Ihara from the 1960s, the zeta function associated to closed paths in a finite graph is rational. By relating graphs to a class of dynamical systems called subshifts of finite type, this result extends, in modified form, to metric graphs (i.e. graphs where each edge is assigned a real positive length). The fundamental group of a graph is a free group and one can interpret these results in terms of length functions on free groups. A naturally analogous problem is then to understand the zeta function for a surface of Schottky type, equipped with a metric of (possibly variable) negative curvature. We will discuss this problem in relation to recent work of T. Morita, showing that such zeta functions have an extension to a half plane including zero.

## Spring Semester 2011

- 4
^{th}Feb 2011**Proofs and Paths**

Ali LloydAbstract (click to view)I will introduce some basic concepts in Type Theory and Homotopy Theory, and show how they are connected by a discovery of Steve Awodey and Vladimir Voevodsky.The connection is rather surprising: Martin-Löf type theory is chiefly known as the benchmark for constructive validity, and as such is motivated primarily by philosophical concerns; on the other hand homotopy theory is decidedly geometric, describing the notion of a continuous deformation between topological spaces. The net effect is that not only can many statements in homotopy theory can be rigorously reformulated and proved, but also geometric intuition can be brought to bear on issues of type theory.

- 11
^{th}Feb 2011**The KW1 Conjecture or: How I Learned to Stop Worrying and Love Lie Algebras**

Lewis TopleyAbstract (click to view)I intend to take you on a quixotic journey through the representation theory of Lie algebras in characteristic p. In contrast to the characteristic zero case, the dimensions of simple modules for Lie algebras over fields of characteristic p are bounded. The KW1 conjecture suggests a rather neat formula for the least upper bound. I intend to introduce the conjecture and explain some affirmative results. In particular I shall briefly walk you through my proof of KW1 for centralisers of nilpotent elements in general linear algebras. I envisage the seminar ending with me inviting the audience to come up and prove the conjecture in full generality cos it's blatantly well easy.

- 18
^{th}Feb 2011**Hilbert's Tenth Problem**

Javier UtrerasAbstract (click to view)In the year 1900, D. Hilbert posed the following problem: devise an algorithm to decide whether a Diophantine equation (i.e. a polynomial equation with integer coefficients) has integer solutions. 70 years later, based on the work by M. Davis, H. Putnam and J. Robinson, Y. Matiyasevic showed that such an algorithm cannot exist. I will sketch this proof (following Davis' 1973 presentation) and briefly present some derived problems over the integers.

- 25
^{th}Feb 2011**Physical Measures**

Rafael Alcaraz BarreraAbstract (click to view)Given a differentiable transformation on a compact differentiable manifold M, it is an important problem to find a measure that shows "ergodic behavior" on a subset of M with positive Lebesgue measure. These measures are called Physical measures. In this talk we will define this concept formally. Using hyperbolic preballs and hyperbolic times, tools developed by J.F. Alves in 2005, we will show the existence of physical measures for some families of transformations.

- 4
^{th}Mar 2011**Conway's Fractran.**

Steve CleggAbstract (click to view)I'll be discussing Fractran, a prime producing machine due to John Conway. The talk will be accessible and will include pictures.

- 11
^{th}Mar 2011**Tame algebras**

Vladimir LukiyanovAbstract (click to view)I intend to go through the basics of representation theory of algebras and quivers, mention some techniques like the Auslander-Reiten quiver, and then look at some examples of algebras that are called tame. Representation theory of algebras is an area of mathematics that combines quite abstract functorial techniques with combinatorics. Tame algebras (over an algebraically closed field) represent one half of a dichotomy as proved by Drozd, the other half being the wild algebras, and the tame algebras have reasonably nice structure while also capturing a fair amount of examples that occur in other areas of mathematics (e.g. the Gelfand-Ponomarev quiver arising in Lie groups).

- 18
^{th}Mar 2011**Sheaves and Model Theory**

James DixonAbstract (click to view)Sheaves and etale spaces are a pretty nifty, but complicated-looking, piece of machinery for doing algebraic geometry. In some sense, though, an etale space is just a fancy product which can also be used to represent various algebraic objects. Model theory is used to dealing with various types of product and results by Comer, Macintyre and Astier show that the Feferman-Vaught technique for analysing products can be used to look at etale spaces.

- 25
^{th}Mar 2011**ζ Functions and Fractal Geometry**

Simon BakerAbstract (click to view)Zeta functions in a general setting are useful tools for uncovering interesting values that relate to an underlying system. In this talk I shall explain how this theory can be applied to Fractal Geometry.

- 1
^{st}Apr 2011**Non-Well Founded Sets**

Philip BridgeAbstract (click to view)Classical set theory demands that no set be allowed to be a member of itself, and more generally, that every collection of sets has a least element, in the membership ordering. This seems a very sensible principle in light of Russell's Paradox, but it isn't really necessary, and in fact alternatives exist that seem just as natural as the classical view. In this talk I will show how to construct such an alternative and explain why it's coconsistent with classical set theory. Also, puns.

- 8
^{th}Apr 2011**The Harmonic Oscillator and Morse Theory**

Graham KempAbstract (click to view)Although simple, the harmonic oscillator is important in physics. One nice relation to mathematics was given by Witten in 1982. In 'Supersymmetry and Morse Theory', Witten perturbs the de Rham complex, mixing in a Morse function. Using Hodge theory he then rederives the Morse inequalities by relating the perturbed Hodge Laplacian to the harmonic oscillator Hamiltonian. I will outline how all of this works, after first explaining the standard (unperturbed) Hodge-de Rham theory.

- Easter Break
**No Seminars Scheduled** - 6
^{th}May 2011**Categories of Fractions**

Philip BridgeAbstract (click to view)In order to continue my quest to explain why Category Theory is useful and you should study it, or at least know something about it, I will give a brief introduction to the idea of a category of fractions. The simple idea is to choose some morphisms in a category, and invert them. In this talk I'll illustrate how many seemingly unrelated ideas can be captured using this construction, such as the localisation of a ring, the sheafification of a presheaf, or the idea of logical deduction from a set of axioms.

- 13
^{th}May 2011**Local Fusion Graphs and Symmetric Groups**

John BallantyneAbstract (click to view)Take a conjugacy class of involutions in a group G as a vertex set, and join two distinct vertices if their product has odd order: we call the resulting graph a 'local fusion graph'. In this talk I'll present some results concerning these graphs when G is a finite symmetric group, and will show how the proofs become relatively straightforward when we consider another, particularly nice type of graph. There will be pictures. But they might be quite boring.

- 20
^{th}May 2011**Dimension and Singularities of Continuous but Nowhere Differentiable Functions.**

David NaughtonAbstract (click to view)In the talk I'll introduce some well known fractals, and the notion of dimension theory in Dynamical Systems. Then I'll introduce a specific fractal, namely a function whose graph is everywhere continuous but nowhere differentiable. This idea was first introduced by Weierstrass in 1872. I will look at the dimension of this function and introduce a new definition of dimension on it. Also I want to look at some singularities of these functions, namely what is known as knot point. The talk will be accessible to all and will not assume any prior knowledge in Dynamical Systems. There will also be pictures, but likely drawn by me so they will almost surely be rubbish. There may also be vegetables.

- 27
^{th}May 2011**TBA**

Alex Hill - 3
^{rd}Jun 2011**TBA**

Dan Vasey

## Previous Seminars

**
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)
**