# 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 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 Autumn 2013 schedule. For the Spring 2014 seminar timetable, please click **here**.

## Upcoming seminars

- 20
^{th}September 2013**The spectra of a real number (An Erdos problem)**

Simon BakerAbstract (click to view)Given an infinite set of polynomials and a real number q, what can we say about the distribution of the set of points obtained when we evaluate this set of polynomials at q? Erdos and others considered this problem in the case of polynomials with positive integer coefficients and positive q. In this talk I shall give an overview of their work and give a "self contained proof" of one of the main results in this area.

- 27
^{th}September 2013**Lines and lines and lines and lines and lines!**

Dave NaughtonAbstract (click to view)If you've ever seen examples of continuous yet nowhere differentiable functions, they you've probably seen something like the classical Weierstrass function, which can be written down explicitly. In this talk I will introduce a parametrized family of functions, constructed in a different way, for which we cannot explicitly write down a formula. In particular I will talk about how we can construct them as invariant sets for some dynamical system, and about non-differentiability in different cases. The construction is dead simple, but discloses in a very elementary way a connection between nowhere differentiable functions and the Cantor-Lebesgue singular functions. Also it's Meatloaf's birthday on Friday, so you should probably come. I asked him to come...he said to let him sleep on it.

- 4
^{th}October 2013**Applications of Graph Theory**

Jamie PhillipsAbstract (click to view)Graph theory is becoming increasingly significant as it is applied to various other areas of mathematics, science and technology. It is used actively in computer science (algorithms), biochemistry (genomics), electrical engineering (networks) and operations research (scheduling). However, the combinatorial methods used in graph theory have also been used to prove fundamental results in pure mathematics.

In this talk I will give graph theoretic proofs of some famous results including Fermat’s Little Theorem and the Nielson-Schreier Theorem, and discuss applications of graph theory to DNA sequencing, computer network security and how the famous Four Colour Theorem is used in the assignment of frequencies in GSM mobile phone networks. - 11
^{th}October 2013**Counting redefined in HD**

Amit KuberAbstract (click to view)Lebesgue, Hausdorff and Haar and many others have defined measures. Is there a common roof under which all these measures come together? Motivic measures are universal in this sense, but one requires heavy machinery to define them. They are not real valued and instead of working with the Boolean algebra of measurable sets, we directly work with the formulas defining those sets.

We will focus our attention on the motivic counting measure. I will begin by giving an `overkill’ construction of the counting measure for identifying essential components of the construction. Then we will gently develop the theory towards motivic counting; we will encounter some logic and algebraic geometry on our way. - 18
^{th}October 2013**Finite Groups, Simplicial Complexes and Representations**

David WardAbstract (click to view)Since the dawn of time (well the 20th century) group theorists have been striving towards the goal of classifying all finite simple groups. These are seen as the building blocks of finite group theory, as in some senses a finite group can be built using these simple groups. The classification theorem is one of the greatest results in pure mathematics and was finally proved in the 1980s, 1990s or 2000s (depending on who you believe.....), and the original proof spanned over 10,000 pages of journal articles.

Since the classification theorem, many people have seen finite group theory as being a dying subject. However, despite the existence of the classification there are many apparently elementary results that are still unknown, such as what the representations of certain finite groups are, and how subgroups of given groups interact.

In this talk we will look at a few such subjects, seeing how determining when two nice subgroups of a group sit inside another subgroup can be extremely tricky, and also seeing how the knowledge of such results can allow us to determine representations of the given group. No prior knowledge of representation theory will be required (after all, it's a Friday afternoon!).

- 25
^{th}October 2013**Boolean rings**

David WildingAbstract (click to view)Any Boolean algebra (think the powerset of a set) can be turned into a ring in which each element is multiplicatively idempotent, and vice versa. Such rings are called Boolean rings. I will describe some of the interesting properties of Boolean rings, and I will explain how I have used the correspondence between Boolean rings and Boolean algebras in my work.

- 1
^{st}November 2013**The abstract theory of topological subspaces**

Christopher TeddAbstract (click to view)A topology on a set X is a family of subsets of X that is closed under finite intersection and arbitrary union. That is, viewed as a poset with subset inclusion as the order relation, it is a complete distributive lattice (per Pure Postgrad Seminar 25 Oct). A continuous function between topological spaces X and Y maps sets from the topology on Y to sets in the topology on X under inverse image; we see that this inverse mapping is order-preserving when considering the topologies as posets. In fact, we have a (contravariant) functor from the category of topological spaces and continuous functions, to a category of a special kind of lattices, called frames, and particular order-preserving maps between them. Even better, we (with a bit of ingenuity) have another functor going back in the opposite direction, and what is more, these functors provide a dual (i.e. contravariant) equivalence between the two categories. This provides the setting for so-called ‘pointless topology’.

It turns out many familiar topological properties can be defined and studied in the category of frames. In this seminar we look at subspaces, which correspond to quotient frames. We take a bit of a digression into the world of logic, and hopefully end up with an ‘interesting observation’ or two by the seminar’s end. - 8
^{th}November 2013**Daily basis dynamics**

Rafael Alcaraz BarreraAbstract (click to view)It is not surprising at all that maths is involved in loads of our daily basis activities. There is nothing as common as cooking and nothing is more relaxing after the seminar than enjoying ourselves at the pub. In this talk, I will mention some basic notions in dynamical systems which can be inspired by "pub activities" and cooking.

- 15
^{th}November 2013**Quantum Algebras and their Semi-classical Limits**

Siân FryerAbstract (click to view)I suspect you all know by now that non-commutative rings (especially ones involving fractions) can be really quite difficult to work with. But what if we could replace the non-commutative structure with a commutative one, while somehow still preserving enough information to get a sense of how the original ring behaves? Do the extra tools we gain by working in a commutative ring outweigh the difficulties? This is the very tip of the deformation-quantization iceberg, and I will talk about how we can apply this to a few specific examples, including (inevitably) the q-division ring but also some quantum groups. In particular, I'll talk about a conjecture made by Goodearl connecting the prime and primitive ideals of various quantum groups to the Poisson prime/Poisson primitive ideals of their semi-classical limits.

For those of you who don't accept the premise that trying to describe the prime ideals of a ring is interesting in its own right, well... (a) you're wrong, and (b) I finally have a good quality video of last year's Domino Computer in action which I plan to show at the end of the talk if there's time, so it's still worth turning up! - 22
^{nd}November 2013**Return of the Jedi**

Alex LongdonAbstract (click to view)Near the end of Star Wars, when all hope seemed lost, the loveable ewoks rose up against the Imperial forces and saved the day (spoilers, I guess). Long time fans of the postgraduate seminar will no doubt remember my previous two talks, in which we met homotopy groups and learnt how good they are (A New Hope) only to find out how difficult they can be to compute (The Empire Strikes Back). In this, the third part of the saga, we will see how stable homotopy theory can emulate the unlikely success of the ewoks and come to our rescue.

- 29
^{th}November 2013**The moduli space of compact Riemann surfaces**

Goran MalicAbstract (click to view)The classification theorem of closed surfaces states that any connected closed surface is determined, up to homeomorphism, by its genus and (non-)orientability. Very often surfaces will carry some additional data, for example a complex structure; in that case we say that two surfaces are isomorphic if they have the same topological "shape" and carry the same data. The moduli space of compact Riemann surfaces is precisely the collection of isomorphism classes of this kind - two Riemann surfaces represent the same class if they are topologically the same and carry the same complex structure.

In this seminar I shall properly say what it means to 'carry the same data' and give a couple of examples of moduli spaces in low genus.

Also, with the holidays approaching, most of you will have brushed up your baking skills by the time the seminar starts. So, bring me a cake! Or else... - 6
^{th}December 2013**A Very Special Seminar**

Matthew Taylor and Nic ClarkeAbstract (click to view)For the end-of-year seminar, Nic and I will be giving a beamer presentation on a mystery topic only known to us and one other person. We won't have seen the slides in advance. The twist is that each slide has been created by a different pure maths postgrad - and they were each only allowed to see the slide immediately before theirs. Expect chaos, laughter and us making complete fools of ourselves.

** For the Spring 2014 seminar timetable, please click here.**

## Previous Seminars

**
List of 2012/2013 seminars (Andrew Davies/David Ward)
List of 2011/2012 seminars (Simon Baker/David Naugton)
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)
**