# Pure Postgraduate Seminars

The Pure Postgraduate Seminar Series provides an informal environment for pure maths postgrads to present mathematical ideas. The academic year 2008/2009 seminars were organised by Ali Everett.

**The seminars are held in the Alan Turing Building, Frank Adams Room 2 (1.212), Fridays from 4pm to 5pm.**

## Summer Seminar Series 2008

- This series was jointly organised with Jacob George.
- 20
^{th}Jun 2008**Lower bounds on the query complexity of non-adaptive quantum algorithms**

Juergen LandesAbstract (click to view)The aim of the talk is to give some very general ideas as to how such bounds can be proved and to give a flavor of the methods used. (Un)fortunately the quantum aspect will all but disappear and hence no knowledge of the quantum world is required.

Expected length of the talk: 25'. - 4
^{th}Jul 2008**Break for Toric Topology Conference/Workshop**

No Seminar Scheduled - 18
^{th}Jul 2008**Symplectic Geometry for Beginners**

Rabah DjabriAbstract (click to view)Our talk will mainly focus on how to get a Poisson structure from a given symplectic manifold. The main properties of the Poisson structure will be proven.

- 1
^{st}Aug 2008**Sheaves for Beginners**

Chelsea WaltonNo Abstract - 15
^{th}Aug 2008**Stacks for Beginners**

Jacob GeorgeAbstract (click to view)The notion of a stack seems to appear in differential and algebraic geometry, theoretical physics and a great number of other areas of mathematics. Indeed, these are (according to the

*n*-Category café) a more natural language than supermathematics for Lie algebroids and differential graded manifolds.

Here, the definition of a stack over a topological space and, given time, more generally over sites will be given. This will necessarily be a very gentle introduction since the speaker has but recently become acquainted with these wonderful objects himself. Assumed knowledge: definition of a topological space, definition of a category. - 29
^{th}Aug 2008**Operads for Beginners**

Hadi ZareAbstract (click to view)My aim is to give a quick introduction to the notions of operads, and then just briefly state the "Deligne Conjecture" in topological settings. This is based on the Paolo Salvatore's last pre-print on arxiv

http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.3904v1.pdf

To do this, I will define operads. I state the origin of this notion which lies in the theory of "infinite loop spaces", and I will try to avoid any technicalities. I will say what I mean by an algebra over an operad. Then I will be in a position to state the Deligne's conjecture. This conjecture is proved by different people, in different settings. If I have some time left, I will say a few words about generalisations of the notion of an operad. - 12
^{th}Sept 2008**Whist for Two Beginners**

Ali EverettAbstract (click to view)Whist is a simple "trick-taking" style card game, in our case for 2 players.

The style is played such that both players are dealt the same number of cards and each round (or "trick") requires each player to lay one card in turn. The designated player to lay first in the round "has the lead", and in order for the other player to "take the trick" (i.e. score a point), they must lay a card of higher rank in the same suit as the first card laid. Otherwise, the player on lead takes the trick.

Now the question becomes: Once the cards are dealt, is it possible to determine the optimal number of tricks taken?

We examine the case when both players have the same number of cards in each suit, and both players have perfect information (i.e. they both know the distribution of cards). Along the way, we shall construct semigroups, intervals and maps in order to determine the outcome of a deal, and prove some nice theorems while we're at it.

## Autumn Semester 2008

- 19
^{th}Sept 2008**Six Things That Rock About Topoi**

Philip BridgeAbstract (click to view)In this talk, I will lay out a short introduction to topoi. My aim is to say what a topos is, show how it fits in with logic and foundations, and supply you with a broad gist of this topic, including any fun things I can think of. Stop laughing, maths has fun things! I don't plan to focus my talk on any particular goal or proposition; I want to supply only a rough intro to this topic.

You should know about functors and natural transformations; I'll bring in additional notions as I go along. - 26
^{th}Sept 2008**Monte Carlo Algorithms for Black-Box Groups: Computing in the vaguest possible way**

Paul TaylorAbstract (click to view)"Black-box" group algorithms work without any knowledge of the particular group being used: they rely on mysterious "oracles" to perform all the group operations. "Monte Carlo" algorithms sometimes get the answer wrong. In this seminar we combine these two concepts, attaining dizzying new heights of non-specificness in algorithm design. We define the concepts, look at an example for centralizer computation, and build it into an algorithm for the more difficult task of normalizer computation. There will be pictures.

- 3
^{rd}Oct 2008**Maths in Popular Culture**(Time Change: 5.15pm)

Katie StecklesAbstract (click to view)Mathematics, a subject we all know and love, recieves its share of attention in the media and Hollywood. I have been looking at the way it is portrayed by TV, film and other media, and have managed to find some interesting bits of mathematics which hopefully at least some of you will not have heard of before, as well as providing some light entertainment.

- 10
^{th}Oct 2008**Theories of Bounded Articles**(Time Change: 5.15pm)

Simon PereraAbstract (click to view)A lot of the tools and techniques of conventional model theory, such as compactness, are no use for studying finite structures. So allow me to lead you out of your collective comfort zone (meaning regular model theory, naturally) to see some of the methods that can be used in finite model theory, including various equivalences we can place on structures, E-F games and pebble games and some mention of more complicated methods. I will use simple examples such as evenness/parity of a set and of a linearly ordered set and properties like connectedness, 2-colourability, hamiltonicity and cyclicity of graphs.

I will also discuss a proposition of MacQuarrie's, the 0-1 Law, how the definability of a class of structures in certain different logics is equivalent to their decidability in different complexities like**P**and**NP**(not that these are necessarily different (I'm still checking my proof of that)), and a surprising result of Shelah's. - 17
^{th}Oct 2008**On Some Chevalley Groups**(Time Change: 5.15pm)

Steve CleggAbstract (click to view)I will attempt to explain a class of groups which occurs often in finite group theory. I may even talk about infinite groups on occasions. To keep the prerequisits to a minimal, the arguments will not be thourough and the focus will be on examples. In keeping with the tradition of the French (although Chevalley is born in South Africa, I understand) I shall enjoy a glass of wine during my talk. Or, maybe a beer, to reminisce of times passed when the seminar would be over by 5.

- 24
^{th}Oct 2008**Synchronizing Automata and the Černý Conjecture**

Elaine RenderAbstract (click to view)Consider a satellite orbiting the moon. We are able to control the satellite when we can see it, but when it is "behind" the moon we lose control of it. It would be useful then to know some sequence of instructions which we could send to the satellite which would reset the satellite to some known state, once it is back in view again. This is an example of a problem which motivates the study of synchronising automata. We will discuss how to identify such a sequence of instructions, some classical examples, and some other motivations. The Černý conjecture is a long standing open problem relating to the length of such reset sequences. We will discuss the definition of the conjecture, and the current state of research relating to it. There will also be puzzles and ghosts, so bring paper and pen! There may even be a prize, if I remember.

- 31
^{st}Oct 2008**The Prisoner's Dilemma**

Richard SimmondsAbstract (click to view)There are many situations in the real world in which cooperation yields a better outcome than competition. A fairly complex example might be manufacturers competing in a buyer's market, when a cartel would be more advantageous (at least from their point of view). The Prisoner's Dilemma is a simple and classic example in game theory in which rational self-interest is not the way to an optimal outcome. Using this as our motivation we look over the basic concepts in game theory.

- 7
^{th}Nov 2008**Reading Week**

No Seminar Scheduled - 14
^{th}Nov 2008**The Generalized Section Extension Property (***G*-SEP)

Wemedh A'EalAbstract (click to view)A

*G*-map*(E*which satisfies a certain_{1},E_{2},B, e)*G*-local extension condition for the*G*-section has the corresponding*G*-global extension property. A new definition that is*G*-section will be introduced and from this definition a new property namely the*G*-section extension property will be got. Also, this property is local property will be shown. More precisely, this property holds provided it holds over every set of a numerable covering*{V*of_{λ}}_{λ∈Λ}*B*. Finally, the relationship between this property and*G*-fiber map will be introduced. - 21
^{st}Nov 2008**Geometry of Surfaces in Odd Symplectic Superspace**

Oliver LittleAbstract (click to view)In the context of supermanifolds there are two counterparts to symplectic structure on a classical manifold - one "even" and one "odd". After reviewing some basic theory about supermanifolds we will introduce odd symplectic structure on a supermanifold and consider constructions related with it. Comparisons can be drawn with both classical symplectic geometry and the geometry of hypersurfaces in Euclidean space.

- 28
^{th}Nov 2008**Super-Decomposable Modules Over Tubular Algebras**

Richard HarlandAbstract (click to view)I should stress that this is nothing to do with superspaces (see last week), and no knowledge of any words in the title is assumed. Apart from "over".

A module is said to be superdecomposable if it has no indecomposable direct summands- in other words, if you write it as a direct sum of two modules, neither of them will be indecomposable.

By their nature, they are difficult to construct, and not obvious to the naked eye. There are, however, techniques to ascertain whether or not they exist, and what properties they may satisfy.

I will use the talk to explain exactly what a tubular algebra is, and whether or not it can support something as horrendous as a superdecomposable module. The talk will incorporate- amongs other things- graph theory, logic, lattices, and representation theory. Hell, I might even say the word "topology" if there's time. - 5
^{th}Dec 2008**Maximality of Subgroups in Janko's First Simple Group**

Ali EverettAbstract (click to view)In a previous seminar, I gave a concrete permutation representation for Janko's first sporadic simple group,

*J*. This time round, I will work with_{1}*J*in a much more abstract setting, looking purely at the structure. This group has seven maximal subgroups (up to isomorphism, which is apparently overrated), and I will explain why there are only seven and why they are maximal._{1} - 12
^{th}Dec 2008**A Special (Super) Class of Blocks**

Stavros ApostolouAbstract (click to view)We will define a class of blocks introduced by Eaton in 2005 behaving similarly to blocks of

*p*-solvable groups. We will go through some examples of such blocks and through some of their main properties.

## Spring Semester 2009

- 6
^{th}Feb 2009**Brackets, Operators and Projective Connections**

Jacob GeorgeAbstract (click to view)The concept of 'projective connection' is a classical one that has emerged as a collection of disparate structures in modern differential geometry. Most recently, it has been used to precisely pin down the correct definition of 'projectively equivariant quantisation' of tensor densities.

Here, I'll the define projective connection, (tensor) densities, brackets and various other structures, before giving some indication of how these are related. The talk will finish with the result that on a projectively connected manifold, symbols of second differential operators on densities can be extended canonically to well defined operators.

As usual, I will assume as little knowledge as possible - hopefully, the talk will evolve interactively around the audience so no-one will be left behind. - 13
^{th}Feb 2009**Pure Injective Hulls of 1-***pp*Types over Commutative Valuation Domains

Lorna GregoryAbstract (click to view)Over a commutative valuation domain, 1-

*pp*types are in objective correspondence with non-decreasing functions (satisfying certain properties) from the value semi-group Γ to its set of cuts Γ. This gives a geometric representation of the 1-*pp*types as the graph of these functions. Using this, various algebraic properties of the hulls of these 1-*pp*types can be reformulated in terms of the geometry of their "graph". I will then describe a geometrical equivalence relation on the "graphs" of these*pp*-types such that the graph of two*pp*-types are equivalent if and only if their pure injective hulls are isomorphic.

I promise to try my best to introduce all necessary concepts and draw some pictures. This is all from papers of Gena Puninski. - 20
^{th}Feb 2009**Polynomial Identity Rings**

Chelsea WaltonAbstract (click to view)A polynomial identity (PI) ring is a ring whose elements satisfy a polynomial relation. For example, consider a commutative ring - its elements satisfy the polynomial identity

*f(x,y)=xy-yx*. Some nontrivial examples include matrix rings over fields. (Can you guess such a polynomial identity for these rings?)

The first part of the talk is a brief historical survey of the appearance of PI rings before the formalization of abstract ring theory. The main focus however is the power of PI theory. In particular we will highlight theorems of Amitsur-Levitzki, Kaplansky, and Artin-Procesi. - 27
^{th}Feb 2009**Geometric Introduction to***K*-Theory

Steve MillerAbstract (click to view)*K*-Theory is a cohomology theory built out of vector bundles over a topological space. It is a finer tool than ordinary cohomology for studying maps between spaces, and was famously used by our very own Frank Adams to derive important results in the homotopy groups of spheres.

In this talk I will review some facts about complex vector bundles, and define the contravariant functor*K(X)*of a topological space*X*. I will show how*K*is used to construct a cohomology theory, and describe some of its useful properties. I will finish by presenting one or two interesting applications. - 6
^{th}Mar 2009*G*-(Co)Homology

Gemma LloydAbstract (click to view)This talk will be a fairly elementary one. We will define simplicial complexes, simplicial homology, linear representations of finite groups and group actions on simplicial complexes. We will then attempt to fit these areas together to form some kind of idea about what I describe as "

*G*-homology". No apologies will be made for the individual basic elements of this talk. Not everyone knows what you know! - 13
^{th}Mar 2009**Something to do with Ramsey and/or Gödel**

Richard HarlandAbstract (click to view)In 1931, Kurt Gödel shot a dagger through the heart of avid proof-theorists and meta-mathematicians, when he proved the "Incompleteness theorems"- which stated that no consistent theory can prove everything there is to know about basic arithmetic.

His proof was good, but it didn't give an interesting example of an unprovable theorem. Since then, however, examples have been unearthed which do satisfy such a curious fact. I will illustrate one, via a journey through graph theory, logic and combinatorics. - 20
^{th}Mar 2009**Modules over Ringed Spaces**(Room Change: Sydney Goldstein Room (MAGIC), 1.213)

Philip BridgeAbstract (click to view)A ringed space is just a topological space with a sheaf of commutative rings on it. A module

*M*over a ringed space*R*is a sheaf of abelian groups on the space such that for each open set*U*,*MU*is an*RU*-module. We ask when the category of such modules has a very nice property that makes it well behaved and really quite friendly. We might even answer this question if I have time. I promise to explain what a sheaf is whenever it becomes relevant.

Promise. - 27
^{th}Mar 2009**2**^{nd}Leeds And Manchester Event

Joint University Mathematics Pure Postgraduate Seminar (The University of Leeds)

Programme - Easter Break
**No Seminars Scheduled** - 24
^{th}Apr 2009**Loop Spaces and Dynamics**

Katie StecklesAbstract (click to view)At LAME:JUMPPS, Bev talked about the fundamental group of a topological space. I will begin by discussing this and the other constructions which involve loops on a manifold, and then introducing the idea of 'relative loops', formed under an automorphism of the space. We will also consider analogues of the spaces of loops under such conditions, and discover how they are related to knot theory and group theory. We will then study the example of the Newtonian planar two centre problem, and calculate the resulting invariants in this case.

- 1
^{st}May 2009**Perfect Isometries (PIs) and Their Group**

Pornrat RuengrotAbstract (click to view)There is a notion of distances in the spaces of generalized characters of group representations. We consider the distance-preserving isomorphisms (i.e. isometries) between these spaces. With some additional properties, these isometries become "perfect". Given two random spaces, perfect isometries between them rarely exist but when they do, some interesting things happen. In fact, if the two spaces are the same, the existence of perfect isometries is always guaranteed and they even form a group!

In this talk I will give the relevant definitions, derive some interesting properties of PIs, some structure of their group and show how information about these PI groups could be used to determine the existence of PIs between different spaces. Broue's abelian defect group conjecture (PI version) will also be mentioned. - 8
^{th}May 2009**Generating Functions of Nestohedra and Applications**

Andrew FennAbstract (click to view)Nestohedra are a type of polytope which naturally form families with a polytope in each dimension. In this talk I will explain why this is useful and why generating functions are a good way to study these families. If there is time I will also demonstrate some applications of these generating functions.

- 15
^{th}May 2009**Character Theory and Finite Groups**

John BallantyneAbstract (click to view)Character theory is a powerful tool in the study of finite groups. In this talk, our aim is to demonstrate just some of the ways this beautiful theory can be applied, in particular to deduce structural properties of groups.

We will introduce the basics of the theory, enabling us to prove a famous theorem of Frobenius. If time allows, we will indicate how these methods have been adapted to deal with more general cases. (You may be pleased to hear that no prior knowledge of character theory is assumed for this talk!) - 22
^{nd}May 2009**Large Cardinals in Set Theory**

Ali LloydAbstract (click to view)Large cardinals are central objects of study in contemporary classical set theory. Essentially they are numbers so large that one cannot prove their existence, but adding them axiomatically furnishes us with many new models of set theory and hence many subtly different backgrounds in which to do mathematics.

I will attempt to survey this field beginning with the very basics of the concept of number, and show that these things are important not just for the mathematical logician. If I have time I might also discuss in what way their actual existence can be asserted.

Applications to aspects of squodge theory should be inferred throughout. - 29
^{th}May 2009**A 2,503,413,946,215-Vertex Graph for***Fi*_{24}'

Ben WrightAbstract (click to view)Let

*G*be the largest of the Fischer sporadic simple groups Fi_{24}. In this talk we will give the construction of a graph structure on*G*which is connected to a certain maximal 2-local geometry on*G*described by Ronan and Smith in 1980. We will also give some justification why studying such a structure might be useful.

This talk is intended to be fairly elementary, and only very basic group theory will be assumed. - 5
^{th}Jun 2009**Local description of the Atiyah sequence**

Rabah DjabriAbstract (click to view)In our talk , we will describe the Atiyah sequence of a principal bundle and write down all the maps involved in local coordinates. Then, We will show how to describe a connection in a principal bundle in terms of its Atiyah sequence.

- 12
^{th}Jun 2009**Braid Monodromy: Fundamental Groups of Algebraic Hypersurface Complements and Film Quiz**

Joel Haddley (The University of Liverpool)Abstract (click to view)I am a mathematician. Therefore, when I am presented with a difficult problem my immediate reaction is to find out if someone else has already solved it and just use their results. That's sort of the motto for this seminar. I will explain how the problem of calculating fundamental groups of hypersurface complements arises in singularity theory, and show how methods of Artin, Hurwitz and co rule out any need for complicated calculations. There will also be a film quiz.

## Previous Seminars

**
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)
**