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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

Autumn Semester 2008

  • 19th Sept 2008
    Six Things That Rock About Topoi
    Philip Bridge
    Abstract (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.

  • 26th Sept 2008
    Monte Carlo Algorithms for Black-Box Groups: Computing in the vaguest possible way
    Paul Taylor
    Abstract (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.

  • 3rd Oct 2008
    Maths in Popular Culture (Time Change: 5.15pm)
    Katie Steckles
    Abstract (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.

  • 10th Oct 2008
    Theories of Bounded Articles (Time Change: 5.15pm)
    Simon Perera
    Abstract (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.

  • 17th Oct 2008
    On Some Chevalley Groups (Time Change: 5.15pm)
    Steve Clegg
    Abstract (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.

  • 24th Oct 2008
    Synchronizing Automata and the Černý Conjecture
    Elaine Render
    Abstract (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.

  • 31st Oct 2008
    The Prisoner's Dilemma
    Richard Simmonds
    Abstract (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.

  • 7th Nov 2008
    Reading Week
    No Seminar Scheduled
  • 14th Nov 2008
    The Generalized Section Extension Property (G-SEP)
    Wemedh A'Eal
    Abstract (click to view)

    A G-map (E1,E2,B, e) which satisfies a certain 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 {Vλ}λ∈Λ of B. Finally, the relationship between this property and G-fiber map will be introduced.

  • 21st Nov 2008
    Geometry of Surfaces in Odd Symplectic Superspace
    Oliver Little
    Abstract (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.

  • 28th Nov 2008
    Super-Decomposable Modules Over Tubular Algebras
    Richard Harland
    Abstract (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.

  • 5th Dec 2008
    Maximality of Subgroups in Janko's First Simple Group
    Ali Everett
    Abstract (click to view)

    In a previous seminar, I gave a concrete permutation representation for Janko's first sporadic simple group, J1. This time round, I will work with J1 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.

  • 12th Dec 2008
    A Special (Super) Class of Blocks
    Stavros Apostolou
    Abstract (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

  • 6th Feb 2009
    Brackets, Operators and Projective Connections
    Jacob George
    Abstract (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.

  • 13th Feb 2009
    Pure Injective Hulls of 1-pp Types over Commutative Valuation Domains
    Lorna Gregory
    Abstract (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.

  • 20th Feb 2009
    Polynomial Identity Rings
    Chelsea Walton
    Abstract (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.

  • 27th Feb 2009
    Geometric Introduction to K-Theory
    Steve Miller
    Abstract (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.

  • 6th Mar 2009
    G-(Co)Homology
    Gemma Lloyd
    Abstract (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!

  • 13th Mar 2009
    Something to do with Ramsey and/or Gödel
    Richard Harland
    Abstract (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.

  • 20th Mar 2009
    Modules over Ringed Spaces (Room Change: Sydney Goldstein Room (MAGIC), 1.213)
    Philip Bridge
    Abstract (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.

  • 27th Mar 2009
    2nd Leeds And Manchester Event
    Joint University Mathematics Pure Postgraduate Seminar (The University of Leeds)
    Programme
  • Easter Break
    No Seminars Scheduled
  • 24th Apr 2009
    Loop Spaces and Dynamics
    Katie Steckles
    Abstract (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.

  • 1st May 2009
    Perfect Isometries (PIs) and Their Group
    Pornrat Ruengrot
    Abstract (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.

  • 8th May 2009
    Generating Functions of Nestohedra and Applications
    Andrew Fenn
    Abstract (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.

  • 15th May 2009
    Character Theory and Finite Groups
    John Ballantyne
    Abstract (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!)

  • 22nd May 2009
    Large Cardinals in Set Theory
    Ali Lloyd
    Abstract (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.

  • 29th May 2009
    A 2,503,413,946,215-Vertex Graph for Fi24'
    Ben Wright
    Abstract (click to view)

    Let G be the largest of the Fischer sporadic simple groups Fi24. 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.

  • 5th Jun 2009
    Local description of the Atiyah sequence
    Rabah Djabri
    Abstract (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.

  • 12th 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.

For the current seminar timetable, please click here.