Mathematics Department Student Seminars, 2009

Organizers: Wendy Barattaand Peter Tingley.

Time and place: Fridays 16:00-17:30, Richard Berry, Room 107

Pure Maths Series

May
8
 Wendy Baratta
Pieri formulas starring MacDonald polynomials with their trusty sidekicks the interpolations.
15
 Josh Howie
Knot Theory and 3-Manifolds.
22
 Martha Yip
Alcove walks and Macdonald polynomials
29
 Steve McAteer
Happy Snaps from {f|f:*C*->*C*} - Visualisations of Complex Valued Functions.
June
5
 Matt Kotros
 A Geometric View of Groups
12
 Thara Supasiti
Buildings, Groups and Symmetries
19
 Anita Poising
The 2-Boundary Temperley-Lieb algebra and a light introduction to the qKZ equation
26
 Nick Davis
Self-similar groups and dynamics
August
7
 Michael Couch
Lax Equations: It turns out algebraic geometry is actually good for something
14
 Zajj Daugherty
Combinatorics and the representation theory of centralizer algebras
21
 Anthony Mays
Something about juggling
28
 Anthony Mays
The joy of juggling (aka yet more stuff about juggling)
September
4
 Wendy Baratta
Latin Squares
11
 Nicholas Beaton
Block Designs
18
 Nicholas Stevenson

Invariant Theory

General Maths Series

October
9
 Sandy Clarke
 Statistics
16
 Anthony Mays Random Matrices: What they are and where they're at
23
 Ellie Button

Fluid Dynamics

Abstracts


May 8: Wendy Baratta, Pieri formulas starring MacDonald polynomials with their trusty sidekicks the interpolations.
Symmetric Macdonald polynomials were first introduced by Ian Macdonald in 1988. Macdonald explored the expansions when two polynomials of this basis were multiplied, the so called Pieri Formulas. Six yours later the more general Nonsymmetric Macdonald polynomials were introduced and there are no known analogous Pieri formulas. By expoiting the theory of the even more general polynomials, the interpolation Macdonalds, I will derive a special case for the Pieri formulas, the product of the first elementary symmetric function and a nonsymmetric Macdonald.

May 15: Josh Howie, Knot Theory and 3-Manifolds.
An introduction to knots and 3-manifolds, focusing on injective and incompressible surfaces embedded in knot complements. And a quick look at constructing an algorithm that can decide if a knot is alternating.

May 22: Martha Yip, Alcove Walks and Macdonald Polynomials.
The Littlewood-Richardson rule for Schur functions is a combinatorial formula involving skew tableaux. The Littelmann path model provides a generalization to all Weyl characters. We use alcove walks to give a Littlewood-Richardson rule for Macdonald polynomials, which are generalizations of Weyl characters.

May 29: Steven McAteer, Happy Snaps from {f|f:*C*->*C*} - visualisations of Complex Valued Functions.
Complex valued functions are much talked about but rarely seen. But their graphs live in (something like) *R*^4, so how can we hope to *see* them? I will discuss one way to do this which produces some quite cute pictures - and can on occasion lead to some degree of insight..

June 5: Matt Kotros, A Geometric View of Groups.
This talk introduces some fundamental notions of geometric group theory. In particular, hyperbolic groups will be discussed, followed by a glimpse at one form of its relative.

June 12: Thara Supasiti, Buildings, Groups and Symmetries
One of the fundamental questions in group theory is the classification of finite simple groups. In general, it is hard to determine whether or not a group (or family) is simple. During the 60's and 70's, Bruhat and Tits published a series of papers on an object which we know as building today. It turns out to be one of the important tools for classification of projective special linear group PSL_n(F_m). I will give an example of a building to demonstrate some of the features of a building.

June 19: Anita Ponsaing, The 2-Boundary Temperley-Lieb algebra and a light introduction to the qKZ equation.
During this talk, you'll find out what qKZ stands for, if you didn't already know, as well as its relationship with the 2BTL algebra and the Yang-Baxter equation. It's a lighthearted mixture of algebra, combinatorics, pretty pictures and polynomials.

June 26: Nick Davis, Self Similar groups and Dynamics.
Since the 1980s, the class of self-similar groups has furnished us with groups having many interesting properties. The most prominent is the Grigorchuk Group, which is an infinite finitely generated torsion group and has intermediate growth. More recently, the connection between self-similar groups and dynamics has been studied. I'll aim to shed light on this connection, using the so-called Basilica Group as an example..

August 7: Michael Couch, Lax Equations: It turns out algebraic geometry is actually good for something.
The Lax Equation \frac{dL}{dt} = [L,M] is a naturally occurring matrix differential equation. Indeed the fundamental equation in quantum mechanics is of this form. Most interesting for mathematicians however is that it is typically equivalent to a non-linear differential equation, indeed, an /emph{integrable}, that is analytically solvable, non-linear differential equation - a rare occurrence. The equation suggests different things to different people, and its subtleties are many. For a certain class of Lax equations, I shall describe a surprising approach to its solution, one winding through the wilds of algebraic geometry.

August 14: Zajj Daugherty, Combinatorics and the representation theory of centralizer algebras.
There are many great examples of algebras which arise as tensor power centralizer algebras, algebras of operators which preserve symmetries in a tensor space. The most familiar example is the connection between the general linear group and the symmetric group, studied by Frobenius and Schur around 1900. The key is that these mutually centralizing algebras have linked representation theory, so the combinatorial tools used to study one algebra can be harnessed to study the other. In this talk, I will introduce some of these combinatorial tools, and talk about how to apply them to the representation theory of many favorite examples of diagram algebras.

August 21: Anthony Mays, Something about juggling
Juggling is the noblest of pursuits, invented by the ancient king of Babylon. These facts might be fabrications, but my talk this afternoon will be full of things with more truth value. Juggling has an interesting description in terms of integer sequences, as well as being fun, so it has at least two things in common with mathematics. I'll look at how some juggling geeks have applied maths to improve their leet skills, and then how some maths geeks have applied juggling to prove some sick theorems. Crowd participation invited - you will be compensated with carbohydrates and trans-fats.

August 28: Anthony Mays, The joy of juggling (aka yet more stuff about juggling)
There'll be some juggling, and some maths. Then probably some more juggling. I plan to cover the stuff I didn't get onto last week, plus some new stuff I've read about in the meantime. Come for the show and stay for the maths. No prior knowledge required. And if that's not enough to get you there, then the promise of artery-hardening treats should do it.

September 4: Wendy Baratta, Latin Squares
I'll be chatting about Latin squares, particularly mututally orthogonal Latin Squares. Looking at existence and generation - it'll be a mixed bag of combinatorics and algebra with some discussion at the end of their applications in the real world - yes, sometimes pure maths does have applications!

September 11: Nicholas Beaton, Block Designs
In the same spirit as Wendy's talk on Latin square last week, I'll be doing a rundown on block designs - what they are, some of their properties, how to construct them, and the sometimes-quite-tenuous applications that combinatorialists claim in order to justify more funding

September 18: Nicholas Stevenson, Invariant Theory
A branch of algebra, was one of the major areas of mathematics in the 19th century and remains of interest today - David Mumford won a Fields medal for his application of David Hilbert's work on invariant theory to algebraic geometry. In this talk, I will sketch Hilbert's proof that the invariant ring of a linearly reductive group is finitely generated and discuss its significance.

October 9: Sandy Clarke, Statistics
If you want breadth in your mathematical education, you really ought to know some stats. The plan is to discuss a few simple but important results, look at some examples of ways in which these can be expanded to make things more interesting and visit a couple of amusing historical applications (and misapplications). You will leave wanting to be a statistian or, at the very least, a better person.

October 16: Anthony Mays, Random Matrices, What they are and where's it at
"Random" is a word that the youth of today have distorted out of all meaning. With their pants practically hanging off and their hair in all manner of disorder, they scornfully strut about butchering the language with impunity. We can't let this continue, and we'll start by taking back the word "random". Join me as we explore the workings of Random Matrix Theory, which has a respectably proper randomness about it. Cowabunga.

October 23: Ellie Button, Fluid Dynamics
Have you ever wondered what those dirty applied mathematicians do? I will introduce the equations governing basic fluid flow, and discuss the very crude assumptions we make in order to solve them. Some simple examples will be given. No knowledge of fluid dynamics is assumed, and you should expect very little maths (this is about the biscuits!).