Mathematics Department Student Seminars


Organizers:Wendy Baratta and Peter Tingley.

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

Pure Maths Series 2009

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 2009

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

Fluid Dynamics

30
Maurice Chiodo

Groups and the Rubik's Cube

November
6
Olivia Madill
Integer Programming
13
Hugh Miller

Why everyone should quit their day jobs and become a data miner

20
Hugh Miller

Why everyone should quit their day jobs and become a professional gambler

27
Wendy Baratta

Getting Macdonald Polynomials into Shape: Prescribed Symmetry Macdonald Polynomials and Constant Term Identities

Pure Maths Series 2010

January
29
Steve McAteer
Some Solvable Nonlinear Differential Equations *sigh* or Another Mysterious and Wonderful Beast Bites the Dust
February
5
Michael Couch
Supersymmetry in Quantum Mechanics
12
Thara Supasiti

Cutting and Gluing Groups

19
Shanil Ramanayake

Visualising the 3-sphere

26
Thara Supasiti and Steve McAteer

The top 6 (yes, 6) graphs of all time!!!

March
5
Michael Payne
Triangulating the rectangle
12
Yi Huang

Counting Things

19
Sandy Clarke and Wendy Baratta

Cryptic Title: Left Making Money, half basics after tomb (5,8) Straight Title: Learn Cryptics

26
Anita Poising

The Razumov-Stroganov "Conjecture"

April
4

Easter Break
9
Matt Kotros

Boundaries of Groups

16
Shaun

Title

23
?

Title

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!).

October 23: Maurice Chiodo, Groups and the Rubik's Cube
The Rubik's cube can provide hours of enjoyment (or frustration) for anyone. But how does it work, why is it hard, and why can some people solve it so effortlessly? In this talk I hope to explain some of the mathematics behind the Rubik's cube, and use the cube as a wonderful example of group theory in everyday life. A beautiful application of pure mathematics... .

November 6: Olivia Omadill, Integer Programming
One of the many techniques used in operations research is integer programming. Come along to learn about some of the problems which can be solved using it, the beauty of IP modelling and why the software to solve these problems costs thousands of dollars.

November 13: Hugh Miller, Why everyone should quit their day jobs and become a data miner
Once viewed as a backwater in statistics for people who couldn't hack theory, data mining has grown into one of the largest and most exciting areas for modern research, with a deep theory and a startlingly large array of important applications. This talk will be reasonably non-technical and aims to explore: what data mining is all about, why people find it interesting, the types of applications and how to become a millionaire as a data miner.

November 20: Hugh Miller, Why everyone should quit their day jobs and become a professional gambler
Gambling in casinos and on sporting events is generally regarded as a "losers game", in that the odds are usually stacked in favour of the house/bookmaker. However, this viewpoint ignores one important fact: there are some professional gamblers out there who have made lots of money. This talk will introduce some probability and information theory, and explore how they apply to sports betting. It will include discussion of Dutch books, as well as optimal betting strategies

November 27: Wendy Baratta, Getting Macdonald Polynomials into Shape: Prescribed Symmetry Macdonald Polynomials and Constant Term Identities.
Macdonald polynomials with prescribed symmetry can be obtained from the nonsymmetric Macdonald polynomials via symmetrisation, antisymmetrisation and normalisation. By computing the explicit form of the normalisation with respect to the constant term inner product we provide a derivation of a special case of a conjectured q-constant term identity.

January 29 Steve McAteer, Some Solvable Nonlinear Differential Equations *sigh* or Another Mysterious and Wonderful Beast Bites the Dust
We are often told that solvable nonlinear differential equations are rare - if you write down a random nonlinear DE, it's an extremely safe bet that it's not going to be solvable. We are then presented with a solvable example and we stand rightly in awe. Where did it come from? How does it have any right to be? In this talk I will follow a discussion presented in a manuscript by M. Noumi & T Takebe. Lots of solvable nonlinear DEs are constructed in a reasonably pedestrian manner. If time permits, I will also discuss other interesting results presented in the manuscript.

Febuary 5 Michael Couch, Supersymmetry in Quantum Mechanics
Mechanics The equation of motion in quantum mechanics is Schroedinger equation, which simply states the equivalence of wave-like energy and particle- like energy (kinetic energy and potential energy). The choice of the potential, a function depending on position in space, determines the system that is being studied. Like all PDE's, solutions for generic systems do not have any elementary form. Supersymmetry is an interesting trick, used in quantum mechanics and quantum field theories used to generate new explicitly solvable potentials from old. Remarkably, these new potentials, while appearing very different, have the same spectrum and eigenfunctions, and scattering data. I will give a short qualitative review of quantum mechanics, concentrating on the mathematics rather than the physics, and explain some of the results of supersymmetry.

Febuary 12 Thara Supasiti, Cutting and Gluing Groups
In the western philosophy, we study an object by focusing on its "smaller" components and how they fit together. So what about groups? The usual way for studying groups -- in particular finite groups -- is to study its subgroups, whether it be normal, characteristics or Sylow. So what about infinite groups? We have group splitting. Natural question is then: when does a group split over its subgroup? In this talk, I will explain exactly what I mean by group splitting and then use its connection to topology to give a vague answer. If time permits, I will draw another connection to graph theory. There will be plenty of pictures if I can draw them.

Febuary 19 Shanil Ramanayake, Visualising the 3 sphere
We can think of the three sphere as an embedded object in 4-dimensional space (R^4). We will attempt to present a method by which we can "visualize" the 3-sphere, by projecting it down to three dimensional space while preserving some of its structure. This will allow us to "see" (in some sense) an object in 4-dimensional space. In particular we characterize the three sphere in terms of the unit quaternions, and the use the Hopf map (A map from the three sphere to the two sphere), to capture some of the structure of the three sphere. Then we project to 3-dimensional space (R^3) via stereographic projection. The nice thing about all of this is that we only need to use a bit of linear algebra to make it work!

Febuary 26 Thara Supasiti and Steve McAteer, The top 6 (yes, 6) graphs of all time!!!
We have, after much deliberation picked our top 6 (yes, 6) graphs of all time. Like 20 to 1; but for nerds ... and less celebrity commentary ... and Thara and Steve instead of Bert Newton ... and instead of 20, it's 6. There will of course be quite a bit of personal bias in the selection, and we would hope that this provokes an all-in Donnybrook. Will your favourite graph be there? (And don't pretend you don't have one.) Will Thara include some obscure pure mathsy thing which he is convinced the universe hinges on? Will Essendon beat Geelong in round 1? (Yes.) There'll be pictures, there'll be videos, there'll be fun and games, there'll be graphs. Sorry people, strictly no 3D.

March 5 Michael Payne, Triangulating the rectangle
Can a rectangle be cut into an odd number of triangles all having the same area? I will explain a neat proof by Monsky that shows that in fact it is not possible.

March 12 Yi Huang, Counting Things
it is widely acknowledged (Wikipedia) that many children, by the age of two, are able to build bijections between a set with n objects and the canonical set of cardinality n (for n small enough). We will attempt to employ these formidable skills in this seminar to count areas - possibly even volumes, of things. P.S.: ability to count up to 25 will probably suffice.

March 19 Sandy Clarke and Wendy Baratta, Cryptic Title: Left Making Money, half basics after tomb (5,8) Straight Title: Learn Cryptics
Ever wondered why an Iron Man is Female (Iron=Fe, Man=male) or how there could be a hint within this (witHIN This)! On Friday Sandy and I will unravel the mysteries of cryptic crosswords in a fun and interactive, that's right, fun and interactive! seminar that will make you (in theme of Sandy's previous talk) a better person.

March 26 Anita Poising, The Razumov-Stroganov "Conjecture"
(otherwise known as the Remarkable Conjecture, or RC) No, it's not edible. This conjecture, written down 10 years ago, describes a bijection between the solution of a lattice model in statistical physics and the enumeration of alternating sign matrices (or fully-packed loops, or 6 vertex models, whichever you prefer). This conjecture has been close to the center of a lot of work in the field of combinatorics and statistical physics, and recent events may mean that it will no longer be called "conjecture" but "theorem" (hence the ambiguous title of this talk). I will present a sketch of the bijection, without any attempt at going through the proof, mostly because I haven't read the new paper yet and wouldn't be game to try even if I had. It will mostly involve drawing pictures and counting: the highlights of Combinatorics.

April 9 Matt Kotros, Boundaries of Groups
One of the key ideas in geometric group theory is that a finitely generated group can be regarded as a geometric object in its own right. In fact, one usually considers a Cayley graph of the group equipped with a canonical metric, which will contain the group as a subspace. To any such Cayley graph (or to a metric space in general), one can associate a space at infinity called the boundary. For certain classes of groups, this boundary is independent of the choice of Cayley graph (a quasi-isometry invariant) and so can be called the boundary of the group. This talk will introduce the boundary for such classes of groups and describe some algebraic information about the group that can be obtained from the boundary.