Gandalf seminar 2013-2014
Gandalf is the postgrad-student-run pure maths seminar. Talks normally take place on Thursday afternoon at 16:00 in whatever room happens to be free at the time. Biscuits are always supplied!
Gandalf stands for the Geometry AND ALgebra Forum, name due to Herbert Gangl. Occasionally it becomes the Radagast seminar, Research And Development in Algebra, Geometry And Sometimes Topology, if the Topologists are feeling particularly left out.
Gandalf seminar archive: 2010-2011, 2011-2012, 2012-2013, 2013-2014, 2014-2015, 2015-2016, 2017-2018, 2018-2019, 2019-2020.
Epiphany 2014 Talks
Organised by Steven Charlton
On the Khovanov Homotopy Type
Dan Jones
Thursay 20 March 2014, at 16:00, in CM219
Abstract: In 2000, Khovanov introduced a knot invariant in the form of a homology theory now known as Khovanov Homology. I will introduce this invariant and go on to talk about a new knot invariant which takes the form of a suspension spectrum, and so is invariant up to stable homotopy. The Khovanov Spectrum was introduced by Lipshitz and Sarkar in 2011 and has been shown to be a stronger invariant than Khovanov Homology. I aim to discuss some questions I have been looking at related to this spectrum.
The spectrum of the 1-form Laplacian on a graph-like manifold
Michela Egidi
Thursay 13 March 2014, at 15:00, in CM107
Abstract: A graph-like manifold is a family of neighbourhoods of thickness \(\varepsilon>0\) of a metric graph shrinking to the graph itself as \(\varepsilon\) approaches zero. In spectral geometry, graph-like manifolds were first introduced by Colin de Verdiere to prove that a manifold of dimension \(n\) greater or equal than 3 admits a metric with the first non-zero eigenvalue of the Laplacian having multiplicity \(n\) and since then, they have been used as a toy model to prove properties of manifolds or disprove conjectures. In physics, they are a model for nano and optical structure and metric graphs are believed to be a good approximation for them since the spectum of their Laplacian is a good approximation of the spectrum of the Laplacian of graph-like manifolds. In my talk I will explore the relation between the spectra of the Laplacian acting on 1-forms on the graph-like manifold and the Laplacian acting on 1-forms on the metric graph with some insights about higher degree forms.
Diophantine approximation and Khintchine's theorem
Robert Royals
Wednesday 5 March 2014, at 15:00, in CM103
Abstract: A look at the classic Khintchine's theorem in diophantine approximation and extensions of it to function fields and number fields.
A Theta Lift in \(\operatorname{SL}(2,1)\) and Locally Harmonic Maass Forms
Jonathan Crawford
Thursay 27 February 2014, at 16:00, in CM219
Abstract: Modular forms of integral weight and half integral weight have many interesting applications in number theory. Shimura in 1973 defined a very important correspondence between the two which can be defined in the framework of theta lifts. More recently harmonic weak Maass forms (generalisations of classical modular forms) and their uses have been studied. In this talk I will discuss these objects and their properties and describe my work on a theta lift which links all of them together.
Quantum Invariants of Knots
Jonathan Grant
Thursay 20 February 2014, at 16:00, in CG85
Abstract: After the Jones polynomial was discovered in the 80's, Jones's methods were generalised repeatedly until Reshetikhin and Turaev described a very general method for constructing polynomial invariants of knots. For any representation of any simple Lie algebra, their method describes a procedure for constructing a knot invariant from it. This procedure involves the so-called 'quantum groups' described by physicists, which (for the purposes of this talk) are 1-parameter deformations of universal enveloping algebras. Even in the simplest cases, these knot invariants are quite poorly understood and are still very actively researched. In this talk I will try and describe (without too many messy details) quantum groups and how knot polynomials are obtained from them, with the construction of the Jones polynomial as an example.
Polylogarithms and Double Scissors Congruence Groups
Steven Charlton
Thursay 13 February 2014, at 15:00, in CM107
Abstract: Polylogarithms are a class of special functions which have applications throughout the mathematics and physics worlds. I will begin by introducing the basic properties of polylogarithms and some reasons for interest in them, such as their functional equations and the role they play in Zagier's polylogarithm conjecture. From here I will turn to a Aomoto polylogarithms, a more general class of functions and explain how they motivate a geometric view of polylogarithms as configurations of hyperplanes in \(\mathbb{P}^n\). This approach has been used by Goncharov to establish Zagier's conjecture for \(n = 3\).
Reductive algebraic groups and parabolic induction
Zhe Chen
Thursay 6 February 2014, at 15:00, in CM105
Abstract: Two prototypes of algebraic groups are elliptic curves and \(\operatorname{GL}_n\)'s, which stand for the projective side and the affine side respectively. Here the focus is on the latter one. Their importance and interests come from many sides, e.g. the deep relations with absolute Galois groups, and the mixture of explicit appearances and complicated structures. Here I will introduce some basic concepts on affine algebraic groups and talk about the parabolic induction approach to their representations.
Mutation-Infinite Cluster Algebras
John Lawson
Thursay 30 January 2014, at 15:00, in CM107
Abstract: Cluster algebras were first introduced by Fomin and Zelevinsky in an effort to provide concrete terms to describe "dual canonical bases" in different settings. Cluster algebras are special in that the final construction of the algebra is rarely interesting, rather it is the process of finding the generators of the algebra which yields fascinating results. Generators are found using an iterative process of mutation on labelled seeds, and I am particularly interested in those mutation classes of infinite size.
What I do
Stephan Wojtowytsch
Thursay 23 January 2014, at 16:00, in CM221
Abstract: A heuristic introduction to my PhD topic. Some geometric measure theory, Gamma convergence of phase field models and pretty pictures.
Michaelmas 2013 Talks
Organised by Jonathan Crawford
Geometry of Periodic Monopoles
Rafa Maldonado
Thursay 28 November 2013, at 17:00, in CM221
Zeta functions and étale cohomology
Zhe Chen
Thursay 21 November 2013, at 16:00, in CM221
Abstract: The work on the Weil conjectures is one of the most exciting stories happened in the 20th century. These concrete statements on counting points over finite fields traced back to some of Gauss' work, and are amazingly encoded in what we now called étale cohomology, which is itself highly interesting (it generalizes Galois cohomology, gives a Hilbert Satz 90 for curves, and has applications to representation theory of algebraic groups, and etc). So this is also a good example on illustrating how an abstract machinery can be used to solve a very concrete problem. In this talk some of these smart ideas will be introduced.
Compactness-like Covering Properties
Petra Staynova
Thursay 7 November 2013, at 16:00, in CM221
Modular Forms and The Kissing Number Problem
Jonathan Crawford
Thursay 31 October 2013, at 16:30, in CM219
An Introduction to Khovanov Homology
Dan Jones
Thursay 31 October 2013, at 16:00 in CM219
Multiple Zeta Values
Steven Charlton
Thursay 24 October 2013, at 16:30, in CM221