Gandalf seminar 2015-2016
Gandalf is the postgrad-student-run pure maths seminar. Talks normally take place on Wednesday at 16:00 in CM221, or 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.
Summer 2016 Talks
Organised by John Lawson. If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email me.
Thomism, Cats and the meaning of Science
Calum Robson
Wednesday 22 June 2016, at 16:00, in CM221
Abstract: In this talk I will analyse what it means to have a mathematical theory of the real world, using the lens of Aristotelian/Thomist philosophy. I will begin by giving an overview of Thomist metaphysics with the help of cat pictures. I will then formulate a definition of a Mathematical theory of physics as a functor- like map, in one of two ways. First, as between a set of mathematical objects and the basic objects (or substances) in a conceptual model of the physical world. Second, as a map between certain numbers resulting from a mathematical model of a system, and certain observable properties (or accidents) of that system. This second map seems to be a kind of, 'probabilistic homomorphism' due to the error bars inherent in such a description. Finding the consistency of the first kind of map corresponds to a philosophical analysis of the physical theory in question, and I will argue that this motivates the conceptual analysis of a physical theory as a way of making progress in physics. Finally I will flesh out the abstract discussion of these issues by applying such an analysis to classical mechanics.
Zeta values and characteristic polynomials
Zhe Chen
Wednesday 15 June 2016, at 16:00, in CM221
Abstract: In this talk we present an introduction to an analogue between number fields and function fields, first discovered by Kenkichi Iwasawa.
Arithmetic & Denominators of Eisenstein Series
Rob Little
Wednesday 8 June 2016, at 16:00, in CM221
Abstract: A modular form \(f(z)\) for \(SL_2(Z)\) with rational coefficieints has a denominator, ie an integer \(D\) exists such that \(Df(z)\) has integral coefficients; this \(D\) is often arithmetically interesting - in particular, when \(f(z)\) is an Eisenstein series.
Any modular form \(f(z)\) has an associated cohomology class for the space \(H/SL_2(Z)\), which will also have a 'de Rham denominator'; we hope that these two denominators are in fact the same!
By an extension of the classical Shintani lift, I shall try to explain a method (so far only partially successful) in showing this conjecture. We give an overview of the methods used, as well as a couple of generalisations that I have been looking at this year. The talk should hopefully give an idea of the enormous arithmetic interest in the area of modular forms, as well as the way that geometric methods may be used in the pursuit of number-theoretic goals.
Topological surgery in nature
Will Rushworth
Wednesday 18 May 2016, at 16:00, in CM221
Abstract: In recent work Lambropoulou and Antoniou characterised a number of natural phenomena in terms of topological surgery. To do so, they augmented the abstract notion of surgery in order to include dynamics and a notion of continuousness. I'll go through this augmented definition of surgery and some examples of natural processes in which it occurs.
On the elliptic Monge-Ampere equation
Daniel Ballesteros-Chavez
Wednesday 11 May 2016, at 16:00, in CM221
Abstract: The Monge-Ampere equation is a fully nonlinear partial differential equation strongly related to the Minkowski problem of hypersurfaces with prescribed Gauss curvature. Topological methods are used to state the existence of solutions by using a priori estimates. We will talk about these methods in the elliptic type case.
Diophantine equations, rep theory and clusters
John Lawson
Wednesday 4 May 2016, at 16:00, in CM221
Abstract: Cluster algebras are known to be closely linked to the study of the geometry of surfaces. Some recent work in this area accidentally gave rise to a number of diophantine equations along with a procedure to compute integer solutions. This might also have links in the geometric study of representations of surface fundamental groups, as well as quiver guage theories and cluster automorphisms. We will discuss ideas around this theme in whichever direction the audience prefer.
Integer valued polynomials
Anna Szumowicz
Wednesday 27 April 2016, at 16:00, in CM221
Abstract: The polynomial \( {X \choose m} \), where \(m\) is a natural number is an example of a polynomial which takes integer values on \(\mathbb{Z}\) even though its coefficients are not integer. A polynomial \(f\in \mathbb{Q}[x]\) with the property \(f(\mathbb{Z})\subset \mathbb{Z}\) is called integer-valued. If \(f\) is of degree at most \(n\), then it is enough to check that \(f({0,...n}) \) takes values in \(\mathbb{Z}\) to know that \(f\) is integer-valued. A finite set \(A\) is called \(n\)-universal if \(f(A)\subset \mathbb{Z}\) implies that \(f\) is integer-valued for every \(f\) of degree at most \(n\). I will talk about \(n\)-universal sets when \(\mathbb{Q}\) is replaced by a number field.
Michaelmas 2015 Talks
Organised by John Lawson. If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email me.
Twisty puzzles and group theory
Steven Charlton
Wednesday 15 December 2015, at 16:00, in CM221
Abstract: Everyone has probably played with a Rubik's cube at some point. Some people might have even learned how to solve it. But wouldn't it be much more satisfying if you could figure out your own solution? Using the ideas of commutators and conjugation from group theory I will explain how you can do this, not only for the Rubik's cube but for various other twisty puzzle you might encounter. He will also bring along plenty of different puzzles for people to play with!
Many Moonshines: Monstrous, Mathieu and M(Umbral)
Sam Fearn
Wednesday 09 December 2015, at 16:00, in CM221
Abstract: Mathieu Moonshine concerns a surprising observation relating string theory to the representation theory of a particular sporadic group, Mathieu 24. This is reminiscent of Monstrous Moonshine in which it was discovered that the coefficients of the modular j-function are related to the representation theory of the Monster group. In this talk we will introduce a topological invariant of string theories compactified on K3 surfaces, called the elliptic genus of K3, and see how Mathieu 24 appears in this context. To this date, the role of the large discrete symmetry M24 in String Theory is not properly understood. We will then discuss Umbral moonshine, which comprises of 23 examples of moonshine in which the Niemeier lattices are used to connect certain mock modular forms to finite groups.
Modular Invariance in String Theory
Richard Stewart
Wednesday 02 December 2015, at 16:00, in CM221
Abstract: Elliptic functions and modular forms are a common feature in certain calculations within string theory. I aim to give a light overview of some aspects of string theory to provide some context, before describing various elements of the calculations involved.
Deligne-Mostow lattices and cone metrics on the sphere
Irene Pasquinelli
Wednesday 25 November 2015, at 16:00, in CM221
Abstract: Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space. One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle. In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with three folding symmetry.
Geometry of representations of finite linear groups
Zhe Chen
Wednesday 18 November 2015, at 16:00, in CM221
Abstract: This will be an expository talk on sheaves-functions correspondence and its applications. I will start with finite Abelian groups and some involved arithmetic problems, and then turn to the characters and representations of Lie type groups.
Virtual Khovanov Homology
Will Rushworth
Wednesday 11 November 2015, at 16:00, in CM221
Abstract: Khovanov homology is a chain-complex valued invariant of links. Virtual knot theory is a generalisation of classical knot theory which considers embeddings of circles into thickened surfaces of genus g>0 (the g=0 case returns classical knot theory). We give an introduction to virtual knots and a quick overview of the definition of Khovanov homology, before going through the process of a generalising it to the virtual case in a picture-heavy and pedagogical way.
Skew Howe duality in Type A quantum knot invariants
Jonathan Grant
Wednesday 4 November 2015, at 16:00, in CM221
Abstract: Both the Jones polynomial and the Alexander polynomial can be viewed as invariants arising from the representations of quantum (super)groups in type A. Skew Howe duality give these invariants particularly nice descriptions in terms of trivalent diagrams. This method is particularly powerful when defining knot homology theories categorifying these polynomials. I will discuss the relationship between representations of quantum groups and the trivalent diagrams appearing in calculations of knot invariants, and describe how this can be used to understand knot homology theories, and progress towards obtaining a 'quantum' categorification of the Alexander polynomial.
Willmore's Energy: Topological Constraints and Diffuse Interfaces
Stephan Wojtowytsch
Wednesday 28 October 2015, at 16:00, in CM221
Abstract: In 1965, Tom Willmore (Durham) first considered a curvature energy for immersed surfaces in \( \mathbb{R}^3 \) which would become widely studied in differential geometry and the modelling of liquid membranes. I will briefly discuss Willmore's energy, an application in biology, and a diffuse interface approach to the minimisation problem. The diffuse model has computational advantages, but makes control of the topology of surfaces more involved. In the last part of the talk, I will indicate how we managed to prescribe connectedness for a limiting problem along with some computational evidence. This talk is based on joint work with Patrick Dondl and Antoine Lemenant.
Primes of the form \( x^2 + ny^2 \)
Steven Charlton
Wednesday 21 October 2015, at 16:00, in CM221
Abstract: Fermat observed that (except for \( p = 2 \)) a prime \( p \) can be written as the sum of two squares if and only if \( p \equiv 1 \pmod{4} \). This result motivates our basic question: which primes does a given quadratic form represent?
To begin to answer this, we will relate the question of primes represented by a quadratic form to questions about ideal classes in quadratic number fields. And we will then be able to study these questions using the powerful tools of class field theory.
The main goal of this talk will to give a complete answer to this question for a specific class of quadratic forms, the so-called principal forms \( x^2 + ny^2 \). In this case the answer has the following form: there exists a polynomial \( f_n(t) \) such that \( p = x^2 + ny^2 \) if and only if \( f_n(t) \) has a root modulo \( p \). And for squarefree \( n \), this polynomial \( f_n(t) \) has an explicit interpretation as the polynomal describing the `Hilbert class field' of \( \mathbb{Q}(\sqrt{n}) \).
The structure of arc complexes
Jon Wilson
Tuesday 13 October 2015, at 13:00, in CM219
Abstract: