Gandalf seminar 2018-2019
Gandalf is the postgrad-student-run pure maths seminar.
Talks 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. Whenever Clare's involved, Gandalf becomes Saruman, for Students Acronymised Research Using Measures And Numbers
Gandalf seminar archive: 2010-2011, 2011-2012, 2012-2013, 2013-2014, 2014-2015, 2015-2016, 2017-2018, 2018-2019, 2019-2020.
Easter 2019 Talks
Knotted Surfaces in 4-Manifolds
Oliver Singh
Wednesday 29 May, 2019, at 14:00, in CM105
Abstract: Topologists like to study smooth embeddings of one manifold into another up to ambient isotopy. Studying embeddings of S^1 into S^3 (or R^3) is what is referred to as knot theory. In general, the phenomenon of `knotting’ occurs when the codimension is 2. High dimensional embeddings are most often understood using surgery, leaving perhaps the most mysterious problems being embeddings of 1-dimensional manifolds (circles) in the 3-dimensional manifold, and embeddings of 2-dimensional manifolds (surfaces) in 4-dimensional manifolds. I will discuss the latter case of knotted surfaces, talk about some recent results in this area, and explain my own result concerning notions of distance between knotted surfaces.
Epiphany 2019 Talks
Finite-dimensional distributions of the height of a renewal model
Clare Wallace
Wednesday 20 March, 2019, at 13:00, in CM105
Abstract:
Suppose we have a collection of blocks with (integer) heights and widths, and we use a random selection of them to build a stick whose total width is $n$.
Working from left to right, we track the cumulative total height at the endpoints of each block. We can linearly interpolate between these endpoints to create a piecewise linear height function for the whole stick.
Under a few assumptions about the distributions of heights and widths of the blocks in our collection, we can write a central limit theorem for the height function at any $k$ points along its width. In particular, we can (almost) prove that the height function, properly rescaled, converges to the trajectories of the Brownian motion.
What's an anagram of "Banach-Tarski"? Banach-Tarski Banach-Tarski
Phil Kamtue
Wednesday 13 March, 2019, at 15:00, in CM105
Abstract: In this talk, I will present two paradoxes involving Axiom of choice. As a warmup, we will go through a few variations of the Prisoners and hats problems, and then discuss the Prisoners and hats paradox. After that, we will switch to the Banach-Tarski paradox, which simply says that a sphere can be partitioned and reassembled into two spheres identical to the original one. We will see a sketch of a proof for this "doubling" process.
Michaelmas 2018 Talks
Organised by Daniel Ballesteros and Rob Little. If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email either of us.
From a PDE to the classical Isoperimetric Inequality
Daniel Ballesteros-Chavez
Wednesday 12 December 2018, at 14:00, in CM201
Abstract: From Vergilius' Aeneid, we find in the story of Queen Dido one of the most famous and ancient mathematical problems: among all curves of given length, how do we find the one which encloses the maximal area? This is the so-called Isoperimetric Inequality Problem, and the usual proofs use geometric measure theory arguments. In 2000 X. Cabré proved the classical isoperimetric inequality in all R^n with a new method, based on Alexandrov's idea of moving planes and using a solution to a Neumann problem from PDEs. In this talk I will present Cabré's proof using only elementary ideas from calculus and partial differential equations.
The Uniform Infinite Planar Triangulation
Clare Wallace
Wednesday 5 December 2018, at 14:00, in CM301
Abstract: The study of planar triangulations has appeared in many different fields, from the 4-colour theorem to quantum gravity. We start with the properties of the uniform distribution on triangulations with n vertices, and ask the inevitable question of what happens when we let n go to infinity. I can also tell you there'll be spheres, planes, hyperbolic planes and even some measures in there.........
Geometric Modular Forms and Picard Modular Surfaces
Rob J. Little
Wednesday 28 November 2018, at 14:00, in CM221 (last minute notice: CM105)
Abstract: The link between the geometry of locally symmetric manifolds and modular forms has been a major theme of arithmetic geometry of the last 50 years. In particular, work of Shimura, Kudla, Zagier, Hirzebruch, Millson, Cogdell and more has been to understand "geometric" maps between spaces of modular forms. In this talk I shall give a small overview of the the development of these ideas, as well as attempting to explain my work on the extension of Kudla-Millson theory in the case of Picard modular surfaces. Indeed, in this case, we wish to use topological and number-theoretical techniques to find modular forms which are also cohomology (or homology) classes on these Picard modular surfaces, and hence to extract arithmetic results on denominators of Eisenstein cohomology.
Let's tile 2x1 dominoes!
Phil Kamtue
Wednesday 14 November 2018, at 14:00, in CM221
Abstract: Can we tile some certain domain (in the Euclidean plane) with the 2x1 dominoes, such that they fully cover the domain, and do not overlap each other? We study one variation of the elementary question above: Given a chessboard with some cells being removed, we would like to figure out in which case we can still tile the rest of the board by 2x1 dominoes. In fact, we show that if the number of removed cells is "small enough" we can always do so. We also generalize this result to a 3 dimensional case, and this problem is somewhat related to Isoperimetric inequality! In this talk, I promise the beginner's level of difficulty.
TQFTs, Skein Modules and Knots in 3-Manifolds
Oliver S. Singh
Wednesday 7 November 2018, at 14:00, in CM221
Abstract: The Jones polynomial can be defined in several ways including as an evaluation of a Topological Quantum Field Theory (or TQFT) and using the Kauffman skein relation. Either generalises to give an invariant for knots in a three manifold. The latter generalisation has a simple description as an element of a ‘skein module’. I will introduce all of these concepts and, motivated by this example, tell you about TQFTs and skein modules. I will also talk about categorification of these objects and talk about applications to 3 and 4 dimensional topology.
Spooky Scary Triangles
John E. Blackman
Wednesday 31 October 2018, at 16:00, in CM221
Abstract: In this talk, we will discuss how one can interpret multiplication of a continued fraction by n some integer as a map between the Farey complex and the 1/n-scaled Farey complex. In turn, this allows us to interpret the integer multiplication of continued fractions as a replacement of one triangulation on an orbifold with another triangulation. We will then discuss how closed curves on these orbifolds directly correspond to periodic continued fractions, and how this correspondence allows us to deduce information about the divisibility of convergent denominators for x, as well as the growth rate of partial quotients of xn^k for x eventually periodic. This work is motivated by a reformulation of the p-adic Littlewood Conjecture, which roughly states that the partial quotients of xp^k become arbitrarily large as k tends to infinity (for p a fixed prime).