# 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.

## Easter 2019 Talks

Organised by John Blackman and Clare Wallace. If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email either of us.

### Knotted Surfaces in 4-Manifolds

Oliver Singh

Wednesday 29 May, 2019, at 14:00, room TBC

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.

### 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).

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## Easter 2018 Talks

Organised by Daniel Ballesteros and Robert Little . If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email any of us.

### Trisections of 4-manifolds and Group Trisections

Oliver S. Singh

Wednesday 6 June 2018, at 16:00, in CM221

Abstract: Any Closed 3-manifold admits a Heegaard splitting, a decomposition of the manifold into two handlebodies. These splittings can be described by systems of curves on surfaces known as Heegaard diagrams, which give a combinatorial, diagrammatic characterization of 3-manifolds. In 2012 Gay and Kirby showed that any 4-manifold has an analogous decomposition, referred to as a trisection, which gives analogous diagrams for 4-manifolds. I will go over some theory about Heegaard splittings, define a trisection and explain the correspondence with group trisections. This allows for a purely group theoretic rephrasing of many 4-manifold questions, such as the smooth Poincaré conjecture. There will be lots of pictures and I wont assume familiarity with 3 or 4-manifold topology.

## Epiphany 2018 Talks

Organised by Daniel Ballesteros and Robert Little . If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email any of us.

### An algorithm to test for potential counterexamples to the p-adic Littlewood conjecture

Matthew J. Northey

Wednesday 7 February 2018, at 16:00, in CM221

Abstract: One of the most important questions in Diophantine Approximation is whether the p-adic Littlewood conjecture (PLC) is true. The conjecture has already been shown to have close relationship to the continued fraction expansion of x: for example any x with partial quotients of unbounded height is known to be a solution to PLC. PLC can be reformulated in terms of the set of continued fractions {p^n x} (n natural), and so it becomes natural to try to understand how continued fractions transform under prime multiplication. During this talk, I will focus on the p=3 case as an example to illustrate the key ideas that lead to a multiplication algorithm for general p, and how we can implement it to test for potential counterexamples to PLC. Time permitting, I will also talk about some potential theoretical applications that this algorithm has.

### Class Field Theory: a re-imagining of fundamental objects in number theory

Salvatore M. Mercuri

Wednesday 24 January 2018, at 16:00, in CM221

Abstract: Class field theory takes classical objects found in algebraic number theory and transforms them into topological objects. In doing so, some long-established and fundamental results follow almost immediately from some basic topological properties. For example, the ideal class group Cl(F) of a number field F has been known to be finite since at least the early 1800s, and in this talk we'll give an alternative proof of this using class field theory and the adele ring.

## Michaelmas 2017 Talks

Organised by Daniel Ballesteros and Robert Little . If you have any suggestions for speakers, or want to volunteer to give a talk yourself, please feel free to email any of us.

### Optimal Transport problem and Ollivier-Ricci curvature notion on graphs.

Supanat Kamute (Phil)

Wednesday 13 December 2017, at 16:00, in CM221

Abstract: In 2009, Yann Ollivier introduced the definition of ''coarse'' Ricci curvature in term of how much small balls are closer (in the sense of Optimal Transport) than their centers are. This definition extends to discrete metric spaces (i.e. graphs). In this talk, we will first familiarize the setup of Optimal transport problem on graphs. Then we define Ollivier-Ricci curvature notion as well as give several examples of graphs with different signs of curvatures. For someone who knows a little bit of Riemannian geometry, we will quickly visit the equivalent versions of Bonnet-Myers and Lichnerowicz theorems. For everyone else, we've got a very cool interactive and easy-to-use Graph Curvature Calculator, thanks to David Cushing. The link is http://teggers.eu/graph/ and I will explain how this application works! If I have a bit of time left I will mention about my current work on this topic as well.

### Suborbital Graphs of Fuchsian Group \mathcal{H}(\sqrt{m})

Wanchalerm Promduang

Wednesday 29 November 2017, at 16:00, in CM221

Abstract: Suborbital graphs are graphs constructed from geodesics in spaces. I'm going to illustrate the work on suborbital graphs from Hecke groups by Keskin and extend it to the groups \mathcal{H}(\sqrt{m}).

### The Geometry of Liu Hui

William Rushworth

Wednesday 15 November 2017, at 16:00, in CM221

Abstract: Liu Hui was a Chinese mathematician, who lived during the third century CE. We'll discuss some of the things he proved and how he proved them, including his estimate of the value of pi, and the calculation of the volumes of solid objects.

### Waring's Problem - An Introduction to the Circle Method

Matthew J. Northey

Wednesday 8 November 2017, at 16:00, in CM221

Abstract: In 1770, Lagrange proved that every positive integer, N, can be written as the sum of four squares. Shortly after this, Waring asked whether it was possible to represent any N as the sum of finitely many k-th powers, where the number of terms, g(k), depends only on the choice of k. Little progress was made towards a solution for over 100 years, until it was eventually proven by Hausdorff in 1909. However, Hausdorff's proof only showed that g(k) was finite for every k, and so a natural question arose as to whether it was possible to get some bound on the size of g(k). Hardy and Littlewood developed a powerful technique known as the circle method, which - for sufficiently large N - gives a "good" upper bound on the size of g(k). The circle method has since been generalised to many different, and is currently a very active area of research in analytic number theory. In this talk, we will use Waring's problem as motivation to introduce some of the key ideas that underpin the circle method, and if time, use these ideas to give an overview as to how one can show that g(k)<2^k+1.

### Deligne-Mostow lattices and cone metrics on the sphere

Irene Pasquinelli

Wednesday 11 October 2017, 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 can be used to build fundamental polyhedra for Deligne-Mostow lattices with 2- and 3-fold symmetry.

### A Geometric Approach to the p-adic Littlewood Conjecture.

John E. Blackman

Wednesday 4 October 2017, at 16:00, in CM221

Abstract: The Littlewood Conjecture is an open problem in Diophantine approximation dating back to the 1930's. In 2004 de Mathan and Teulié proposed a related conjecture called the mixed Littlewood conjecture. A specific case of the mixed Littlewood conjecture is the p-adic Littlewood conjecture (p-LC), which has been tackled by several mathematicians over the last 13 years. Notably: Kleinbock, Einsiedler, Bugeaud and Badziahin. The majority of the research into the problem has been based on analytic number theory. An interesting question (and potential solution) which arises from p-LC is how to multiply continued fractions by prime numbers. Following the work of mathematicians, such as Artin and Series, continued fractions can be viewed as geodesics intersecting the Farey triangulation in the upper half plane. I am currently using this geometric interpretation of continued fractions to tackle the p-adic Littlewood conjecture. In particular, I have been investigating the prime multiplication of continued fractions using geometric methods. In this colloquium, I will introduce the basics of these topics and present some interesting findings of my own research.

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