# Gandalf seminar 2017-2018

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 **G**eometry **AND** **AL**gebra **F**orum, name due to Herbert Gangl. Occasionally it becomes the Radagast seminar, **R**esearch **A**nd **D**evelopment in **A**lgebra, **G**eometry **A**nd **S**ometimes **T**opology, 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.

## 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 TeuliĆ© 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.