# Gandalf seminar 2019-2020

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. Whenever Clare's involved, Gandalf becomes Saruman, for **S**tudents **A**cronymised **R**esearch **U**sing **M**easures **A**nd **N**umbers

Gandalf seminar archive: 2010-2011, 2011-2012, 2012-2013, 2013-2014, 2014-2015, 2015-2016, 2017-2018, 2018-2019, 2019-2020.

## Epiphany 2020 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.

### An unusual characterisation of doubly ruled surfaces

Guillermo Zara-Cobos

Wednesday 5 February, 2020 at 15:00, in CM225a

Abstract: Doubly ruled surfaces are surfaces which can be covered with straight lines in two different ways. Examples include the one-sheeted hyperboloid and the hyperbolic paraboloid (and, surprisingly enough, no more!). In this talk I am going to focus on the following result: "Given 3 skew lines in space, the set of lines that intersect those 3 lines will always be one of the two sets of lines of a doubly ruled surface." This fascinating and mysterious result is stated and proven in Hilbert and Cohn-Vossen's book Geometry and the Imagination. I will talk about a different approach one could take about proving this, this time using some geometric structures defined in the space of lines in Euclidean 3D space. No prerequisites, come along I think you will enjoy it!

### Unramification of number fields

Dan Clark

Wednesday 22 January, 2020, at 15:00, in CM107

Abstract: The aim of this talk will be to give a short overview of the topic of my masters dissertation. The subject of algebraic number theory is in some sense, the study of Extensions of Algebraic Number Fields and a key idea in this study is the idea of 'ramification'. As such, I will give the definition of ramification, a little intuition on why it is in some sense 'bad' and, if there is time, overview some of the known results about the study of the 'Maximal unramified Extension' of a given number field K.

## Michaelmas 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.

### Mystery Gandalf

Daniel Ballesteros-Chavez

Wednesday 30 October, 2019, at 14:00, in CM105

Abstract: It's a mystery!

### Limit Theorems for Trajectories of a Renewal Random Walk

Clare Wallace

Wednesday 23 October, 2019, at 14:00, in CM105

Abstract: In this talk, I present a discrete-time renewal process in which each renewal event is associated with an integer mark. This model may also be considered as a random walk on the 2-dimensional integer lattice, whose increments have strictly positive horizontal component. Under the assumption that the increment distribution has exponential moments, we prove Central and Local Central Limit Theorems for the (intermediate) heights of the trajectories of this walk. We also establish the weak convergence of the distribution of the trajectories to the distribution of the Brownian motion. We next place a large deviations condition on the height of the trajectory at its endpoint; under this condition, similar Central and Local Central Limit Theorems for the (intermediate) heights exist. We further establish weak convergence of the distributions of the conditional trajectories to the law of the Brownian bridge.

### A geometric algorithm for multiplying continued fractions with applications to the p-adic Littlewood conjecture

John Blackman

Wednesday 25 September, 2019, at 14:00, in CM107

Abstract: In recent years, the problem of deducing the continued fraction expansion of px from the continued fraction expansion of x has become a topic of increased interest. This is due in part to a reformulation of the p-adic Littlewood conjecture – an open problem in Diophantine approximation – which pertains to the behaviour of the partial quotients of xp^k as k tends to infinity (for p a fixed prime). In this talk, we will discuss how one can interpret multiplication of a continued fraction by some integer n as a map between the Farey complex and the 1/n-scaled Farey complex. In turn, this allows us to interpret integer multiplication of continued fractions as a replacement of one triangulation on an orbifold with another triangulation. Using this geometric setting, we will then construct a reformulation of the p-adic Littlewood Conjecture and discuss how one can improve on known bounds for this problem.