Postgraduate Seminars Autumn 2019

These seminars (unless otherwise stated) will take place on Thursdays at 12pm in Drayton Jevon's Lecture Theatre B20 on an almost weekly basis.

See the map for further details. Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

03 October 2019

Speaker: Angela Wu
Supervisor: Dr J Evans

Title: Lagrangian Knot Concordance

Knots, being one dimensional, are the boundaries of various two dimensional surfaces. If two knots bound a cylinder, we say they are concordant. If that cylinder has the added benefit of being a Lagrangian submanifold of a symplectic manifold, then we say that we have a Lagrangian Knot Concordance between two Legendrian knots. Determining which knots are (Lagrangian) concordant to which others turns out to be a highly non trivial problem. This talk aims to precisely define these ideas and to explain a bit about what is currently known about Lagrangian knot concordance.

10 October 2019 in Maths room 706

Speaker: Sean Jamshidi
Supervisor: Prof ER Johnson

Title: Long, nonlinear waves on coastal fronts

An ocean front is a region where there is a sharp lateral change in some physical property such as density. The simplest mathematical model of an ocean jet such as the Gulf Stream is as a step-change in potential vorticity (PV -- a fundamental quantity in geophysical fluid dynamics), and so there is much interest in the motion of PV fronts. In this talk I will present a mathematical model for long waves on a PV front near a coast. The resulting equation is similar to the KdV, and analytic results relating to wave propagation can be obtained using the technique of `dispersive-shock fitting’ discovered by El. The talk will focus on the mathematical aspects of El’s technique, and no knowledge of fluid dynamics will be assumed.

17 October 2019

Speaker: Francesco Di Giovanni
Supervisor: Prof J Lotay

Title: Ricci flow and symmetries

A year ago I gave an introductory talk about Ricci flow focussing on the reasons why I don't like algebraic geometry :)

This year I will expand on that, trying to describe how, in some sense, Ricci flow is not only able to detect and preserve symmetries but, in fact, also able to enhance and create them. The emphasis will be on some examples of such phenomenon I have been working on for the last year.

I will draw some nice pictures to voluntarily hide the not-always-nice analysis and theory supporting the property above. However, I will dedicate the last 10/15 minutes of the talk to write some actual maths to hopefully make the talk somehow beneficial to people that have some familiarity with geometric analysis, riemannian geometry and pdes.

24 October 2019 in Maths room 706

Speaker: Bradley Doyle
Supervisor: Dr E Segal

Title: Geometry and categories

Given a geometric object one can associate to it many different invariants, many of these are categories, or are described using category theory language. Some of the well known geometric objects can themselves be described as solutions to problems from category theory (for example, projective space).

I will attempt to explain some of the connections between geometry and categories. I will start by briefly introducing and motivating categories. I will avoid technical definitions, trying to focus on the main ideas and motivation, where possible, I will try to draw some pictures as well. Finally, if there is time I will explain a problem from geometry (resolving singularities) that has a categorical interpretation.

31 October 2019

Speaker: Xiaoshu Sun
Supervisor: Prof T Betcke

Title: Numerical framework on approximating the log determinant via low-rank representation with implementation in computing Casimir energy

Keywords: log determinant, low-rank approximation, Casimir energy, Bempp Library

Since late 1940s, the Casimir effect was proposed and the advanced study on vacuum effects in quantum electrodynamics has been developed deeply and fast. The stress-tensor integration and path-integral approach are two traditional ways to compute the Casimir energy and Casimir force. In both cases, they need the information on the results of boundary element equations.

Since 2007, we start to use BEM to solve the scattering problem and use the result inside path-integral formula. The key step is to evaluate BEM matrix elements between localised basis functions and compute the log determinant of the matrix. However, since the previous study directly compute the dense $LU$ decomposition to evaluate the log determinant, which has high storage requirements and running time, we figure out a new fast direct way to compute the log determinant of the matrix corresponding to the discretized linear operator (such as electric field boundary operator).

In this talk, I will present our new fast direct algorithms on computing the log determinant of the matrix by introducing the main idea on low-rank approximation. Moreover, the numerical experiments of the new log determinant approach and the performance of implementing this new algorithm on computing the Casimir energy using the Bempp software package will also be presented.

07 November 2019 in Maths room 706

Speaker: Giordano Giambartolomei
Supervisor: Dr N Sidorova

Title: Time-dependent balls and bins model with positive feedback

Balls and bins models are classical probabilistic models where balls are added to bins at random according to a certain rule. The balls and bins model with feedback is a non-linear generalization, where the probability of a new ball choosing a bin with m balls is proportional to m^a, with a being the feedback parameter. It is known that if the feedback is positive (i.e. a>1) then the model is monopolistic: there is a finite time after which one of the bins will receive all incoming balls. A time-dependent version of this model, where s_n independent balls are added at time n instead of just one, has recently been considered by Dr. Sidorova, with whom I work, and it has been shown that if a>1 then one of the two bins gets all but a negligible number of balls. (A phase transition in the growth of s_n between monopolistic and non-monopolistic behavior has also been identified). When a=1, it has been shown that no dominance occurs, that is, each bin gets a non-negligible proportion of balls eventually.

In the talk, I will firstly introduce the audience to the two bins model, explaining the results above (along with some of the main analytic and probabilistic ingredients necessary to the proof). Then I will talk about my first PhD year's results on a further generalization of the model to any number of bins.  In doing so I will often focus on the more visual case of three bins, which allows nice representations of the states space in lower dimensions, so I will be able to draw at the board and ease and speed up everybody's understanding.

I will make sure that nothing beyond undergrad analysis, linear algebra and measure theoretic probability will be required to understand the talk.

14 November 2019

Speaker: Benjamin Aslan

Supervisor: Prof J Lotay

Title: Minimal Surface in S4

The soap film which is formed when dipping a wire frame into a soap solution is a physical realisation of a minimal surface. Solving this variational problem has become an important task in differential geometry. We will give a very gentle introduction to the subject and outline how complex analysis can be used to find minimal surfaces. In the end, we will expand on this idea and describe the twistor fibration of S4 explicitly to give many examples of minimal surfaces in S4.

21 November 2019 in Maths room 706

Speaker: Albert Wood
Supervisor: Prof F Schulze

Title: Soap Films and Geometric Measure Theory

Abstract:When Joseph Plateau played around with soapy water and wire in his 1800’s kitchen, he could not have imagined that the problem that resulted would inspire new mathematics for 200 years to come... In this talk I will show how historical developments in complex numbers, measure theory and functional analysis has enabled mathematicians to defeat one of the longest-standing problems in physical geometry - the existence of soap films - with copious pictures to keep everyone entertained!

28 November 2019

Speaker: Francisco Berkemeier
Supervisor: Prof KM Page

Title: TBC