# Talks

Below I give a summary of the talks I have given recently. This will be updated on a regular basis as required

- MasDoc Seminar Talk
- CCA Conference Talk
- Post-Graduate Seminar Talk
- Applied PDEs Seminar Talk
- CCA Conference Talk
- MasDoc Retreat Talk

A summery of other MasDoc talks can be found on this page.

On the 17th October 2012 I gave a talk for the MasDoc semainer based on the work I did for my MSc summer research project (see here). The talk gave a brief introduction to density functional theory and some of the techniques associated with it and considered the ideal gas and a specific case of the hard-core gas as motivating examples. The abstract for this talk is given below

"*This talk will cover some of the material used for my MSc dissertation, supervised by Christoph Ortner. The topic of the dissertation was density functional theory, specifically a classical consideration of the one-dimensional hard-core gas in the canonical ensemble.*

*The talk begins by introducing some basic techniques required for density functional theory, after which we develop the ideal gas as a motivating example. We conclude the talk with a brief look at one of the methods used to consider the one-dimensional hard-core gas.*

*Density functional theory is a topic of interest in mathematics, physics and in chemistry. This talk reviews material required for a specific problem in the hope of demonstrating some of the techniques and problems encountered in density functional theory. This talk should hopefully be accessible to anyone with a reasonable degree of mathematical literacy*."

The slides for this talk are given here.

On the 26th March 2013 I gave a talk as part of the joint CCA-MasDoc conference. This talk is based on my first year PhD work and bridges the gap between DFT used in my MSc and the PFC model used in my PhD. The abstract for this talk is given below

"*This talk will review the paper [Berry et al, 2007] which uses density functional theory (DFT) to provide a justication for the simpler and more numerically tractable theory of phase eld crystals (PFC). The talk begins by introducing some basic techniques required for DFT, after which we develop the ideal gas as a motivating example and a starting point for PFC.*

* We then proceed to derive the PFC functional by using a functional Taylor expansion and a gradient expansion. We conclude by demonstrating some initial results. DFT is a topic of interest in mathematics, physics and in chemistry whereas it is hoped that PFC will be more easily treated numerically . This talk reviews the link between these two interesting areas, it should hopefully be accessible to anyone with a reasonable degree of mathematical literacy."*

The slides for this talk are given here.

On the 22nd May 2013 I gave a talk for the Warwick post-graduate semainar. This talk is based on my first year PhD work and again summaries the bridge between DFT used in my MSc and PFC model used in my PhD and summarises some of the results I have obtained using PFC. The abstract for this talk is given below

*"In this talk we will consider the relatively new area of Phase Field Crystals (PFC) and its applications. We begin by motivating the PFC model by deriving it from the older and more extensively studied area of density functional theory (DFT). This can be considered as an exercise in reducing the number of independent variables from a very large number to two and thus PFC is much simpler and hopefully more numerically tractable than DFT. This follows the method of [Elder et al, Physical Review B, 75, (2007).]*

*Having derived PFC we present a new method for minimising the functional of PFC in a two dimensional rectangular domain with periodic boundary*

*conditions. This technique relies on translating our equation into Fourier space and using the Fast Fourier Transform.*

*We conclude by presenting some of the results of our numerical simulations. We concentrate on produce a unit hexagonal cell and demonstrate how this*

*can be used to simulate a lattice and various defects. This talk is designed to be accessible to anyone with a reasonable degree of mathematical literacy."*

The slides for this talk are given here.

On the 4th June 2013 I gave a talk for the Warwick Applied Pdes semainar. This talk is based on my first year PhD work and summaries two different numerical methods for simulating the PFC model and reviews the results of these two methods . The abstract for this talk is given below:

*"In this talk we consider the Phase Field Crystal (PFC) model and several ways of simulating this model. PFC is a recent model for crystallisation proposed in [Elder, Grant, PRE,2004], this model consists of a relatively simple functional which when minimised exhibits a phase transition dependent on two parameters and in one case the minimiser can be chosen to be a periodic lattice.*

*We consider two numerical approaches for the minimisation of our PFC functional which lead to two partial differential equations. The first is a review of the method of [Elsey, Wirth, M2AN, 2013] which gives a numerical approach to simulating the PFC equation. The second is a new approach which numerically solves a modified gradient flow of the PFC functional. Both methods rely on solving the equation in Fourier space and use periodic domains so that the fast Fourier transform can be used.*

*Finally we compare the two methods and show that they both obtain the expected minimisation lattice. We also give some possible applications for PFC emphaising the applications in the physics of crystals. This talk should be accessible to anyone with a reasonable degree of mathematical literacy."*

On the 18th March 2014 I gave a talk as part of the joint CCA-MasDoc conference. This talk is based on my second year PhD work and covers a variety of methods for minimising the PFC functional investigated during my PhD. The abstract for this talk is given below

*In this talk we review a series of numerical methods associated with the Phase Field Crystal (PFC) model. This model was ﬁrst proposed in [1] as a simple way of modelling crystal growth. The advantages of this model are that it preserves the elastic properties of crystals and exhibits a phase transition in two dimensions between a constant state (taken to represent a liquid) and a state with hexagonal symmetry (taken to represent a crystal). We begin by introducing the model through its deﬁning functional and giving its ﬁrst and second variations. This allows us to consider gradient ﬂow as a method of minimising the functional. Two numerical methods of minimising the functional through a convex-concave splitting are reviewed ([2] and [3]). We then introduce a method with similarities to Newton’s method in that the Hessian inﬂuences our choice of gradient ﬂow. We suggest both a static and variable metric version of this method. We brieﬂy summarise the convergence and stability results for this method. We conclude with a brief summary of some numerical results and a review of our current work and future aims. This talk should be accessible to anyone with a reasonable degree of mathematical literacy.*

**References**

*[1] K.R. Elder, M. Katakowski, M. Haataja, M. Grant. Modelling Elasticity in Crystal Growth, Physical Review Letters, Volume 88, Issue 24, 245701, (2002). *

*[2] S.M. Wise, C. Wang, J.S. Lowengrub An Energy-Stable and Convergent Finite Diﬀerence Scheme for the Phase Field Crys- tal Equations, SIAM Journal of Numerical Analysis, Volume 47, Issue 3, pages 2269-2288, (2009). *

*[3] M. Elsey, B. Wirth, A Simple and Eﬃcient Schemer for Phase Field Crystal Simulation, ESIAM: Mathematical Modelling and Numerical Analysis, Volume 47, Issue 5 , pages 1413-1432, (2013).*

On the 19th May 2014 I gave a talk as part of the Masdoc Retreat. This talk is based on my second year PhD work and covers a numerical issue discovered in methods for minimising the PFC functional. There is no abstract for this talk.