# Postgraduate Seminar

Organiser: Stephen Cantrell

#### Term 1 2017-18 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 1: Wednesday 4th October

**Alex Wendland **- Generalising Cayley Graphs

Cayley graphs form one of the most well understood families of vertex transitive graphs, in this talk we talk about one possible generalisation to enlarge this family. The talk will consist of a soft introduction to Cayley and vertex transitive graphs in general, then will discuss some easily stated problems in this field to motivate why generalising this definition is a natural construction. The talk will include copious hand waving and trippy graph drawing.

Week 2: Wednesday 11th October

**Ellie Archer - **Random Walks on Fractals

The aim of this talk is to introduce random walks on fractals via some key examples. It is not a priori clear how to even define a random walk on a fractal so we start by discussing random walks on finite graphs. We will then introduce electrical network theory, which can be used to define random walks on fractals by essentially taking scaling limits of random walks on sets which approximate the fractal. We will illustrate this with the example of the Sierpinski Triangle.

We will then move on to random fractals, focusing mainly on the case of random trees and looptrees, and explain how the theory developed for deterministic fractals can be applied to the random case. Time permitting, we will conclude by discussing some properties of random walks on the Continuum Random Tree and on looptrees.

Week 3: Wednesday 18th October

**Philip Herbert - **Particles in Biomembranes

The shape of a membrane is important in many biological mechanisms, such as trafficking or signal detection. One wishes to study how constraints affect the shape. This talk aims to introduce the problem of a membrane with embedded proteins and will discuss some of the tools being used to model this problem supposing that any deformations are small.

Week 4: Wednesday 25th October

**Trish Gunaratnam - **Transport of Gaussian measures under Hamiltonian PDE dynamics

Transport of Gaussian measures under Hamiltonian PDE dynamics

In this talk, we look at the transport properties of a Gaussian measure that is naturally associated to linear wave dynamics. In particular, we are interested in a recent result of Oh and Tzvetkov about the quasi-invariance of this measure under the dynamics of the cubic nonlinear Klein-Gordon equation on the two-dimensional torus.

We start gently and explore the transport properties of this measure under linear dynamics, leading up to the Cameron-Martin theorem and also to invariance under the wave equation. We then turn our attention to the dynamics of the nonlinear Klein-Gordon equation. The focus will be on understanding the result, its limitations, corollaries and associated results. Time permitting, we will look at some ideas behind the proof.

Week 5: Wednesday 1st November

**Augustin Moinat - **Coming down from infinity in $\Phi^4_d$.

We present a new method for obtaining estimates in stochastic differential equations, through the example of the $\Phi^4_d$ equation:

\[(\partial_t-\Delta)u=-u^3+\xi\]

where we prove bounds independent of initial conditions. The method involves looking at regularity through bounds on mollified objects.

Week 6: Wednesday 8th November

**Natalia Jurga - **Dimension of Bernoulli measures for non-linear countable Markov maps

It is well known that the Gauss map $G: [0,1) \to [0,1)$

$$G(x)= \frac{1}{x} \mod 1$$

has an absolutely continuous invariant probability measure $\mu_G$ given by

$$\mu_G(A)= \frac{1}{\log 2} \int_A \frac{1}{1+x} dx$$

Kifer, Peres and Weiss showed that there exists a 'dimension gap' between the supremum of the Hausdorff dimensions of Bernoulli measures $\mu_{\mathbf{p}}$ for the Gauss map and the dimension of the measure of maximal dimension (which in this case is $\mu_G$ with dimension 1). In particular they showed that

$$\sup_{\mathbf{p}} \dim_H \mu_{\mathbf{p}} < 1- 10^{-7}$$

They also proved analogous results for maps $T$ arising from $f$-expansions with a corresponding absolutely continuous measure $\mu_T$, under the condition that the digits of the $f$-expansion were dependent with respect to $\mu_T$.

However, sometimes the absolutely continuous measure is not known or the above condition is difficult to verify. Instead, we consider the underlying geometric cause of the dimension gap; in particular we show that under an explicit non-linearity condition on the map $T$ we obtain a dimension gap.

Week 7: Wednesday 15th Novermber

**Livia Campo - **Walking on a zig-zag: a journey on Sarkisov links

Wanna go from an algebraic variety to another? Without getting lost in the path, if it's possible... you know, no-one likes to be trapped in front of a canyon...

But if you listen to Sarkisov's advices (and you follow the rays!) you'll find your way! However you have to take into account that your perception of the world might be flipped...

If this adventure hasn't teased you enough, and you'd like to have some more technical information, than you might want to know that Sarkisov links are useful tools to construct maps between Fano 3-folds looking at the ambient space where they sit inside. Since Fano 3-folds are still far away from being completely classified, the big aim is to connect known ones to undiscovered ones via birational maps (for example Sarkisov links).

Week 8: Open Day Talks - Wednesday 22nd November

Talk 1 - **Bartlomiej Matejczyk - **Macroscopic models for ion transport in nanoscale pores

'You only see what your eyes can see' Do you? What if we can use mathematical reasoning to see objects and shape that are too small to see even with a microscope?

During the seminar, we will discuss particles flow through nanoscale structures. We will present different approaches to modelling of the electrochemically driven flow and its application in biology and physics. We show how to use the models to recover a shape and physical properties of objects that are too small to observe even using the best available microscopes.

The talk covers an introduction to the Poisson-Nernst-Planck system of differential equation and its applicability to the shape identification problems connected to Ionic transport through confined geometries.

Talk 2 - **Mattia Sanna - **Siteswap: the math behind Juggling

The oldest record of juggling dates back to almost four thousand years ago in a tomb of an unknown Egyptian prince. However a big question among the juggler was does exists a method to learn and develop new tricks in an easy and fast way? The answer came in 1981: Siteswap. Siteswap is a mathematical notation system which can be used to notate juggling tricks & patterns. It has often been described as, "Sheet music for jugglers". As well as being a very useful notation system used for descriptive purposes it is also a very elegant mathematical model of a juggling pattern. A model which can be analysed & manipulated in many ways to create new patterns.

Week 9: Wednesday 29th November

**Yiwen Chen - **Morse theory for Hamiltonian functions and application to convexity theorems

Hamiltonian group actions on symplectic manifolds have their origin from Hamiltonian mechanics: the phase space of a mechanical system is a symplectic manifold and the groups represent the symmetries of the system. A Hamiltonian action is by definition an action with an associated function whose critical points correspond to the fixed points of the action. Thus it becomes natural to study the action with Morse theory, which reveals the topological structure of a manifold using the information of differentiable functions on it.

During this talk we will review the classical application of Morse theory to Hamiltonian systems and see how it provides a unified proof for some seemingly unrelated convexity theorems. As yet another application, we will talk about a combinatorial description of a certain kind of Hamiltonian actions if time allows.

Week 10: Wednesday 6th December

**Wojciech Ożański -** An invitation to convex integration

The method of convex integration, which goes back to the celebrated work of Nash (1954 & 1956) on the isometric embedding problem, is a modern method that can be used to yield existence of irregular solutions to various problems in the area of partial differential equations and elsewhere. It has recently been applied in the context of the Euler equations by De Lellis & Székelyhidi Jr. (2009, 2012), and, after a number of significant contributions to the theory, led to the resolution of the Onsager conjecture by Isett (2016) and Buckmaster, De Lellis, Székelyhidi Jr. & Vicol (2017). Moreover, very recently this method was used by Buckmaster & Vicol (2017) in the context of the 3D Navier–Stokes equations to show nonuniqueness of weak solutions, a result very closely related to one of the six outstanding Millennium Problems.

In the talk we will discuss a counter-intuitive theorem regarding divergence-free vector fields whose proof demonstrates the main steps of the method while being elementary. Furthermore, we will explain how the method can be adapted to the 3D Euler equations to yield existence of infinitely many solutions with any prescribed energy profile.

### Term 2 2017-18 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 1: Wedneday 10th January

**Pavlos Tsatsoulis -** Exponential loss of memory for the dynamic $\Phi^4_2$ with small noise

We consider the dynamic $\Phi^4_2$ model formally given by the SPDE

\begin{equation*}

(\partial_t - \Delta) \phi_\varepsilon = - \phi_\varepsilon^3 + \phi_\varepsilon +\sqrt{2\varepsilon} \xi

\end{equation*}

where $0<\varepsilon<<1$ and $\xi$ is a space-time white noise in $2+1$ dimensions. Our goal is to analyze the long time behavior of solutions started from suitable initial conditions based on Large Deviation Theory and discuss how one can derive a strong coupling argument. The basic obstacle is that classical solution theory for SPDEs is not applicable here since renormalization is required to compensate the divergences of the non-linear term. This destroys the dissipativity of the system and a deeper analysis on the level of the so-called perturbative term is required.

Along the way we introduce the necessary terminology and tools to deal with the problem, including a solution theory for the equation and the Large Deviation Principles for the law of the solutions. (ongoing work with H. Weber)

Week 2: Wednesday 17th January

**Simon Etter - **Material Science and Localisation of Matrix Functions

If you want to build a bridge, an airplane or a car, you have to know and possibly tune the properties of the materials you are working with,

and for both understanding as well as designing materials it can be very helpful to be able to simulate the atoms and electrons constituting the

material in question. *Density functional theory* is the most widely used model to perform such simulations, and its key mathematical step is to

compute trace$(f(H))$ where $H$ is a large but sparse matrix and $f$ is an analytic function in a neighbourhood of the spectrum of $H$. It turns out

that while $f(H)$ is a dense matrix, most of its entries are negligibly small. This allows to significantly reduce the computational costs of

evaluating trace$(f(H))$ and has important physical implications, e.g. it explains why perturbing an atomic configuration at one location does not

affect the material properties at a location far away from the perturbation. Mathematically, this *localisation* of $f(H)$ is qualitatively

understood, but a precise quantification of the decay rates is still outstanding, and designing fast algorithms to exploit this property is a

topic of ongoing research.

After providing a brief introduction to quantum-mechanics-based materials simulation, this talk will review a known localisation result and

present a new algorithm which translates this result into computational savings. I will then point out some gaps in the theory and discuss ideas

on how to fill them.

Week 3: Wednesday 24th January

**Dylan Madden - **Cyclic vs. Mixed Homology

In modern mathematics, one extremely common notion is that of a chain complex, which is a sequence of modules $A_0, A_1, A_2, \ldots$ connected by homomorphisms $d_n: A_n \rightarrow A_{n-1}$ such that all compositions are $0.$ These maps are known as differentials. By reversing the arrows, one obtains the dual notion of a cochain complex. Much less commonly considered is the notion of a mixed complex: these are sequences of modules which are simultaneously chain and cochain complexes.

Chain complexes and cochain complexes have, respectively, homology and cohomology; these are sequences of modules obtained by quotienting the kernels of the ''outgoing'' maps by the images of the ''incoming'' maps. Clearly, mixed complexes have both homology and cohomology, but they also have ''mixed homology'', which takes both the upward and downward differentials into account. Moreover, the self-dual structure of mixed complexes allows one to ''perturb'' the complex using polynomials. In this talk, we set up and state a theorem which, under certain nice conditions, describes what this perturbation process does to the mixed homology. No prior knowledge of mixed complexes will be assumed. This is based on joint work with Ulrich Kraehmer.

Week 4: Wednesday 31st January

**Matthew Dickson - **Brownian Bridges and Bosons

Bosons are weird. They are quantum particles that - at low temperatures - occupy the lowest single-particle quantum state at the macroscopic level. This phase transition happens even for the non-interacting Bose gas, demonstrating that the cause of this transition is not energetic in nature. Instead ''Bose-Einstein condensation" is a combinatorial phenomenon arising because of the symmetry of the bosonic wave-function, and it is this symmetry that allows an equivalence at the level of the partition function to be made with a point process of Brownian bridges.

This talk shall first clarify how this equivalence can be drawn, before exploring how it allows us to answer questions about the Bose gas. Many of these require a result about Brownian bridges that is simple to state and yet is not found in standard textbooks. We will see how Doob solved our problem for us in 1949.

Week 5: Open Day Talks - Wednesday 7th February

Talk 1 - **Paul Colognese -** An introduction to rational billiards and translation surfaces

Consider a game of billiards/pool/snooker. If we assume that the ball is a moving point and that there is zero friction, we can consider the long term dynamics of a trajectory. One way of studying this problem is by unfolding the table to get a closed surface known as a translation surface. In this talk, I'll provide a very brief introduction to the subject, focusing on the basic geometry as well hopefully providing some insight into how this perspective can be fruitful when solving problems about billiards.

Talk 2 - **Quirin Vogel -** Why you should do a PhD in Statistical Mechanics

Statistical Mechanics is without doubt the most sexy and fascinating area of modern mathematics. In this talk, I aim to introduce the audience to some of the question statistical mechanics seeks to answer. I will explain the prominence of the Ising model and talk about (Gaussian) Free field in various dimensions. In the passing we will get to know concepts such as phase transitions, thermodynamics limits and random walk representations.

Week 6: Wednesday 14th February

**George Vasdeskis - **Non-Reversible Markov Chain Monte Carlo

The aim of this talk is to give an introduction to Markov Chain Monte Carlo (MCMC) algorithms. We will discuss how the Bayesian methods in Statistics create the need to simulate from a given probability measure, which can be achieved by MCMC. We will introduce the Metropolis-Hastings algorithm and explain why its time-reversibility can make the algorithm inefficient. We will then discuss a way to break this reversibility condition and make the algorithm work faster. This will lead to a promising, new stochastic process which is called Zig-Zag and seems to be working well in the context of big data. Time permitting, we will discuss some recent results on the convergence properties of Zig-Zag.

Week 7: Wedneday 21st February

**Luke Hatcher - **Lipid Rafts on Biomembranes

Abstract: Lipid rafts are small microdomains on biomembranes which are believed to play an important role in many cellular processes. A suitable phase field model will be derived, and existence and uniqueness results shown for the corresponding surface PDEs.