# 2017-18

Organiser: Stephen Cantrell - S.J.Cantrell@warwick.ac.uk

#### 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

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.

Week 8: Wednesday 28th February

**George Andriopoulos - **Convergence of blanket times for sequences of random walks on critical random graphs

Consider a simple random walk on a finite, connected graph. The blanket time is defined as the first time at which all nodes have been visited at least a positive fraction of time as expected at stationarity. Ding, Lee and Peres (2011), resolving a conjecture posed by Winkler and Zuckerman (1996), showed that for any graph, the blanket and cover times are within a constant O(1) factor. To tackle this problem they first proved that the cover time of any graph is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian Free Field on the graph, rescaled by its total number of edges.

Motivated by this strong connection between the blanket and cover times we demonstrate how we can prove asymptotic distributional bounds for the blanket times of random walks on sequences of graphs that converge in a suitable Gromov-Hausdorff sense as metric measure spaces equipped with a continuous function. Finally, we establish a scaling limit for the blanket times of random walks on critical Galton-Watson trees. This example is pivotal in order to prove scaling limits of the blanket times of random walks on a few models of critical random graphs such as the Erdős-Rényi random graph and the configuration model.

Week 9: Wednesday 7th March

**Josha Box - **Finding all cubes and squares of a given difference using the modularity theorem

Just for a moment, try to relive your childhood years. Which structures would you build with $8$ cubic blocks? I bet the $2\times2\times2$ cube is one of them. Now add one extra block, and notice the blocks can be rearranged into a nice $3\times3$ square. If you had started with a $3\times3\times3$ cube instead, then adding one block does not allow you to reshape the blocks into a perfect square, since $3^3+1$ is not a square number. In this talk, I will explain why in fact the $2\times2\times2$ cube is the unique one with this property, and how this can be proved using the modularity theorem (formerly the Taniyama-Shimura conjecture), in a way similar to the proof of Fermat's Last Theorem.

Week 10: Wednesday 14th March

**Quirin Vogel - **Why *Machine Learning does work

*maybe. In this talk we will divulge further prepossessing features of statistical mechanics. We will introduce the audience to universality and scaling laws and reveal why mean field theory is rubbish. We also have to talk about renormalization and conclude that machine learning is just a crude rip off.

#### Term 3 2017-18 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 1: Wednesday 25th April

**Alex Wendland - **Facially restricted graph colourings

Arguably one of the best-known theorems from combinatorics is the four colour theorem, stating that every planar graph can be coloured using at most four colours such that no edge connects two vertices of the same colour. In this talk, I will discuss variants of this theorem in particular list colouring's and facial restriction's on the colouring. In this, I present the method of discharging in Graph Theory, used to finally prove the four colour theorem nearly 140 years after it was first stated, which has been used to prove theorems elsewhere in Mathematics.

Week 2: Wednesday 2nd May

**Ronja Kuhne - **Algebra meets Topology - from Frobenius Algebras and TQFTs to Knot Theory

In this talk, we are going to explore one of the various interplays between algebra and topology: the relation between Frobenius algebras and 2D topological quantum field theories. This talk will involve many pictures and does not assume any prior knowledge, so come along to discover that algebra is not as dry as you always thought and that topologists do not only wave their hands about coffee mugs and doughnuts being the same.

After having made peace with Frobenius algebras and TQFTs we will advance to an even more exciting topic: knot theory, and especially a knot invariant called Khovanov homology. Defining Khovanov’s knot homology theory will now be a piece of cake and we will explore its connections to the famous Jones polynomial as well as discuss applications of both of these knot invariants.

Week 3: Wednesday 9th May

**Aidan Browne - **The Willmore energy of perturbed geodesic spheres

There are many well known results relating the local and global geometry of a manifold, for example the Bonnet-Myers theorem or the Cartan-Hadamard theorem, both of which use curvature information (local) to deduce something topological (global).

In this talk we will consider how conditions on the local Hawking mass of a 3D manifold might allow for global conclusions, by analysing the Willmore energy of perturbed geodesic spheres. Specifically, we want the following "rigidity" result: If the local mass is non-positive everywhere, then (with some further standard assumptions borrowed from General Relativity) the manifold is globally Euclidean.

Week 4: Wednesday 16th May

**Maciej Buze -** Lattice Green's function in the anti-plane crack geometry

In order to prove qualitatively sharp regularity estimates for solutions of discrete equilibration problems employed to model crystal defects, one needs to first establish existence of a corresponding lattice Green's function and prove its decay properties. Following a short general introduction to the discrete setup, in this talk we will discuss how to do it in a spatially homogeneous setup, explain why it is non-trivial in a non-homogeneous setup of a crack defect and present a way of tackling this problem.

Week 5: Wednesday 23rd May

**Shannon Horrigan - **Phase Transitions in Delaunay Potts Models

To begin, we will discuss the basic goal of Statistical Mechanics - bridging the gap between the microscopic and macroscopic descriptions of large systems of interacting particles. We will introduce the central concept of Gibbs measures and their interpretation as the macroscopic 'states' of the system in question.

With the key concepts in place we will introduce a particular class of 2-dimensional continuum models, Delaunay Potts models. We will use a Random Cluster Representation to show that for a specific particle interaction we have a phase transition: the existence of multiple Gibbs measures. The Random Cluster Representation is a powerful tool which allows us to complete the proof by showing that percolation occurs in a related process.

Week 6: Wednesday 30th May

**James Plowman - **K-theory and Intersection Theory

Algebraic K-theory has its origins in Grothendieck's reformulation of the Riemann-Roch theorem in which the group $K^0(X)$ of classes of vector bundles on a scheme is introduced. It was expected that there should be higher groups $K^i(X)$ extending the Grothendieck group $K_0(X)$ to a cohomology theory. Despite the topological analogue of higher K-theory having been constructed by Atiyah relatively quickly, the definition in algebraic geometry remained elusive for a long time. In this talk we will give an incredibly brief non-complete overview of the history of intersection theory in order to tell a story leading up to $K_0$. We will then skip ahead to describe Quillen's construction of higher K-theory and time permitting make some connections back to intersection theory e.g. Bloch's formula and Gersten's conjecture.

Week 7: Wednesday 6th June

**Ifan Johnston - **Fractional heat equation and Aronson estimates

In 1967 Aronson showed that the fundamental solution $p(t, x, y)$ of the heat equation for a second order uniformly elliptic operator in divergence form $L$ \[\partial_t u(t,x) = L u(t, x),\]

can be bounded above and below by the fundamental solution of the standard heat equation:

\[c_1 t^{-d/2} \exp\left(-c_2\frac{|x-y|^2}{t}\right) \leq p(t, x, y) \leq c_3 t^{-d/2} \exp\left(-c_4\frac{|x-y|^2}{t}\right),\]

for $x, y \in \mathbb R^d, t > 0$. Probabilistically speaking, this means that the movement of diffusion processes is comparable to the movement of Brownian motion.

In this talk we will look at fractional analogues of the heat equation, for example the time-fractional (standard) heat equation

\[\partial^\beta_t u(t, x)= \Delta u(t,x)\]

where $\partial_t^\beta$ is the Caputo fractional derivative in time of order $\beta \in (0, 1)$. Probabilistically, the fundamental solution of this time-fractional heat equation is the transition densities of a nice (non-Markovian) time-changed Brownian motion. After introducing all these concepts, we will show how to obtain Aronson-type estimates for the fractional (in time and space) heat equation. Along the way we will encounter many important (and very closely related) objects from fractional calculus and probability, including: Mittag-Leffler functions, $\beta$-stable processes and pseudo-differential operators.

Week 8: Wednesday 13th June

**Matt Collins -** Phase Field Approximations of Free Boundary Problems

Free Boundary problems are a class of PDE's where one solves not just for a unknown function but also an unknown domain. Through some classical examples we introduce the phase field method which involves approximating the original problem by smoothing the unknown domain.

Week 9: Wednesday 20th June

This PgS was cancelled.

Week 10: Wednesday 27th June

**Katharina Hopf - **Condensation phenomena in nonlinear drift-diffusion equations

For the abstract please see the posters on the postgraduate office doors.