# Postgraduate Seminar 2016-17

**Organiser**: Alex Wendland

### Term 3 2016-17 - The seminars are held on Wednesday 12:00 - 13:00 in MS.04 - Mathematics Institute

Week 10: Wednesday 28th June

**Marco Caselli** - On the proportion of everywhere locally solvable plane quartics.

Given a random rational plane quartic, we want to compute the probability that it has points when embedded in any completion of $\mathbb{Q}$. i.e. the reals and the $p$-adic fields.

In order to tackle the solubility densities at the non-archimedean completions, we classify and count by reducibility and singularities the quartics over $\mathbb{F}_q$. Through an interpolating method, we show these quantities are polynomials in the cardinality of the base field. Thanks to the Hensel Lemma we are able to determine the solubility in most cases, for the remaining ones we adopt a recursive method which leads to linear relations between these probabilities.

By a different approach, looking at the cone of definite real polynomials and using Hilbert's 17th problem, we can estimate the real solubility density problem are involved.

Week 9: Wednesday 21st June

**James Plowman** - GRR: The origins of algebraic K-theory in intersection theory

Arguably the first result in algebraic K-theory was the Grothendieck-Riemann-Roch theorem, which describes the correction factors required to make the Chern character map from the Grothendieck groups $K_0$ to the Chow groups with denominators commute with proper pushforwards in the non-singular case. The aim of this talk is to explicate this result. In the non-singular case, the Chow groups form more than just a group - they are intersection rings. We will begin by describing the construction of this intersection product following Fulton.

Week 8: Wednesday 14th June

**John Herman **- An Introduction to Feller Processes and Linear Evolution Equations

Brownian motion and Lévy processes are often used as models in areas such as physics, biology and finance. However, as these processes are spatial homogeneous, they can be inadequate for modelling certain phenomena. This leads us to study Feller processes, Markov processes which are locally like Lévy processes, but are spatially inhomogeneous. In this talk we will review some of the basic theory of Feller processes and discuss the connection of these processes to certain partial integro-differential operators, providing several examples. In particular, we will consider Feller processes taking values in a state space with a boundary, with focus on processes taking values in the half-line. After discussing certain methods for constructing Feller processes with a given generator, we will then provide a well-posedness proof for a reflected diffusion process with certain Lévy-type jumps restricted to the half-line.

Week 7: Wednesday 7th June

**Rosemary Taylor **- Constructions using unprojections

The Graded Ring Database uses numerical data to create a list of certain varieties which could exist. One method of proving their existence is to use unprojections to construct them from 'smaller' varieties that we understand. This talk will give a light introduction to the topic of type I, or Kustin-Miller type, unprojections and will focus on their application rather than their theory. In particular, I will provide several examples of constructing type I unprojections in low codimensions.

Week 6: Wednesday 31st May

**Calvin Khor** - Connections between the incompressible Euler equation and the SQG equation

The incompressible Euler equation in three dimensions is a well known partial differential equation (PDE) that models the velocity of an ideal fluid that has been studied for hundreds of years. Despite this, it is still the subject of active research. In particular, global existence and regularity for the Euler equation in 3D is an open problem, as is the problem for the related Navier-Stokes equation.

This talk will aim to introduce this equation gently to the less initiated, with some exploratory computations to provide basic intuition. Then I will define the Surface Quasi-Geostrophic (SQG) equation and describe some of its many striking similarities to the Euler equation.\\

Time permitting, I will also describe the problem of rigorously describing the evolution of thin vortex filaments for the Euler equation, and some work in this direction motivated by the SQG equation.

Week 5: Wednesday 24th May

**Lorenzo Toniazzi** - Probabilistic solution of fractional heat equations

Consider the standard heat equation

\[

\frac{1}{K}\partial_tu(t,x)=\frac{1}{2}\Delta_xu(t,x),\quad u(0,x)=\phi(x),\quad \ t>0,\ x\in \mathbb R^n,\ K\in\mathbb R^+,

\]

solved by $u(t,x)=\mathbb E[\phi(B^x(Kt))]$, where $\{B^x(Ks)\}_{s\ge 0}$ is a time-changed Brownian motion. Now substitute the first derivative in time $K^{-1}\partial_t$ with a fractional derivative $D^{(K)}_{t,0}$ (of generalised Caputo-type) to obtain a fractional heat equation. This equation is used in the physics and engineering literature to model so called anomalous diffusion phenomena, and its solution is also obtained from a time-changed Brownian motion.

In this talk we first motivate probabilistically the fractional derivative $D^{(K)}_{t,0}$ and the associated fractional heat equation. Secondly, we present a probabilistic proof of wellposedness and stochastic representation for the solution (joint work with Hern\'andez-Hern\'andez and Kolokoltsov). The proof also works for the standard heat equation (with no loss of intuition with respect to the fractional case). Therefore a basic understanding of the standard heat equation is sufficient to follow this talk.

Week 4: Wednesday 17th May

**Esmee Te Winkel **- Why surfaces are just glued pants

The Teichmüller space of a surface $S$ is the space of hyperbolic metrics on that surface. It admits an action of the mapping class group and the quotient is the moduli space of Riemann surfaces homeomorphic to $S$. This allows us to interpret the Teichmüller space as the universal cover of this moduli space.

Our aim is to coordinatize the Teichmüller space following Fenchel and Nielsen, using so-called pants decompositions of the surface. If time permits, we will introduce the pants graph, which is quasi-isometric to the Teichmüller space.

Week 3: Wednesday 10th May

**Elia Bisi** - Point-to-line polymers and orthogonal Whittaker functions

The 1-dimensional *KPZ universality class* is a wide family of probabilistic models, such as random growth processes and interacting particle systems, characterized by the fluctuation scaling exponent $1/3$ and certain long time limiting distributions from random matrix theory (*Tracy-Widom distributions*). In the last few years, it has been possible to study a few special models, thanks to their rich algebraic structure which allows obtaining exact formulas for the distribution of certain observables. One of these is the *log-gamma directed polymer*, which is a lattice path model in an inverse-gamma distributed random environment. In this talk, we consider the log-gamma polymer in three different geometries: (i) point-to-line, (ii) point-to-half-line and (iii) point-to-line restricted to a half-plane. We provide integral formulas for the Laplace transform of the partition functions in these geometries, using two main algebraic tools: Kirillov's geometric Robinson-Schensted-Knuth correspondence, and orthogonal Whittaker functions. We also outline the route to obtain KPZ asymptotics from

these formulas.

This is joint work with Nikos Zygouras.

Week 2: Wednesday 3rd May

**John Sylvester** - Minimal separators of fixed radius balls in random graphs

We begin with an introduction to the binomial (Erd\H{o}s-R\'{e}nyi) random graph, the graph on n vertices with each edge present independently with probability $p$, giving some history of the model and it uses. We shall then look at separators, where by a separator we mean a subset of the vertices whose removal disconnects the graph. In particular Bollob\'{a}s \& Thomason showed that, with high probability, the size of the smallest non-trivial separator coincides with that of the smallest non-trivial neighbourhood of any vertex in the graph. Thus, the ''cheapest" way to disconnect a random graph is to simply isolate the vertex of smallest degree by deleting the vertices connected directly to it. We will extend the classic problem by insisting that at least two of the components separated must contain a graph-metric ball of radius $r$, for some fixed $r\geq 1$. We then sketch a proof that, with high probability under this new restriction, the size of the smallest non-trivial separator is that of the smallest non-trivial $r+1$ neighbourhood.

Week 1: Wednesday 26th April (in B3.01)

**Juan Garza** - Construction and deformation of varieties via commutative algebra

Many problems in classification of algebraic varieties are approachable using Gorenstein rings; in this talk I'll introduce this concept from a geometric point of view, using a couple of fun and basic examples that will serve also as a guide to discuss more interesting cases, in particular I'll explain how to deform some interesting families of surfaces of general type.

### Term 2 2016-17 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 10: Wednesday 15th March

**Jordan Venters** - Combinatorial Nullstellensatz

Many problems in combinatorics may be rephrased as questions about polynomials and in particular the zeroes of these polynomials, one powerful tool in this direction is the combinatorial nullstellensatz developed by Noga Alon which gives us conditions under which a polynomial restricted to some set is not the zero polynomial.

I will state the combinatorial nullstellensatz and give several quick applications of it to additive number theory and graph theory.

Week 9: Wednesday 8th March

**Matthew Eggington** - Relating Newtonian and Boltzmann models for a Dilute Gas

We will give a gentle introduction to two models on differing scales for a dilute gas, the Newtonian particle description, and the mesoscopic Boltzmann equation. The physical validity of both models will be discussed, as well as the relationship between the two in the Boltzmann Grad limit for the special case of a tagged particle interacting via some potential with a background media.

Week 8: Wednesday 1st March

**Giannis Moutsinas** - Hyperfunctions: What they are and why we care

Hyperfunctions are a class of generalized function introduced in 1958 by Mikio Sato. His idea is very intuitive: a hyperfunction can be thought as the difference of 2 analytic functions over the real line. Unfortunately he chose to expose his theory in a very abstract manner. This kept the theory of hyperfunctions relatively unknown.

I will introduce the concept of hyperfunctions in an intuitive and informal way. We will go through the definition, important examples and basic properties. The aim of this talk is to show that there can be a class of generalised functions for which the derivative is exactly what you would expect a derivative to be.

Week 7: Wednesday 22nd February

**Katerina Hristova** - On representations of locally profinite groups

In this talk we consider smooth representations of a certain class of topological groups usually referred to as locally profinite. We give examples of such groups as well as an outline of the basics of their representation theory. We focus our attention on a Theorem by Bernstein which gives a bound on the projective dimension of the category of smooth representations of certain types of locally profinite groups. Even though the result looks purely algebraic, surprisingly the proof relies on the geometry related to those groups. We finish by applying Bernstein's Theorem in the specific cases of $SL_n(\mathbb{F})$ and $GL_n(\mathbb{F})$, where $\mathbb{F}$ is a non-archimedean local field, and explain how the theorem can be extended to more general cases.

Week 6: Wednesday 15th February

**Elena Camacho Aguilar** - Catastrophes in Cell Differentiation

Have you ever wondered how our bodies grow? How from the fertilised egg that divides into two cells and then into four and so on, such a variety of tissues and organs appear? This is what developmental biology studies and it divides this process into four developmental stages: pattern formation, morphogenesis, cell differentiation and growth. In particular, cell differentiation is the process by which unspecialised cells become different types of cells by adopting and maintaining a particular pattern of gene activity, which determines which proteins are synthesised. It is a key stage of development and its understanding could have potential applications, for instance in regenerative medicine.

This talk builds on the approach introduced by F. Corson and E.D. Siggia in 2012, in which they took advantage of the geometric theory of dynamical systems to develop a new geometric method to modelling cell differentiation, and they illustrated this by a model of vulval development in C. elegans. The mathematical form of their model was strongly motivated by the three different fates observed for the cells involved but it’s behaviour is somehow difficult to predict.

In this talk I will introduce the biological process and will explain a new approach to its modelling by taking advantage of catastrophe theory. It is possible to classify a rich class of dynamical systems and the bifurcations between them using singularity theory, which allows us to deeply understand the system. Defining a catastrophe manifold for the model and mapping signals into the parameters, I will show that the we can reproduce the topology of vulval development studied. In particular, we are able to fit the biological data for the wild type case and several mutants.

Week 5: Wedneday 8th February

**Natalia Jurga **- Dimension of measures for a class of planar self-affine sets

We study Käenmäki measures supported on planar self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula which gives an explicit formula in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure.

**Daniel Rogers **- Maximal subgroups of simple groups

Understanding maximal subgroups of a group gives us insight into their structure. In this talk we will give some reasons why maximal subgroups are useful, and describe some theorems which classify the maximal subgroups of some simple groups.

Week 4: Wednesday 1st February

**Alex Torzewski** - Cohomology of Varieties

We shall define what a cohomology theory should be and introduce some of the most common cohomology theories. We go on to discuss what properties they encode and attempts to fit the many different cohomology theories into a common framework. We build evidence for this viewpoint through several worked examples.

Week 3: Wednesday 25^{th} January (in D1.07)

**Wojtek Ozanski** - How to construct a solution to the Navier–Stokes inequality that blows up?

The 3D incompressible Navier--Stokes equations are the central model of fluid mechanics and, despite a significant effort since the celebrated work of Leray (1934), the fundamental question of existence of global in time strong solutions remains unsolved. Although we do not know whether singularities occur, we are able to estimate the Hausdorff dimension $d_H(S)$ of the possible set $S$ of singular points, that is the dimension of the set of points $(x,t)$ at which the velocity field $u$ blows up. This is possible thanks to the partial regularity theory developped by Caffarelli, Kohn \& Nirenberg (1982). This theory is concerned with velocity fields satisfying the weak form of the Navier--Stokes inequality,

\[

u\cdot (u_t - \Delta u + (u\cdot \nabla )u +\nabla p ) \leq 0,

\]

and it provides sufficient conditions for boundedness of $|u|$ in small cylinders in space-time, which implies the bound $d_H (S) \leq 1$. It turns out that this result is sharp, which is demonstrated by the work of Scheffer (1985 \& 1987), who constructed weak solutions to the Navier--Stokes inequality with internal blow-ups. The blow-up in these solutions occurs on a set $S=(S',T_0)$, where $S' \subset \mathbb{R}^3$ either consist of one point or is a uniform Cantor set with $d_H(S') =\xi$ for any preassigned $\xi \in (0,1)$. In the talk we will discuss some of the details of these constructions and we will see that they provide unique insights into the structure of the Navier--Stokes equations and suggest that the question of the existence of global in time strong solutions might have a negative answer.

Week 2: Wednesday 18^{th} January

**Matt Bisatt** - Getting to the root of elliptic curves

One of the first things you learn about elliptic curves is that they have the structure of a finitely generated abelian group but computing the rank of this group is computationally difficult. Instead we shall consider an invariant called the root number which is (conjecturally) dependent only on the parity of the rank; i.e. it tells us if the rank is odd or even. I will discuss how the root number is properly defined and how one might calculate it for a given elliptic curve and their higher dimensional analogues modulo some hand-waving. No prior knowledge of elliptic curves will be assumed.

Disclaimer: No curves were harmed in the writing of this talk.

Week 1: Wednesday 11^{th} January

**Taísa Martins** - *An Introduction to Hypergraph Containers*

Many problems in combinatorics can be formulated in terms of independent sets in hypergraphs. One example is the question of how many subsets of $1$ to $n$ contain no $k$-term arithmetic progression, which may be formulated as the number of independent sets of the $k$-uniform hypergraph with vertex set $\{1, \ldots, n\}$ and hyperedges being all k-term AP's contained in $\{1, \ldots, n\}$.

A solution for the above problem (and many others) can be obtained using the recent technique developed by Balogh–Morris–Samotij and Saxton–Thomason known as the method of hypergraph containers.

The container method offers a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. Informally, it consists of a procedure to obtain a relatively small collection $C$ of vertex subsets of a $k$-uniform hypergraph $H$ satisfying certain properties, such that every independent set of $H$ is contained within a member of $C$, and no member of $C$ is large.

This talk will give an introduction to the topic and show some applications of the method.

### Term 1 2016-17 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute

Week 10: Wednesday 7^{th} December

**Ronja Kuhne** - *On conditionally hard problems in the theory of knots and 3-manifolds*

In this talk I will give an introduction to the theory of knots and 3-dimensional manifolds, focusing on so-called "conditionally hard" decision problems in both areas. To this end we will review the basic notions of complexity theory and outline how classical problems in graph theory are related to problems in low-dimensional topology.

Week 9: Wednesday 30^{th} Noverber

**Ollie Dunbar** - *From Soap to Simulation - A phase field story*

The soap: Surface active agents are molecules which cause a reduction in surface tension of the fluid. They are used extensively in industry in fields such as cosmetics, detergents, agrochemicals and emulsion polymerisation to name but a few. Traditional models derived from physical laws to model systems of multiple fluids often have difficulty (numerically and analytically) in modelling a diverse range of phenomena.

The simulation: I would like to construct a mathematical model of these surfactants in multiple fluids with sufficient detail to show the aforementioned behavioural changes. I will create this using a mathematical tool known as phase field modelling, they are particularly attractive as they follow similar constructions to that of the physical models. Time permitting I may further discuss discretization schemes and present simulations created in the DUNE package resulting from this phase field description.

Week 8: Wednesday 23^{rd} Noverber

**Daniel Rogers** - *The extension problem; or why groups are like Lego and not a jigsaw puzzle*

The role played by finite simple groups when considering all finite groups is similar to the role played by prime numbers when considering all integers. However there are some crucial differences; in particular, given a list of ''factors'' for a group, determining all groups with those factors is a very difficult problem, known as the extension problem. This talk will give a brief introduction to this problem.

**Katie Vokes** - *Mapping class groups and how to study them*

In the study of surfaces, a topic of much interest is that of mapping class groups. The elements of these groups are isotopy classes of self-homeomorphisms, which can be thought of as symmetries of a surface. I will describe some properties of these groups and some of the tools we can use to study them.

Week 7: Wednesday 16^{th} November

**George Kenison**- *Equidistribution asymptotics*

Let \((a_n)_{n=0}^\infty\) be the sequence with terms \(a_n = n\alpha\) where \(\alpha\) is irrational and let \(\{a_n\} = a_n - \lfloor a_n \rfloor\) denote the *fractional part* of \(a_n\). A well-known theorem of Weyl's says that \((a_n)_{n=0}^\infty\) is *equidistributed modulo 1*, i.e. given \(a,b\in[0,1]\) with \(a<b\) we have a frequency asymptotic

\begin{equation*}

\lim_{n\to\infty} \frac{\#\{k\in \mathbb{Z}^+ \colon k\le n \text{ and } \{a_k\} \in (a,b) \}}{n} = b - a.

\end{equation*}

There is a natural connection between Weyl's equidistribution theorem and concepts in dynamical systems and ergodic theory. Following this introduction, we present equidistribution asymptotics for a dynamical system related to a free group.

Week 6: Wednesday 9^{th} November

**Theodoros Assiotis** - *Asymptotic Representation Theory and Dynamics on Branching Graphs*

I will discuss an approach initiated by Vershik and Kerov for studying the characters of "big" groups like the infinite symmetric and infinite-dimensional unitary group via the study of the associated branching graph and its boundary. I will explain how consistent stochastic dynamics on these graphs give rise to Markov processes on the boundary and in a different direction they allow the computation of the correlation functions of certain random growth models. Time permitting I will talk about some very recent results of purely probabilistic origin.

Week 5: Wednesday 2^{nd} November

**Vandita Patel **- *Perfect powers that are sums of consecutive $k$-th powers*

Let $k$ be an even integer such that $k$ is at least 2. We give a (natural) density result and show that for almost all $d$ at least 2, the equation $(x+1)^k + (x+2)^k + \ldots + (x+d)^k = y^n$ with $n$ at least 2, has no integer solutions $(x,y,n)$. For this talk I will focus on the case $k=2$, with plenty of examples and explicit calculations.

This is joint work with Samir Siksek (University of Warwick).

Week 4: Wednesday 26^{th} October

**Pavlos Tsatsoulis** - *Spectral Gap for the Stochastic Quantisation Equation on the $2$-dimensional Torus*

Consider the stochastic quantisation equation on the 2-dimensional torus formally given by

\begin{equation}\label{eq:SQE}

%

\left\{

\begin{array}{lcll}

\partial_t X & = &\Delta X - \sum_{k=0}^n a_k X^k + \xi \\

X(0,\cdot) & = & X_0

\end{array}

\right., \tag{SQE}

%

\end{equation}

where $\xi$ is a space-time white noise, $n$ is odd and $a_n>0$. It is already known that the above equation admits a well-posed theory only as a

renormalised problem which provides Markov solutions evolving in a space of distributions of suitably negative regularity.

We discuss recent results on the long time behaviour of solutions to \eqref{eq:SQE}, including a strong dissipative bound independent of the initial

condition and the strong Feller property for the associated Markov semigroup. We then argue on how, given a support theorem, these results imply

exponential mixing. (joint work with H. Weber)

Week 3: Wednesday 19^{th} October

**Jack Skipper** - *Energy Conservation of the Incompressible Euler Equations*

I will give a gentle introduction to the topic above. Classical methods can be used to prove local and global energy conservation for strong solutions that are continuously differentiable in time and space. I will explain why we cannot apply these methods to weak solutions. When we consider equations without boundaries, energy conservation still holds for sufficiently nice weak solutions. I will outline the methods used and discuss the problem when boundaries are included.

Week 2: Wednesday 12^{th} October

**Matthew Spencer** - *Brauer relations in positive characteristic after semisimplification*

We will talk about the kernel of the map which given a finite group takes a $G$-set to a permutation module. Specifically we will analyse the case where the target of the map is the ring of semisimplified $F_p[G]$-modules. We will describe precisely what we mean by semisimplification in the talk, and our aim will be to classify the elements of this kernel.

Week 1: Wednesday 5^{th} October

**Lorenzo Toniazzi** - *Models for interrupting (random) long jumps/long memories*

We present models for stochastic processes with long jumps/long memory that take into account the behavior with respect to a barrier in their state space. Such models arise naturally in probability theory and their usage is widespread in a variety of applied fields.

The central object of the talk is a class of integro-differential operators viewed as the (Markov) generators of jump-processes "interrupted" on the attempt to cross a boundary.

We will (i) motivate these operators from different viewpoints and (ii) present well-posedness results for a selection of (deterministic) boundary value problems that involve these operators.