QuIPS, the postgraduate seminar, allows PhD students to present their research (or any other interesting maths) to each other in a friendly space. Talks are usually on Wednesdays from 1630-1730 and a free dinner is provided afterwards. Roughly once a month we have a 'pizza QuIPS' instead. Everyone is welcome: QuIPS talks are designed to be accessible to all maths PhD students!
Inspired by Scott Aaronson's essay "Who can name the Bigger Number?" (link is external), we begin the talk with a game - everyone was given 30 seconds and a piece of paper, and asked to write down the biggest number they could. The discussion proceeded to an attempt to discuss some real-life big numbers - the number of cigarettes smoked in the world in a year, the number of letters in the university library, the number of seconds in a human lifetime, the number of elementary particles in the observable universe, etc.
We then went on to discuss some really big numbers - we defined Knuth's Up Arrow Notation, and made an attempt to understand quite how big Graham's number really is. Then we moved onto *really* big numbers. After a (brief) discussion of Turing Machines and the Halting Problem we defined the Busy Beaver, and the Busy Beaver shift function, which provably grows faster than any computable function.
We had a first QuIPS problem session towards the end of last term. This is our second attempt.
People are encouraged to bring along open problems that they think other members of the group might be interested in working on, or able to help with. It can be something related to your own research, or something completely different. Hopefully we can get some useful collaborations going.
Colin will try to show the following: let G be a finite group such that p is the largest prime dividing the order of G, and every Sylow subgroup of G is generated by at most d elements. Then G has a nilpotent normal subgroup of index bounded by a function of p and d. (In this context, nilpotent means a direct product of groups of prime power order.) In particular, there are only finitely many primitive groups with these parameters. He will explain the group theory terminology at the start for those who are unfamiliar with it.
The Sharkovsky Theorem- An ordering of the natural numbers corresponding to periodic orbits of continuous maps on the real line.
Any undirected graph can be viewed as an adjacency matrix, and the eigenvalues of such a matrix is known as the graph's spectrum. I will talk about various aspects of a graph's structure which are encoded by its spectrum, and present two proofs of the well-known theorem which says that if all a graph's eigenvalues are simple, then its automorphism group is an abelian 2-group.
In two dimensional percolation problems, all exact results and most bounds for percolation threshholds make use of planar duality. We will look briefly at this and then consider methods of bounding threshholds without duality.
We will talk about a method of how to visualize bifurcation
structure using transient processes for one dimensional maps.
Further, applying this technique, we will show how to detect
unstable periodic orbits of maps and flows.
We introduce the notions of vector fields and differential forms for Euclidean space using both a local and coordinate free approach. We define the exterior derivative of a smooth function and look at its unique extension to an operator on the exterior powers of the module of differential forms. We then discuss the basics of de Rham cohomology. We finish with Euclidean space by reformulating the familar notions of div, grad, curl, and the Laplacian in terms of the exterior derivative using Maxwell's equations as an example of an application.
We then recall the notion of a differential manifold and show how the differential form approach to multivariable calculus generalises to this more general setting almost painlessly.
Finally, we will mention a little about the generalisation of all of this to the setting of noncommutative geometry.
Families of subsets of {1,2, ... n} -- or equivalently subfamilies of the n-dimensional hypercube Q_n -- are one of the main objects of study in extremal combinatorics. The kind of question we are typically interested in is: what is the maximal size of a family F contained in Q_n with a given property P?
Recently there has been some interest in answering similar questions when F lives inside a different combinatorial space. In this talk, I will review the classical results of Erdos-Ko-Rado and Sperner, before considering how they might generalise to the so-called separated hypercube.
We consider the standard model of a discretised rotation in the limit where the rotation number goes to zero, and find that orbits approach integral curves of a piecewise-constant Hamiltonian vector field. Furthermore we define an ordering on the set of possible orbit topologies and make a conjecture about the existence of so-called 'minimal orbits'.
I will introduce some basic notions from Game Theory and explain how players can “learn” to play a game by repeatedly playing it. I will
show how such a learning process can be modelled mathematically by a differential inclusion.
One such learning dynamics is known as “Fictitious Play”. I will demonstrate some of the remarkable features of this dynamical system and some of the mathematical questions it gives rise to.
Seminar series:
Queen Mary Internal Postgraduate Seminar
We give a very rough introduction to some aspects of three-dimensional
computer graphics, and discuss the applicability of the associated
hardware to general parallel computation. We then consider a problem in
computational group theory and speculate about parallel solutions. No
particular programming knowledge will be assumed.
Recent studies on quantum turbulence have shown that the velocity statistics
of a tracer particle obey power law statistics ( Paoletti et al. [ Phys. Rev. Lett.
101 154501 2008], White et al. [Phys. Rev. Lett. 104, 075301 2010 ]) rather than
Gaussian statistics as observed in classical turbulence ( Vincent et al. [ J. Fluid
Mech. 225 1 1991 ] , Noullez [J. Fluid Mech. 339 287 1997]. A superstatistical
model Beck [Phys. Rev.Lett. 98, 064502 2007] is constructed for a Lagrangian
tracer particle in a quantum turbulent flow. The result is in excellent agreement
with a recent experiment done by Paoletti et al. [Phys. Rev. Lett. 101 154501
2008] to observe the motion of tracer particles in quantum turbulent flow of
superfluid 4He.
Origami has come under considerable mathematical attention over the last 50 years. I will give the outlines of a proof of the so called "fundamental theorem of origami" that more or less states that it is possible to fold any model. The proof will be constructive, so in other words, it will tell you how to design any model yourself!
A Langevin equation with dry friction is studied by using the path integral approach and the
saddle-point approximation. Firstly the pure dry friction case is investigated in details and
its corresponding direct and indirect paths are also derived. And then the Langevin equation
obtained by regularizing the dry friction force with a smooth hyperbolic tangent force are
analysed. An interesting bifurcation phenomenon for the solutions of optimal paths are
observed in the regularized case.
The representation theory of the symmetric group is an old and rich subject. The modern perspective was developed by Gordon James in the 1970s. The Iwahori-Hecke algebra of type A is a deformation of the symmetric group, and one motivation for studying it is that it provides a bridge between the representation theory of the symmetric and general linear groups.
The Specht modules are a family of modules of fundamental importance for the Iwahori-Hecke algebra, and an open problem is to determine which Specht modules are decomposable. I will be discussing my attempts to answer this using a very new approach via Khovanov-Lauda-Rouquier algebras. These algebras at first glance seem completely different from the Iwahori-Hecke algebra, but an amazing result of Brundan and Kleshchev is that the Iwahori-Hecke algebra is isomorphic to a certain KLR algebra. I will explain how using the KLR algebra approach makes Specht modules easier to understand.
I will introduce the topic of topological field theories with some low dimensional examples. I will then work towards some kind of classification of these low dimensional cases. Time permitting, I may say something about how this generalises.
Random Matrix Theory (RMT) is a rich topic with many applications, for example in physics, multivariate statistics and number theory. After giving a short introduction into this field I will focus on the supersymmetry method which is a powerful tool to reduce the number of integrals involved in various RMT calculations.
The Tangled Nature Model of evolution is an individual based, stochastic model, which describes, with good agreement with actual observations, the evolution of a simple ecology. Its dynamics alternates periods of meta-stable configurations and periods of hectic transitions, where the model does not show clear occupancy patterns and the population is spread randomly across the type space. In this talk I will briefly introduce the model, explain the feature of its dynamical evolution and discuss the possibility of forecasting its characteristic transitions by using a deterministic approach.
Random motion is a powerful and simple model to describe processes characterized by stochastic activity, and it is widely used in many fields of science. In particular, random walks have been largely used over the last few years to investigate the dynamical properties of real world networks.
Here we focus on degree-biased random walks, a particular class of walks in which the transition probability from a given node to a neighbour depends on a one-parameter function of the degree of the destination node.
We analyze a small set of characteristic quantities, the Mean Return Time (MRT), the Mean First Passage Time (MFPT) and the Mean Coverage Time (MCT) and we show numerically and analytically the interplay between topological quantity and the dynamical properties of the motion.
We also compare the prediction of the mean-field approximation for synthetic models to the exact numerical solutions calculated for a dataset of 20 real networks.
In 1999 Zhang and Yeung showed that you can send information through a network more effectively than by the routing method that is used in the internet today. In this talk I will show how that is possible. I will also define entropy, which is a measure of how much information a random variable contain, and I will define the entropy cone $\overline{\Gamma^*_n}$, which is an attempt to describe how n random variables can be related to each other. We know what this cone looks like for n\leq 3 but already for n=4, we do not have any good description of the cone. I will talk about what we know about the cone, and how the cone is related to network coding.
Dynamics based on replication and mutation form the core of game theory and population dynamics. Its outcome is often not an equilibrium, but can include oscillations and chaos. In this talk I will introduce a class of models of immunological interest and discuss a method for computing all their Lyapunov exponents.
In graph theory, A graph is called Hamiltonian if there is a cycle which contains every vertex of the graph. Thomassen conjectured that "Every 4-connected claw-free graph is Hamiltonian". In this talk, I will introduce Tutte's technique which is useful for proving some class of this Hamiltonian problem.
Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems. As a consequence of the complicated dependence on the parameters, scattering is quite often of chaotic nature, which allows to tackle many problems within the framework of Random Matrix Theory (RMT). After a short introduction into the general theory of quantum chaotic scattering we discuss the basic ideas of two different RMT-approaches (called Heidelberg and Mexico approach) and show a proof of their equivalence
Cops and Robbers is a perfect information graph game in which a set of cops $C=\{C_{1},\ldots,C_{m}\}$ tries to a catch a robber, $R$. This can be used as a simple model of network security.
Given a graph $G$, the cop number of the graph, $c(G)$, is the minimum number of cops that can guarantee to catch $R$ on $G$. We will look at some bounds on the cop numbers of some simple graphs before moving on to look at the cop number of random geometric graphs.
We will look at a very brief overview of the representation theory of the Iwahori-Hecke algebra, before putting it into a modern, graded setting. In this setting, we will look at Specht modules and the homomorphisms between them. We will discuss new results, extending column and row removal for homomorphisms into this graded setting. We conclude with a brief application of this result.
We will be looking at enumeration problems relating to walks on a lattice with a boundary. These can be used to model polymers interacting with a surface.
Generating function techniques can be a powerful tool for solving these, and we will demonstrate this, focusing first on the simpler case of Dyck paths before moving onto a triangular lattice. We will also present some bijective results between walks, including an open problem.
Topological stacks are just jazzed up topological spaces. I will talk about the following question: We have homology/homotopy groups of topological spaces, so can we extend this notion to topological stacks? I will give lots of examples and explain situations where topological stacks naturally arise.
Suppose you know the distribution of some discrete random variable X, and you want to learn the specific value x that it takes, by asking yes/no questions. Each question cost you £1, how much will it cost you on average to learn x? If you could buy questions that can have 3 different answers instead of only 2, then how much should you be willing to pay for such a question? If you could buy yes/no questions that cost you £2 when the answer is "no" but only £0.50 when the answer is "yes", would that be better in the long run? How much is a question worth if there is a probability that you get the wrong answer? In this talk, I am going to answer these questions, to give an introduction to information theory. I will then give a short introduction to cryptogenography (the study of how to leak information without revealing yourself), where I present the idea and some results.
In this talk, I will briefly introduce the model: Brownian motion with dry friction. And then give remarks on how to solve a first-passage time (FPT) problem. After that, I will show that the FPT problem of a pure dry friction model and a full force model can be solved analytically. I will also show some interesting phase transition phenomena related to the FPT problem.
The dynamics of the Lorentz gas is deterministic but, when one deals with an ensemble of particles, macroscopic properties arises in such way that we can explore microscopic details and understand macroscopic properties of the system.
In this talk, I'll introduce the concept of billiards, or Lorentz Gas, and explain how they are useful as a start to explore diffusion. Then, I will explain the case of billiards systems with soft potentials and a summary of relevant investigations in the field.
We consider the dynamics of small tracer particles for example, electron, ion, solid hydrogen or solid deutterium particles in quantum turbulent flow. The complicated interaction processes of quantized vortices, the quantum mechanical constraints on vorticity and the varying influence of both the superfluid and the normal fluid on the tracer particle effectively lead to a superstatistical Langevin-like model. The model in a certain approximation can be solved analytically. An analytic expression for the probability density function of Lagrangian velocity $v$ of the tracer particle is derived that exhibits not only the experimentally measured $v^{-3}$ tails but also the correct behaviour in the neighbourhood of the center of the distribution. The predicted PDF has an excellent agreement with experimental measurements and numerical simulations.
Spline models are piecewise polynomials that are optimally smooth, but they are practically unworkable in higher dimensions and hard to fit over non-standard regions. Smooth supersaturated models are high degree polynomial models that can be fitted over non-standard regions, portray spline-like behaviour and, as polynomials, are more tractable than splines. This allows us to apply optimal design theory over non-standard regions. We use orthogonality to simplify the fitting of smooth supersaturated models via three (and a half) different basis concepts: A monomial basis, an orthogonal polynomial product (with an example using Legendre polynomials), and a multivariate orthonormal polynomial basis. A simulation study demonstrates smoothing over arbitrary regions using a multivariate orthonormal polynomial basis.
In this talk, I will present a fairly natural and easy introduction to deform a (commutative) associative algebra. Given an associative noncommutative algebra, we analysis the classical structures (like Poisson bracket, connection, etc) of which is quantisation. This approach is very different from the algebraic deformation theory of associative algebras introduced by Gerstenhaber (1964). In my case, I will focus on deformation of classical differential graded algebras (a super-commutative super Hopf algebra) over Poisson-Lie group. A large class of examples of such noncommutative deformation will be given by pre-Lie algebra, an (not necessarily associative) algebra with product such that (ab)c-(ba)c=a(bc)-b(ca).
To study the global behaviour of Einstein equations and their solutions conformal geometry has proved its usefulness in existence stability results. A useful way to get intuition of the geometric meaning of a conformal transformation in general relativity is the construction of Penrose diagrams. In this talk, the general procedure of constructing such diagrams will be outlined by 3 examples: the stereographic projection, the Penrose diagrams for the Minkowski and Schwarzschild spacetimes. With the perspective given by those examples we will discuss the definition of asymptotically simple spacetimes and the main difficulty for deriving a conformal version of the Einstein field equations.
I examine two online social networks where nodes are connected by directed and signed links. The analysis of degree-degree correlations in such networks provides insightful view on how the nature of interactions influence the peculiar correlation patterns observed in many social networks. Positive correlation are associated to a collaborative relation between nodes, but other kinds of interaction generate different correlation patterns. A simple theoretical model is introduced in order to reproduce the distinct degree correlation patterns emerging from positive and negative networks.
We will start with an easy introduction to Hamming Codes- a code that is both optimally error-correcting and optimal as a covering code. We will then discuss an application to an extremal problem in the hypercube. This talk is aimed at a general audience; in particular all non-standard terms in title and abstract will be defined.
I will explain how the the asymptotics of functions can be obtained by the use of the saddle point method. To gain intuition, I will show how Stirling's formula can be obtained in this way. Afterwards, I will explain the case of two coalescing saddle points, with which I am dealing in my PhD project.
I will give an introduction to (some of) proof theory, emphasising links with combinatorics. Beginning with Hilbert's definition of a formal proof and Godel's incompleteness theorems, I will focus on Gentzen's 1936 proof of the consistency of arithmetic. This introduced a proof language called sequent calculus, and the idea that proofs have a natural dynamics given by cut elimination. This leads to linear logic and the hard problem of finding good invariants for the dynamics of proofs (which computer scientists call 'denotational semantics').
We consider a one-parameter family of invertible maps of a two-dimensional lattice,
obtained by applying round-off to planar rotations.
We let the angle of rotation approach $\pi/2$, and show that the system exhibits a
failure of shadowing: the limit of vanishing discretisation corresponds to a piecewise-affine Hamiltonian flow,
whereby the plane foliates into invariant polygons with an increasing number of sides.
Considered as perturbations of the piecewise-affine flow, the lattice maps assume
a different character, described in terms of strip maps, a variant
of those found in outer billiards of polygons. Furthermore the flow is nonlinear
(unlike the rotation) and a suitably chosen Poincare return map is a twist map.
We show that the motion at infinity, where the invariant polygons approach circles,
is a dichotomy: there is one regime in which the nonlinearity tends to zero, leaving only the perturbation,
and a second where the nonlinearity dominates.
In the domains where the nonlinearity remains, numerical evidence
suggests that the distribution of the periods of orbits is consistent with that of random dynamics,
whereas in the absence of nonlinearity, the fluctuations result in intricate discrete resonant structures.
A number of network growth mechanisms have been suggested to explain how social connections are forged and severed over time. Among these mechanisms, a key role is played by homophily, namely the principle that similarity breeds connection. However other studies have pointed in the opposite direction. For example, economists have suggested that similarity can lead to competition for scarce resources.
This talk will examine to what extent homophily appears to govern communication in an online social network.
Percolation Theory is the study of the structures of clusters and connectivity in (infinite) random graphs. The percolation threshold is the point at which there first appears global (or infinite) components. Percolation thresholds have important applications in many physical and real world problems, such as material science and epidemiology. In all but a few simple cases, the exact threshold of most models are not known, and so a lot of work is put into estimating them.
I will give a general introduction to percolation theory before moving on to looking at how to gain rigorous confidence intervals for two and three dimensional lattices.
A scattering process can completely be characterized by its K-matrix. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory, where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. I will show that both approaches are equivalent for a broad class of unitary invariant ensembles of random matrices, using correlation functions of products and ratios of integer powers of characteristic polynomials.
We will discover that for orthogonal invariant ensembles one needs instead correlation functions of half-integer powers, and I will present results for a few of these correlation functions in the limit of large GOE-matrices along with some further examples where these objects also arise.
Imagine two people are each given a set of rods, and instructions on how to connect these rods together with joints at their endpoints. Assuming they both follow the instructions correctly, is it possible that they could still end up with different structures?
If the given constraints (in this case the lengths of the rods and the rules on how to connect them) are sufficient to guarantee a unique realisation, then we say that the structure is globally rigid. In this talk we'll look at the global rigidity of frameworks in two dimensions, and see to what extent this can be determined by their underlying graph.
I will introduce some (hopefully) interesting ideas from intersection theory. I will talk about the question `where should I intersect stuff?' which will lead us from school maths to something a bit more fancy!
I will introduce dynamic algorithms for maintaining all pairs shortest paths (APSP) in undirected graphs with real-valued edge weights. I will highlight the central role played by shortest paths in the analysis of real-world networks.
Because of the high volatility and the increasing size of today's network data sets, the development of efficient methods for the real-time analysis dynamic networks is becoming more and more crucial.
Lastly, I will try to turn on the discussion by showing some (blurred) ideas in connection with networks dynamics, group theory, and quantum field theory.
We consider a game where n people have a randomly (but not independently) chosen natural numbers on their foreheads, and no two player have the same number. The players are not allowed to communicate, but they all know the distribution of the numbers. After seeing the numbers on all the other peoples foreheads, each player can choose two numbers, i and j between 1 and n. The player now wins £1 if the number on his forehead is the i'th smallest, but looses £1 if it is the j'th smallest. We want to choose a distribution such that the players do not win too much in expectation. How well can we do?
We will see that no matter the distribution, the players will always be able to ensure a positive expectation, but for any epsilon, we can choose the distribution such that they have expectation at most epsilon. However, to do this we would need to use numbers as large as 2^{2^{\dots 2^{k_n/\epsilon}}}, where the height of the tower is n-2.
The presentation is based on joint work with Troels B. Sørensen and Vincent Conitzer.
Representation theory of the symmetric group enables us to study the symmetric group via linear algebra. We seek to understand the irreducible representations of the symmetric group over any field. Over a field of characteristic 0 (say, the complex numbers), the modern construction of these irreducible representations, the Specht modules, was developed by G. D. James using combinatorics in the 1970s. In 2009, it was shown that the symmetric group algebra is non-trivially graded, and subsequently that the Specht modules are also non-trivially graded, a significant development in modular representation theory (the theory of representations over a field of positive characteristic). I will give an outline of the representation theory of the symmetric group and construct the Specht modules.
In 1954, Frobenius, Robinson and Thrall introduced a combinatorial formula, the hook length formula, for the dimension of the Specht modules over any field. Recent results have enabled us to define the graded dimension of the Specht modules combinatorially. One hopes to obtain an analagous graded hook length formula; I will discuss my developments on this problem.
Random walks represent a paradigmatic model to study the diffusion properties of complex networks, and have also been extensively employed in the last decade as a tool to characterise the centrality of nodes and to identify densely connected subgraphs or communities. Biased random walks are a particularly interesting class of walks for which the probability to jump from one node to one of its neighbour depends on a function of a chosen topological property of the destination node, usually its degree, and can be therefore tuned at will in order to systematically prefer (or avoid) to move toward nodes having certain characteristics. We present here an analytical treatment of biased random walks on multiplex networks, i.e. complex networks whose units are connected by means of a variety of different relationships, represented by M separate yet interacting layers corresponding to different communication channels, and study how the entropy rate of such processes is affected by the presence of inter-layer degree correlations and other structural properties of the considered systems.
Nonequilibrium systems are characterized by the presence of macroscopic currents. Steady states conserve the average currents, but their statistics can be complex, with anomalous phases and strong fluctuations. No general theory allows to describe it without solving the microscopic dynamics.
We present the large deviation framework for macroscopic observables and we use it to find the current statistics in a time-correlated zero range process (ZRP). We derive the exact stationary solution and a mean-field approximation on the one-dimensional lattice. Analytical and numerical calculations show that, while the steady state
corresponds to that of a Markovian ZRP with effective interaction, the probability of rare currents differs significantly from the Markovian case, with memory-induced dynamical phase transitions playing a central
role.
The results are also interesting for problems that usually fall outside the domain of the fundamental physics, such as congestion-avoidance strategies in data streams and stochastic modeling of strongly correlated biological systems. In fact, in these cases, the rare events can be more important than the typical ones.
In this talk we will briefly motivate/introduce a conformal formulation of the Einstein field equations that has been used to obtain global non-linear stability results for spacetimes as de-Sitter and Minkowski. We will discuss how to use these equations to analyse the behaviour of black hole spacetimes. In particular, we will pose an asymptotic initial value problem (initial data "at infinity") and analyse the evolution equations. It will be pointed out the mechanism in the equations that is responsible for the formation of singularities.
Geometric cluster models are a class of physical toy models used to mirror the behaviour of systems such as polymer chains or cell membranes. In order to understand their physical properties depending on external parameters such as the temperature, one reaches to know their partition functions (or generating functions) and analyse their asymptotic behaviour in the limit of large system sizes.
I will start by giving a few examples for geometric cluster models and discuss some generic properties of them. At the example of Dyck paths, I will then show how their asymptotic behaviour can be investigated.
A well-studied graph invariant is the Tutte polynomial, which can be used to calculate structural properties of a given graph. While the Tutte polynomial extends to matroids, it cannot be applied to a superclass of matroids called polymatroids. I will give an introduction to the Tutte polynomial and to matroids, and give a candidate for a Tutte-like polynomial for polymatroids.
Non-equilibrium networks or growing networks are network models in which new vertices are continuously added in time to the graph and are connected to the already existing vertices according to some attachment rule. In order to find out the degree distribution of such models a general approach is to define a master equation for the system under study. I will show how to use the master equation approach to solve two classical network models, the growing exponential network and the Barabasi-Albert network model.
General Relativity is one of the most successful physical theories ever devised. But as well as providing accurate physical predictions, it has a mathematical elegance which makes it interesting in its own right. Assuming no previous knowledge of general relativity, I will explain the basics of the theory, and describe how symmetries of physical solutions are represented naturally in terms of Killing vectors. Then, I will describe how general relativity can be set up as an initial value problem; and finally, prove the Licnerowicz theorem, which states that any static spacetime which possesses an initial hypersurface with Euclidean topology cannot contain black holes.
The combination of conformal geometry with general relativity has proven immensely useful in analysing the global properties of spacetimes. One of the most relatively recent ideas is to write these equations as a system of wave equations. The first parts of this talk will very quickly show how trying to understand the evolution of Einstein's equations leads to them being written as a system of wave equations. The next part will cover the basics of conformal geometry, why one needs conformal geometry and how this motivates the conformal field equations, (with a brief explanation of how one derives the equations). Once we have covered this prior knowledge, we will see how these two ideas can be combined to derive a system of wave equations for the variables of the conformal field equations. Included in this part will be the list of wave equations that have been derived, those for Einstein-Maxwell spacetimes and for the Conformal field equations in a vacuum. Included will be the derivation of some of the simpler wave equations. Finally, to conclude we will look at the future of this subject, namely how one extends this to include matter models.
Random perturbations in dynamical systems are an ubiquitous phenomenon in many fields of science. These perturbations, e.g. represented by noise, lead to an unpredictable movement of the trajectories of the system. It is of great interest to control such a random motion or other noise-induced effects.
The first part of the talk consists of an introduction to the theory of dynamical systems and related concepts as stability analysis and bifurcation theory. Then I will proceed with the investigation of dynamical systems subjected to Gaussian white noise. In the last part, I will show how noise effects can be modulated by time-delayed feedback control.
A graph G=(V,E) is d-sparse if each subset X\subseteq V with
|X|\geq d induces at most d|X|-{{d+1}\choose{2}} edges in G.
Laman showed in 1970 that a necessary and sufficient condition for a
realisation of G as a generic bar-and-joint framework in 2-dimensions
to be rigid is that G should have a 2-sparse subgraph with
2|V|-3 edges. Although Laman's theorem does not hold when d>2,
Cheng and Sitharam recently showed that if G is generically
rigid in 3-dimensions then every maximal 3-sparse subgraph of G
must have 3|V|-6 edges. We extend their result to all d\leq 11 by
showing that if G is generically rigid in d-dimensions then every
maximal d-sparse subgraph of G must have
d|V|-{{d+1}\choose{2}} edges.
In his seminal paper of 1931, Dirac posited the existence of the magnetic monopole in an attempt to explain the quantisation of electric charge. Since then, the monopole has played a central role in questions related to inflationary cosmology, particle confinement and electromagnetic duality, to name just a few. On the other hand, the subject of topological solitons has been one of great interest in various branches of pure mathematics, from integrable systems to algebraic geometry. In this talk I will describe the topological characterisations of the monopole, touching on the mathematics of fibre bundles and the related Chern invariants, and address the question of which gauge theories allow for their existence. If time permits, I will discuss various generalisations of the magnetic monopole, such as instantons, and some exact solutions of the governing field equations.
Network science is very useful when analysing complex structures like the human brain. I will be talking about community detection (which is a method for partitioning networks) and the mathematics and algorithms it involves.The main measure of the community structure I will discuss is called "flexibility", which gives information to how a nodes connections to other nodes changes as time progresses. I will then present some results obtained after applying these methods to a data set from Professor Bullmore’s lab at Cambridge. The data consists of parcellated fMRI scans of a group of control subjects, and a group of patients diagnosed with schizophrenia, under two different drugs.
We will survey the concept of Decision Tree Complexity. Loosely speaking, it is the study of how much of a function's input do you need to know to determine its output. Time permitting, we will discuss some old conjectures in the area and touch briefly on connections with topological fixed point theorems.
A surprisingly large number of physical theories (including general relativity) can be written as a Lagrangian system, for which Noether's theorem provides a powerful relation between symmetries and conservation laws. In this talk, I will introduce the Lagrangian formulation and illustrate the theorem for some simple Lagrangians. Then I will outline how the theorem can be used to find conserved quantities in general relativity, and characterise black hole spacetimes ("black holes have no hair").
One of the first questions in rigidity theory was: Given a structure made of bars of fixed length, which are connected at their endpoints by flexible joints, when is the resulting structure flexible? This is surprisingly difficult to answer in 3-dimensions, and is still open. However, much more is known in 2-dimensions.
We can model such a 2-dimensional structure by a framework (G,p) where G is a graph whose vertices represent joints, and whose edges represent bars; and p which maps the vertices of G to the plane. So long as the coordinates in p are algebraically independent over the rationals, we can determine whether our structure is rigid or flexible by just looking at the graph. However, most real-world structures have a large amount of symmetry, and so it is not sensible to assume that p is algebraically independent over the rationals.
In this talk I will give an overview of recent work in characterising rigidity and flexibility for symmetric frameworks. And, time allowing, explain how I am currently trying to extend these results to model the symmetry of structures built in CAD-software, where edges can have constraints on their angle as well as their length.
The combinatorics of partitions play a crucial role in understanding the symmetric group and its representations. To each partition, we can associate a Specht module, which is a representation of the symmetric group. In positive characteristic, there are many open questions about these Specht modules. In this talk we will touch on classifying the "blocks" of the symmetric group and determine when two such modules belong to the same block by Nakayama's Conjecture.
The Einstein field equations can be thought in different ways. In the first instance, they are a system of non-linear
second order partial differential equations for the metric coefficients with no evident structure i.e. they are not
a priori hyperbolic nor parabolic nor elliptic. In 1952 Yvonne Choquet-Bruhat taught us that choosing an harmonic
coordinates one can recast the Einstein field equations as a system of quasilinear wave equations (i.e. hyperbolic)
for the metric coefficients. Since then, great efforts have been done to use the notions established in Choquet-Bruhat's seminal work to address general questions about solutions to the Einstein field equations, in particular existence and stability. In this regard, one of the main ways (but not the only one!) to make inroads in these type of problems is to use the so-called vector field methods. In this talk we will first motivate the ubiquitous presence of wave equations in General Relativity and then we will introduce some of the main ideas used in vector field methods.
The conformal field equations have proven to be an extremely useful tool in research in general relativity, especially in analysing properties of spacetimes at infinity. A currently ongoing area of research is whether or it is possible to formulate conformal systems described by the conformal field equations as an initial value problem. It turns out that the answer to this question is yes, at least in the case of vacuum spacetimes, as one can recast the vacuum version of the Conformal field equations as a system of wave equations (and wave equations are essential in formulating an initial value problem for any system). The question of whether or not it is possible to do the same thing with matter is still an open problem.
We'll begin the talk by briefly reviewing and summarizing the key ideas in General Relativity including some of the problems in GR and how these are solved with wave equations and conformal methods. From there we'll look at how one can derive a set of quasilinear wave equations for conformal spacetime models coupled to trace-free matter. The next part will be to show that any solution to these wave equations is also a solution to the field equations, which is achieved using a technique called the propagation of the constraints. Finally we'll discuss some applications of this method, including stability analysis of spacetimes coupled to trace-free matter.
I will start by defining the locally free semigroup of n generators and show that it has the algebraic structure of so-called heaps of pieces, where the pieces are dimers. After this I will outline how the generating function of this special kind of heaps can be found and will explain the connection between heap configurations and the weights of states in the stationary state of the asymmetric simple exclusion process (ASEP), a stochastic process describing the transport of particles along a discrete line.
"Do not worry about your difficulties in Mathematics, I can assure you mine are still greater.”
Einstein never agreed with quantum mechanics, right up until the day he died. As of 1924 he was tormented with the knowledge that, on a subatomic level, space appeared 'fuzzy'. This was due to quantum effects and was encapsulated by the famous Heisenberg uncertainty principle. For him, this meant that the mathematical concept of a point in space and time does not work, the geometry of the real world appears quantum in structure. A search for the correct structure of space and time leads one into the realm of non-commutative geometry.
Non-commutative geometry aims to extend classical notions of geometry to situations where the underlying algebra is non-commutative, as is the case for the matrix algebra that occurs in quantum mechanics.
In my talk I aim to give a brief introduction to a form of non-commutative geometry known as deformation quantisation. With this I will show how one can solve the problem of quantising Riemannian and other differential geometries to first order deformation (in a Planck-scale parameter). I also hope (time permitted) to discuss how this method of 'semiquantisation' could be used to quantise parallelizable manifolds, namely the 3-sphere and the 7-sphere.
In this talk I will introduce an area of combinatorics called Ramsey Theory. Questions in Ramsey Theory are usually of the form:
"Suppose we are going to take some mathematical structure and cut it into pieces. How big must this structure be to guarantee one of the pieces has some property?"
This talk will include some nice colourful diagrams and assume no prior knowledge of Ramsey Theory.
The visibility graph is a method of turning a time series in to a graph. I will be explaining the method and motivation, talking about some cool and interesting results, with some application to financial time series. I'll also be talking about Feigenbaum graphs, which are super amazing visibility graphs related to the logistic map.
Algebraic geometry has a reputation of being incredibly abstract, dealing with increasingly intricate and subtle objects in order to solve harder and harder problems. Tropical geometry blew this apart around ten years ago by showing these abstract objects could be reformulated as very real (even drawable) polyhedral complexes that could solve certain problems algebraic geometers could not previously. In this talk, I'll give an introduction as to how these complexes are constructed and why they're so nice to work with. No previous knowledge required, expect lots of drawings of polygons and spider webs.
Partial Differential Equations are, in general, notoriously difficult to solve exactly, with many solution
generating techniques applicable only to a small class of problems. In this talk I will describe an algorithmic
approach to generating PDE solutions, called the "Symmetry Method", and discuss some of the issues which
arise in its application to covariant equations; in particular, the equations of General Relativity.
No previous knowledge required!
There are many motivations from physics to construct quantum group. You can think quantum group as a notion of quantum symmetry, for which it is an algebra with additional structures that can be the role of symmetry such as duplication. It turns out that quantum symmetry has a Hopf algebra structures. Thus algebraically speaking, quantum group is a family of non-commutative non-cocommutative Hopf algebras.
In this talk, I will describe how we construct Hopf algebra from usual algebra, and give a simplest example of quantum groups.
The 27-dimensional exceptional Jordan algebra (or Albert algebra) of 3x3 Hermitian matrices over the octonions turns out to be a fruitful object for some exceptional groups of Lie type. It was shown that the group of the automorphisms of the Jordan algebra over any field is isomorphic to the Chevalley group F_4 over the same field. The stabilizer of the determinant, which is represented by a certain cubic form, is actually a group of type E_6, and its twisted version is obtained by considering those elements of E_6 over the quadratic field, which preserve certain Hermitian form. In this talk we are going to look at the constructions of these finite groups defined by the Albert algebra.
Since its origins in the study of fluid dynamics, dimensional analysis has been a powerful tool for obtaining remarkable insights about the behaviour of physical systems. In this sense, I will present Buckingham Pi Theorem as a useful (and some people would say 'magical') algorithm to obtain physically relevant quantities in cases where dynamical equations are not known. Also, I will discuss it briefly in more modern mathematical terms involving Lie Algebras and rescaling groups. For the sake of clarity, I will work out several examples where this method can be applied, ranging from the Pythagorean theorem to black holes.
Modelling the motion of agents in a system as random walks is often a useful approximation. Modelling a set of agents that are in some sense connected as a network is also useful in a number of settings (such as modelling the spread of diseases). Given some notion of this ‘connectedness’ between random walkers we can attempt to create a network, and hopefully gain some insight into the system we are modelling from its properties and how they change over time. I will attempt to show how a network of such Brownian walkers can be formed and improved, then show how long one might expect some properties of the network to persist for.
Equation-free methods make possible an analysis of the evolution of a few coarse-grained or macroscopic quantities for a detailed and realistic model with a large number of fine-grained or microscopic variables, even though no equations are explicitly given on the macroscopic level. This will facilitate a study of how the model behaviour depends on parameter values including an understanding of transitions between different types of qualitative behaviour. These methods are introduced and explained for emergence of oscillatory pedestrian counter flow in a corridor with a narrow door. In addition, the concept of control-based continuation is combined with equation-free techniques.
McDonalds sell chicken nuggets in boxes of 6, 9 and 20 - what is the largest number of nuggets you can't buy in one transaction? This seemingly simple and silly question has been studied for over 100 years using a variety of techniques such as polyhedral geometry, semigroup theory and combinatorial optimization. We will instead use a relatively new approach via the commutative algebra of lattice ideals and see exactly how homological data can help you with your fast food purchases.
Often when considering games played on graphs it can be difficult to prove results which are applicable to all graphs (see for instance the notoriously difficult conjecture of Meyniel concerning the Cops and Robbers game). In these situations it can be instructive to think instead about the behaviour on a 'typical' graph. That is, to choose a graph randomly and investigate the likely behaviour.
I will discuss some combination of the Cops and Robbers game and my work on the Revolutionaries and Spies game using no probabilistic machinery more difficult than a union bound.
Questions like:
-why identical twins aren't genetically identical?
-can we quantify how much time does the virus will take to infect the
cell once transmitted to the body
are something which have haunted researchers for a long time. The answers
to all these questions are hidden in the fundamental biological phenomenon
called gene expression which is an intrinsically stochastic process. An
expression of gene can simply described as the journey of DNA to proteins.
Modelling gene expression using the tools of stochastic process can shed
some light to the above questions. To be more precise, we will be talking
about Fano factor(variance/mean), stochastic-di erential equation(SDE), mas-
ter equation, Fokker-Planck equation(FPE) and mean rst passage time(MFPT)
in the context of gene expression(without delving into the intricate biological
details).
Piecewise-smooth dynamical systems have attracted a lot of interest in the last decade. Whereas deterministic models have been studied intensely, their stochastic counterpart is still in its infancy. Systems with dry friction subjected to stochastic perturbations are prominent examples of piecewise-smooth stochastic systems. There are only a few cases known, where exact results can be obtained.
In the first part of my talk I will give a short introduction of concepts and methods from statistical physics, e.g. the Langevin equation and the Fokker-Planck equation. Then I will talk about dry friction models subjected to Gaussian white noise. Finally I will present some recent results for a dry friction model and coloured (exponentially correlated) noise.
Simplicial complexes can be thought of as a generalisation of networks able to encode interactions occurring between two or more nodes via links, triangles, tetrahedra etc. They are emerging as a tool for describing networks with an abundant numbers of short loops and large clustering coefficients and and have already been used to describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. They are also ideal mathematical objects for the discretisation of geometry and so may also open up possibilities for uncovering the hidden geometries of networks.
In this talk I’ll give a brief introduction to the field of networks, explaining the challenges faced and giving some motivation for the use of simplicial complexes. I’ll then introduce two ‘maximum entropy’ models of simplicial complexes based respectively on the placing of hard and soft constraints on their local structural properties.
The models are investigated from a statistical mechanics perspective, and an interesting relation between their entropies is identified. This relation allows us to obtain an expression for the total number of simplicial complexes that satisfy a set of constraints on their structure.
Braided Hopf algebra is a generalisation of quantum groups (also known as ordinary Hopf algebra), means that it is a Hopf algebra that live in braided monoidal tensor categories, and they are commute and cocommute up to a braiding map. Associated to braided group B in category of A-module over dual quantum groups, we construct a new dual quantum groups U(B^(op), A, B*), where "op" means the multiplication is braided-opposite, and B* is the dual of B. Application to this construction is to get the description of the basis of Cq[SL2], which is a very hard combinatorial problem.
In this talk, I will recall the definition of ordinary Hopf algebra and some related definitions and examples, then introduce the concept of braided Hopf algebra. Afterward, on the short time, I will state the new construction of dual quantum groups, and some related result.
A key feature of many classical theories is that a complete knowledge of the present state of a system is sufficient to know the exact past and future behaviour of the system. As a classical theory of gravity, is this also true in general relativity? This question is at the heart of one of the major unsolved problems in mathematical relativity, the so-called strong cosmic censorship hypothesis. In this talk, I will introduce the main ideas of general relativity, show how initial data is represented, and explain the role of the strong cosmic censorship hypothesis in preserving determinism.
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids.
The talk will present a proof of the Alon-Boppana result due to A.Nilli, providing a motivation for the definition of Ramanujan graphs. This will be followed by a sketch of the result due to Marcus, Spielman and Srivastava on the existence of infinite families of Ramanujan graphs of arbitrary degree.
The octonions are an 8-dimensional normed real division algebra which are often overlooked due to their non-associativity. This talk will hopefully give an intuitive introduction to them and some of the challenges related to working with them, specifically in relation to their projective plane.
Most of us learnt at a young age that the game of 'Tic-tac-toe' (or 'noughts and crosses') ends in a draw, at least if both players play with a modicum of sense. At that stage, most of us probably wrote the game off, but perhaps we shouldn't have been so hasty. There are several natural generalisations of the game which are not so easy to play, and many basic questions about these are still unsolved. We will discuss some of the theory behind these and other positional games, such as 'Hex' and 'Sim', including providing answers to questions like 'how do you steal a strategy?', 'why is avoiding harder than achieving?' and 'who or what is Snaky?'
Tropical Geometry arose in the late 90s as a way to studying algebraic geometry as polyhedra. However, it made news headlines in 2007 when economist Paul Klemperer used it to design an auction to protect banks and building societies against the financial crisis. Since then, it has found huge applications in multiple disciplines including combinatorics, game theory and, unsurprisingly, algebraic geometry. We'll give a quick introduction to the subject then focus on a couple of applications it has in others.
Discrete autoregressive processes (or DAR(p)) provide a simple way of generating time series with a controlled amount of memory, and so can be used to build time varying networks that themselves have memory. If we take these time varying networks and run processes on top of them we can then look at how these processes react to a change in the influence of memory. Here this network process will take the form of a simple epidemic spreading model and we will see that memory can either speed up or slow down the infections passage through the network.
The generalised sum of remainders map over a finite non-empty set of positive integers is a map which computes the sum of the remainders upon dividing its argument by every number in the set. The dynamics of this map, i.e., the behaviour of the sequence of numbers generated by iterating the map, is being studied. This talk will present some of the recent results, which are hopefully understandable even to audiences with minimal prior knowledge of dynamical systems and number theory. This is a joint work with Ryan Kasyfil Aziz.
Covariate-adjusted response-adaptive (CARA) designs use available responses to skew the treatment alloca-
tion in an ongoing clinical trial in favour of the treatment arm found at an interim stage to be best for a
patient's covariate pro le. There has recently been extensive research on CARA designs mainly involving
binary responses. Though exponential survival responses have also been considered, the constant hazard
property of the exponential model makes the mean residual life for patients constant, making it too restric-
tive for wide-ranging applicability. To overcome this limitation, designs are developed for Weibull distributed
survival responses by deriving two variants of optimal designs based on an optimality criterion. The optimal
designs are based on the doubly-adaptive biased coin design (DBCD) in one case, and the ecient randomised
adaptive design (ERADE) in the other. The observed treatment allocation proportions for these designs con-
verge to the expected targeted values, which are derived based on constrained optimization problems. The
merits of these two optimal designs are also discussed. Given the treatment allocation history, response his-
tories, previous covariate information and the covariate pro le of the incoming patient, an expression for the
conditional probability of a patient being allocated to a particular treatment has been obtained. To apply
such designs, the treatment allocation probabilities are sequentially modi ed based on the history of previous
patients' treatment assignments, responses, covariates and the covariates of the new patient.
The ERADE is preferable to the DBCD when the main objective is to minimise the variance of the al-
location procedure. However, the former procedure being discrete tends to be slower in converging towards
the expected target allocation proportion. Since the ERADE provides a design with minimum variance,
it is better than the CARA design based on the DBCD as far as the power of the Wald test for testing
treatment di erences is concerned. An extensive simulation study of the operating characteristics of the pro-
posed designs supports these ndings. It is concluded that the proposed CARA procedures can be suitable
alternatives to the traditional balanced randomization designs in survival trials, provided that response data
are available during the recruitment phase to enable adaptations to the designs.
Representation Theory is one of the areas of mathematics in which a wide range of techniques can be applied, from combinatorics and differential geometry to category theory. Different abstract algebraic objects can be understood by representations of them, namely matrices (linear transformations of vector spaces). Thus, we revise the problem of studying abstract algebra into the considerably easier problem of studying linear algebra.
We will start with a quick review on the purpose and applications of Representation Theory. Following this, we will address the representations of finite groups and one of its current open problems: describe and understand the composition factors of its modular representations. Then, we will focus on the modular representation theory of S_n and an object of significant importance for its study over an arbitrary field: the Specht modules.
General Relativity constitutes the better theory we have so far to understand how gravity works at large
scales. Mathematically, it is formulated using the language of differential geometry and topology, but PDE theory and analysis come into play when the dynamics are studied. In this talk I will give a brief introduction to the initial value problem in relativity and how conformal methods provide with a nice approach to it. In particular, the anti-de Sitter spacetime will be taken as an example of what happens when initial data is not sufficient to establish a well-posed problem. Finally, some comments will be made about why theoretical physicists are so interested in understanding it, even though it does not represent a physical world.
Given a type of object, a common goal in mathematics is to classify them. Geometry aims to do this via a moduli space, a parameter space for the objects that "reflects the geometric data in some nice way". Such hand-wavy intuition turns out to be hard to define, and spaces often become incredibly messy very quickly. As a result, the study of moduli spaces has a (kind of fair) reputation as an impenetrable mess. In this talk, we'll cut through the formal definitions with lots of concrete examples and give some insight into why such spaces can very quickly spiral out of hand. We'll conclude with an example of a particularly nice (ie. drawable) moduli space: the space of metric trees.
We derive bounds on the size of regular induced subgraphs in certain regular graphs.
Model theory is a branch of mathematical logic which is concerned with interpreting mathematical statements in different algebraic structures. It has a reputation for being abstract and difficult to grasp, but this talk will aim to show the power of considering a model theoretic perspective. I will give an introduction to the language of model theory then give an example of a proof made much simpler using these techniques. If time permits I will also give an idea of what aspects of model theory I use in my work.
A novel approach is being considered for dose escalation in phase I-II clinical trials.
Along with efficacy and toxicity as endpoints, the pharmacokinetic (PK) and pharmacodynamic (PD) information is also being considered. Patan and Bogacka (2011) conducted some simulation studies taking into account such information. However, they only considered fixed effects models for the PK/PD effects. We are considering mixed effects model for the PK/PD.
The population Fisher information matrix for PK/PD model is found by following
Bazzoli, Retout, and Mentré (2009). For the dose-response model, the approach of
Zhang, Sargent, and Mandrekar (2006) is followed, who consider a trinomial response
y = (y0; y1; y2)T for each patient, where y0 is a neutral response, y1 is an efficacy endpoint and y2 is a toxicity endpoint. Following the assignment of the lowest dose to a cohort of patients, the trinomial response is observed for each patient. PK and PD responses are measured at the D-optimal time points.
The dose-response curves are updated sequentially. For the next cohort, the dose is
selected in such a way so that the estimated probability of efficacy is a maximum, subject
to the condition that the estimated probability of toxicity is smaller than a pre-specied
value and also the efficiency of estimation of the PK/PD parameters is at least a desired
level. When the trial is finished according to some rules, a complete analysis of the data
is carried out and the dose is chosen to recommend for further studies. Thus, an adaptive design is being implemented.
An opportunity to meet other PhD students in the department!
Let G be a multigraph without loop and F be a subgraph of G. An F-Tutte trail of G is a trail H of G such that
(i) Each component of G\V (H) has at most three edges connecting it to H.
(ii) Each component of G\V (H) containing a vertex of F has at most two edges connecting it to H.
In this talk, I will show that 2-edge connected plane graph has a Tutte trail. I also talk about the relation between Tutte trail and Hamiltonian problem.
We consider a one-parameter family of invertible maps of a two-dimensional lattice, obtained by applying round-off to planar rotations. We let the angle of rotation approach π /2 and show that the limit of vanishing discretisation is described by an integrable piecewise-smooth Hamiltonian flow, whereby the plane foliates into families of invariant polygons with an increasing number of sides. The round-off perturbation introduces KAM-type phenomena: a positive fraction of the unperturbed curves survives, and locally this fraction converges to a rational number strictly less than 1.
In Graph Theory, for d ≥ 1, s ≥ 0, a (d, d + s) - graph is a graph whose degrees all lie in the interval {d, d + 1, . . ., d + s}. For r ≥ 1, a ≥ 0 an (r, r + a) -factor of a graph G is a spanning (r, r + a) - subgraph of G. An (r, r + a) - factorization of a graph G is a decomposition of G into edge -disjoint (r, r + a) - factors.
In this talk, I will provide upper and lower bounds for the simple graph (r, s, a, t) - threshold number σ (r, s, a, t), and for the multigraph (r, s, a, t) - threshold number µ(r, s, a, t). We also determine the pseudograph (r, s, a, t) – threshold number π(r, s, a, t).
In Graph Theory, for d ≥ 1, s ≥ 0, a (d, d + s) - graph is a graph whose degrees all lie in the interval {d, d + 1, . . ., d + s}. For r ≥ 1, a ≥ 0 an (r, r + a) -factor of a graph G is a spanning (r, r + a) - subgraph of G. An (r, r + a) - factorization of a graph G is a decomposition of G into edge -disjoint (r, r + a) - factors.
In this talk, I will give more general results and different techniques used to find the simple graph (r, s, a, t) - threshold number.
The importance of the probabilistic method in Combinatorics can hardly be overstated. Introducing the method, Erdős demonstrated its value by giving non-constructive proofs of the existence of graphs with certain exceptional properties using little more than elementary probability. Some of the results obtained have scarcely been improved in the intervening 60 years. Assuming no familiarity with graph theory I will give a gentle introduction to the method, surveying some early approachable results, before talking briefly about some recent work of Conlon, Fox and Sudakov on Sidorenko's conjecture.
In 1917 Sōichi Kakeya asked the following question: "What is the least area needed to continuously rotate a needle by 360°?". If this is a question you have not came across before then you may enjoy trying to see how well you can do (by a needle we mean a unit line segment, and we may move the segment in 2d). We will look at Besicovitch's (1928) arguably surprising answer to this question and consider some related questions of a similar flavour. In particular we consider subsets of d-dimensional space which contain a unit line segment in all directions and we replace area (or more accurately measure) by some other measure of size such as Minkowski dimension or, in the case of the finite field analogue, cardinality. Time permitting we may discuss the resolution of the Kakeya conjecture for d=2 or talk about Zeev Dvir's recent (2008) solution of the finite field analogue to the Kakeya conjecture or even have time to look at a gif of a line segment rotating.
In this talk I will give an introduction to the topic of Extremal Graph theory. We will start by looking at Mantel's theorem, which tells us the maximum number of edges a graph with no triangles can have. We will then see how this generalises to Turán's theorem, which deals with larger complete graphs than triangles. If we have enough time, we may also talk about the Erdős–Stone theorem or Dirac's theorem,
No prior knowledge of graph theory will be assumed.
What does saturation mean in the context of graph theory? We say a big graph G is saturated with a small graph F if the graph G doesn't contain a copy of F, but adding any edge to G creates a copy of F. A lot is known about the maximum number of edges a saturated graph can have, but what about the minimum number of edges? It turns out this number is a slippery beast, and I'll talk about why. And then, because why not, we'll see if we can generalise these ideas to hypergraphs.
In this talk I shall give an introduction to noncommutative Riemannian geometry based on quantum groups and its proposed role in quantum gravity. In doing so I shall discuss the notion of a differential anomaly and how we can resolve this by increasing the dimension of the cotangent bundle beyond the classical. One such way this is done is by a mechanism known as 'spontaneous time generation', where we demonstrate that if time did not exist we would be forced to create it if space is noncommutative.
This talk is based on work completed in the mid 2000's by my supervisor S. Majid.
C*- algebras were introduced, as natural abstractions of matrices acting on Hilbert spaces, by Murray and von Neumann in 1935 in order to explain certain physical observables of quantum mechanics. Today, they form an integral part of pure mathematics as they are able to generalize and improve on many of the notions we have for classic operators: eigenvalues, rank, projections etc. In this talk I will prove some spectral ("eigenvalue") theoretic results for Banach and C* - algebras.
I will start off by providing plenty of examples of C* and Banach algebras, as well as their spectral properties. Then I will move on to prove one of the most important spectral theorems for Banach algebras. If time permits, I will prove the Spectral Mapping Theorem and the existence of a continuous function calculus for self - adjoint operators of C* - algebras.
I aim to make the talk accessible for all attending, but knowledge of basic Functional Analysis will make you a little happier.
Nowadays it is well known that there is no general solution by radicals for a polynomial f with arbitrary coefficients (lying in a field characteristic 0) when deg(f)>4. This was proved by Abel using Galois Theory. We will start by giving a brief introduction to this theory which will then use to show the cases in which a polynomial of degree 5 is solvable by radicals. If time allows it, we will also work out a characterisation on solvable quintics in terms of the resolvent and the discriminant associated with the polynomial.
The modelling of the spread of epidemics forms a large and active area of research. The onset of a zombie apocalypse seems like an ideal time to make use of this work… if we are not prepared then how can we hope to survive? But zombies don’t seem to work like the flu, so we have to make some adaptations to the older models. I will go through some cases of how traditional epidemics are studied, and then how the models can be adapted to help forecast the end of the world.
Santa is lost in Manhattan, and needs to escape to deliver gifts to Queen Mary Maths PhD students before the night is out! Will he make it?! Join us on an exploration of strange traffic-inducing Xmas traditions, a city that literally goes on forever, and some gentle algebra representation theory, as we journey to the very centre of (the) Manhattan (algebra)
The Ricci flow is a powerful tool in geometry and topology. Most famously, it has been used by Perelman to resolve the Poincaré conjecture. While the Ricci flow can be defined in any dimension, I will focus on the case of surfaces. In this setting, the Ricci flow successfully deforms any given surface into a surface of constant curvature.
After briefly reviewing the geometry of surfaces, I will define the Ricci flow and explain how it achieves this on the sphere, following a proof of Hamilton.
Finally, I will connect this to my own attempts at understanding flows closely related geometric flows on surfaces with more singular behavior.
Several popular linear machine learning algorithms can be adapted to a
nonlinear setting by means of the kernel trick. The resulting kernel
methods operate implicitly in a high-dimensional feature space without
the need to carry out expensive computations in said space. However,
they still require the computation, storage, and manipulation of a
matrix whose size scales with the size of the dataset.
As more data is readily available to researchers than ever, techniques
have been developed to work with a low-rank approximation of the kernel
matrix instead, sampling only a small subset of relevant columns. This
talk shall serve as an introduction to these methods.
Abstract: In the late 1950s models were introduced to study random graphs and through the probabilistic method understand properties of deterministic graphs. It wasn't until the mid 2000s however that a similar treatment was first given to study the topological properties of large random simplicial complexes. We will begin with a quick overview of the Gilbert random graph before looking at several models of random simplicial complexes and surveying some (hopefully) interesting results.
Abstract: Finding the symmetries a manifold possesses is of great interest as they can provide with some insights into its global properties. Remarkably, Noether's theorem states that each symmetry generates a conserved quantity; in particular, these symmetries are encoded by the so-called Killing vectors. This results particularly useful for mechanical systems as one is able to integrate the equations of motion via quantities like energy or angular momentum.
A review of the basic notions of differential geometry, making emphasis on flows on manifolds, will lead to the concept of Killing vectors and how they are related to the symmetries of a spacetime. Additionally, I will briefly discuss my work on understanding these kind of vectors in the context of General Relativity, specifically when conformal transformations are performed.
Abstract: Given a finite point set X in some Euclidean space one can build a simplicial complex on X (in different ways) by fixing a scale constant ‘r’. The homology groups of the simplicial complexes obtained from all possible scales ‘r’ can be used to characterize X. In fact it is possible to compute the ranks of the homology groups (for a fixed homological degree k) for all ‘r’ with an efficient algorithm, whose output is known as a persistence diagram. These diagrams can be considered ‘’good summaries’’ of the topological and geometric information of X. In my talk I will go through the procedure used to obtain persistence diagrams, hopefully give a motivation of to the claim of them being ‘’good summaries’’, and present the research questions I am studying.
Given a locally compact Hausdorff group G and a unitary representation $\pi$, we can extend it to the *-algebra $L^1(G)$, consisting of all absolutely integrable functions with respect to the left Haar measure, to get a *-representation. In this way, we can define two classical group C*-algebras related to G. As a result, the C*-algebras of a group encodes all the information about unitary representations of a group. In general, the two group C*-algebras are different. Actually, we can construct a distinguished homomorphism between them two and this homomorphism is an isomorphism if and only if the group is amenable.
I will introduce the definitions of some relevant concept, give some examples and special cases to help understand the concept and finally present the central result without proof.
Extremal combinatorics is typically about extremising combinatorial properties of constrained set systems. The most recent problem I have been working on is minimising the number of distinct unions we can achieve when choosing a fixed size family of k-subsets of [n]. Though not the most efficient description of my results, I will give a chronological account of my research, in the hope that it gives a more expressive exposition of the problem and that it might be interesting to you/cathartic to me to have the true nature of my research honestly accounted. I will also include a rapid introduction to the compression method and its limitations so that people can come away from the talk with some useful mathematics.
The tropical semiring is the real numbers with the operations min and plus. This odd choice of ring crops up surprisingly regularly in many fields, especially optimisation and computer science. When we try to do geometry over this ring, some surprising geometry and combinatorics falls out. In this talk we shall examine arrangements of hyperplanes over the tropical semiring and discover a link to classical graph theory via matchings. No prior knowledge required (or expected!).
I will give an overview of the results Peter Cameron, David Ellis and I have got on the following, self-contained, problem (and some extensions): what is the smallest a subspace of F_2^n can be such that its cyclic shifts cover all of F_2^n? The talk will contain combinatorics, basic representation and Galois theory, proofs, and (ever popular) counterexamples.
Thermodynamic uncertainty relation and bounds on the time-integrated current fluctuations are the recent advances in non-equilibrium thermodynamics but have, so far, been derived only for Markovian processes. We explore the validity of one particular result (which states that the entropy production rate bounds statistical errors in current fluctuations) and discuss some open questions in the context of a simple non-Markovian toy model - a discrete-time asymmetric random walk on a ring with one-step memory.
We will start off by discussing what is probably the most famous example of sequential optimisation problem - the secretary problem, and then naturally moving on to more sophisticated, and, inevitably, more complex problems that are of a particular interest in the field. The latter are approached by means of dynamic programming, which often lead to Bellman optimality equations with no closed-form solutions. Consequently, we will be reviewing asymptotic properties of the solutions to selected problems.
I will introduce the notion of a groupoid, and using lots of examples I will demonstrate how many classic objects in mathematics (such as groups, graphs, dynamical systems, relations ...) give rise to groupoids. I will then try to convince you that, algebraically, groupoids are very easy to classify, and so in order to have a richer theory I will introduce the étale topology on groupoids. If time permits I will briefly (and very gently) discuss how one can construct C*-algebras from groupoids and how this all relates to my current work on Cartan subalgebras of C*-algebras.
In this marvellous talk, I will give a simple proof of the Riemann Hypo... ehm, take two, action!
In this astonishing seminar, I will answer one of the most crucial questions of our life: why is the pizza round? Then I will give some intuition for how to solve the same problem from the point of view of a not-so-hungry alien living in another universe.
Category theory can sometimes have a reputation for being abstract and complicated; this talk will aim to provide an easy introduction with no prior knowledge needed nor assumed. I will try to give motivation for learning a little bit of category theory, then define relevant notions from scratch (categories, functors, natural transformations). The main goal is for us to understand the first big result: Yoneda's Lemma.
You may know what 'factorizing a cube' usually means, but forget all that number theory now: this is graph theory, where a 'cube' is the graph formed from the vertices and edges of an n-dimensional cube, and a 'factorization' is a partition of the edges into spanning subgraphs. I will give an introduction to factorizations of graphs, focusing in particular on 1-factorizations on the complete graph and the cube. This will include some nice results (the wonderful Walecki construction!), some less nice results, and some open problems. There will also be a surprise. We will finish with a (literal) sketch of my recent results on factorizing the cube.
Named after the famous mathematicians Issai Scary and Hermann Wail, Scary-Wail
duality describes a positively ghoulish relationship between the representations
of the symmetric group S_n, and the general linear group GL(k). I will attempt
to explain this nightmarish affair, with the ever-present spectre of the
Manhattan algebra looming in the shadows.
Luckily, during this talk, we will not be buzzing around but we will focus on a way more interesting problem: how likely is it that your trousers' zipper
inevitably decides to mock you in a crowd and comes naturally down!?
From a first sight this looks like a bland problem to face (one can always start dressing in trousers with buttons!), but it will instead be an interesting introduction
to the fascinating world of statistical mechanics as part of theoretical physics.
We will study how entropy, and its convexity, are related to the zipper problem, realizing that in the (thermodynamic) limit of an infinite long zip, a phase transition
appears, leaving us without trousers, and the need to formulate a new calculation tool for replacing the Legendre transform.
Recently, seamless phase or combined phase clinical trials have become more popular for reducing the estimated time it takes to complete the development of drugs. A seamless phase II/III clinical trial's main purpose is to compare a number of drugs or doses in a single trial conducted in two stages. The first stage studies all of the experimental
doses or drugs and selects the population with the largest sample mean. This selected treatment will continue to the second stage for further analysis. The problem is to obtain the best estimator of the mean of the selected population. In the analysis of two-stage trials, the issue of estimation bias introduced by treatment selection has long been known. The sample mean for the selected population and the maximum likelihood estimator (MLE) are biased estimators of the corresponding population mean, due to combining data from both stages. To correct for the bias efficiently, the uniformly minimum variance conditionally unbiased estimator (UMVCUE) has been derived for trial designs with normally distributed data and unequal stage one and stage two sample sizes. Moreover, formulae for the variances of the MLE and the UMVCUE
have been obtained and are compared. Finally, simulation results for the bias of the MLE are presented.
Linear programming is a well studied branch of optimisation where everything is linear. Linear is usually code for 'easy', but this isn't quite the case for linear programming, as fast algorithms tend to break for some special examples. This led Smale to list the complexity of linear programming as one of the 17 most important problems in modern mathematics. In this talk I'll outline some of the main algorithms and results from linear programming, and show how a new branch of maths is shedding light on this old problem in surprising ways.
Heavy particles suspended in turbulent fluid flows, so-called turbulent aerosols, are common in Nature and in technological applications. A prominent example is rain droplets in turbulent clouds. Due to their inertia, ensembles of aerosol particles distribute inhomogeneously over space and can develop large relative velocities at small separations.
We use statistical models that mimic turbulent flow by means of Gaussian random velocity fields to describe these systems. Compared to models that involve actual turbulence, our statistical models are simpler to study and allow for an analytical treatment in certain limits. Despite their simplicity, statistical models qualitatively explain the results of direct numerical simulations and experiments.
In my talk, I will discuss how methods of non-equilibrium statistical mechanics and large deviation theory are used to study statistical models of heavy particles in turbulence. The one-dimensional versions of the models will serve as simplified playgrounds to create intuition for, and give important insights into the behaviour of higher dimensional particle systems.
As the name 'controllability' suggests, we want to control something. Basically, we want to take the solution of the considered ODE from its initial state to a chosen final state, using a suitable control. I will show that this problem is equivalent to the notion of observability, and derive a condition specifying when the ODE is controllable.
Control theory has applications in many other subjects such as Physics, Biology, Economics etc. But, don’t worry, I will not address any of those in the talk.
In this talk we will study the dynamical behaviour of a conservative, nonlinear DE system of coupled oscillators governed by a "special" potential derived from Hollomon's law. This characterizes the phenomenon known in engineering as "Work Hardening". We will start with the quite famous simple harmonic oscillator and then proceed with generalizing this model, to obtain its solution in the form of generalized trigonometric functions.
As an application, we study a two degrees of freedom problem. To better understand the underlying dynamics, we produce and study the corresponding Poincaré sections for a specific energy value.
This talk is focused on how a simple ODEs generalization (mathematicians just love doing that) can open the door to applications in other fields.
[1] Wei, Dongming, and Yu Liu. "Some generalized trigonometric sine functions and their applications". (2012).
[2] D. Shelupsky, "A generalization of the trigonometric functions", The American Mathematical Monthly, vol. 66, no. 10, pp. 879-884, December 1959.
[3] F. D. Burgoyne, "Generalized trigonometric functions", Mathematics of Computation, vol. 18, pp. 314-316, 1964.
In this talk we will survey the rich interplay between representation theory of the symmetric group $S_n$ and combinatorics of partitions of $n$. The last happen to be in bijection with the irreducible representations of $\F{S_n}$ when $\F$ is an algebraically closed field of null characteristic. Unfortunately, this may not be true when $Char(\F)>0$. We will have a look at what information about the simple modules of $\F{S_n}$ in positive characteristic it is possible to obtain by looking at partitions of $n$. Only elementary group theory will be needed to follow the results given here.
Linear operators and isometries are two important mathematical notions which are widely used in many branches of mathematics, especially in analysis-related branches. It's natural to consider the relation between them. In my talk, I will prove a theorem which shows you how far a linear operator on a Banach space is from an isometry. If time permits, I'd also like to give some cases in which an isometry is linear.
A good command of knowledge of real analysis and functional analysis will make it easier for you to understand the talk well.
Markov Chain Monte Carlo (MCMC) continues to be a significant application of Bayesian Statistical thought 50 years after its development at the dawn of the nuclear arms race. Modern revolutions in computing, including artificial intelligence, machine learning and stochastic optimization are founded upon this simple but powerful concept. Therefore MCMC continues to be a versatile tool of scientists, as its applications are limited only by the computing power available and the imagination of the Statistician, with Moore's Law largely negating the first limitation with time.
This presentation attempts to engage an audience who may be unfamiliar with MCMC by providing a concise introduction into the theoretical foundations of MCMC, such as "simple" Monte Carlo integration and relevant Markov Chain concepts such as Reversibility, Ergodicity, and the Detailed Balance equation. Furthermore, the discussion will seek to briefly elucidate upon common methodologies of MCMC in practice, such as the Metropolis-Hastings Algorithm and Gibbs Sampling, along with a brief exposure to practical implementation issues such as idealized acceptance rates and convergence.
In this talk we show that variables defined by a generic chaotic system can be used to approximate a stochastic process with a Gaussian distribution.
If we regard the initial values as random variables, any deterministic dynamical system generates a stochastic process, but generally this process is not a Gaussian, and is characterised by non-vanishing higher-order correlation functions. Those correlations can be described by sets of simple graphs, such as N-ary trees. We use a simple chaotic map to illustrate that the system gives a Gaussian probability density in a perturbative way.
A little knowledge of Bernoulli shift in dynamical systems and probability theory would be helpful, but the talk is friendly to everyone.
I will introduce the notion of topological groups, and in particular abelian locally compact Hausdorff groups (a.l.c.h.g.). Since general a.l.c.h.g.'s are a bit fiddly to work with and a mouthful to say, I'll focus on discrete and compact groups, and the important examples of the reals, the integers and the unit circle.
The dual of a topological group is its space of characters, which is related to the Fourier Transform where it is the frequency space. Pontryagin Duality states that every a.l.c.h.g. is canonically isomorphic to its double dual, which I will attempt to prove for compact and discrete groups.
Complexity arises in many systems where the interaction of their components results in evolutionary processes and often emergent behaviour at a macro level. Examples are our society, the brain, the financial market and ecosystems. The heterogeneity of the spatial organisation of these systems quite often carries important information about their function as a whole. A particularly interesting problem in this area is the quantification of spatial segregation, i.e., the tendency of people to cluster around uniform patches of spatial settings. Despite the vast existing literature on the topic, quantifying segregation is still problematic, mainly due to the granularity of the data used, the spatial scale of the neighbourhoods or measures with the presence of one or more free parameters. An alternative formulation is to quantify the heterogeneity of the distribution of classes across a city by looking at the statistical properties of the trajectories of a Random Walk over the city graph.
In my talk, I plan to give a short introduction to random walks, move on to applications on networks and finally introduce my research problem with some fresh results.
Without delving too much into the language of categories and Hopf algebras, we will look at how the search for Knot Invariants, and invariants of Ribbons in a general setting feed into the structure of Quantum Groups. The Jones polynomial was discovered in the 1980s as a stronger invariant of Knots compared to the Alexander polynomial. However, in the 90s, after the discovery of Quantum groups, it was seen that the Jones Polynomial can be realised via the representation theory of quantum groups. The reason behind this theory is now understood much more generally and relates to Topological Quantum Field Theories (whatever that is!). By looking at what TQFTs in lower dimensions are, we will rediscover the axioms of Hopf Algebras, and Tensor Categories all together!
A system of equations involving differential forms on a manifold is called an exterior differential system. Equivalently, an EDS can be understood as a differential ideal of the algebra of differential forms. Many interesting partial differential equations can be translated into this geometric framework to great effect. We shall introduce the most important notions in the theory of EDS, discuss the fundamental Cartan–Kähler theorem, and consider some simple applications.
Counting the number of trees on a labelled vertex set is a classical result called Cayley's formula that has been known since the 1800s, but counting of their generalisations has a much more recent history. The first question is how do you even generalise a tree to higher dimensions? As with cat skinning there are multiple ways but I will focus on just two: Q-acyclic simplicial complexes and minimal connected covers, with a particular emphasis on the latter. We will end the talk with a few results that do give Cayley type formulae for these generalisations.
A brief introduction to the concept of General Relativity is outlined. The aim is to give a description satisfying the mathematician (rather improbable...) and yet accessible to the intuitive learner. The emphasis will not be on differential geometry and tensor calculus, though this will be used, but rather on the physical ideas of the theory that is essential to understand. If time allows, I will briefly mention my own research within the framework given. Warning for the pure mathematician: this talk will be highly contaminated by applicable material.
Financial institutions form a highly interconnected system and its disruption has serious consequences for the economy and the society.
I will describe the 'PD model', a dynamical model of the financial system as a network of banks (nodes) characterized by their total assets, equity and probability of default PD per unit time. The edges of the network represent credit exposures between banks (for example loans). The contagion mechanism is an increased PD for nodes in a neighbourhood of a defaulted bank. The network is also characterized by a correlation matrix that is linked to the tendency of the nodes to default during the same time period. The results show the existence of a 'strong contagion' regime where a lower correlation between nodes is associated to higher risk. This is in sharp contrast with the standard assumption in financial credit risk where a more diversified portfolio (lower correlation) is considered less risky.
The 'PD' model unifies credit risk techniques with network theory and allows measuring systemic risk (risk that a considerable amount of the network is disrupted) and to assign a systemic risk rank to each node.
No previous knowledge of Finance is required.
In this talk we will be recalling some fundamental notions of category theory, in particular the notion of a limit. We shall then define an elementary topos which will be followed by some important examples, such as the category of presheaves (and if time permits, a "light" discussion of sheaves).
Periodic functions appear in various areas of mathematics - including Number Theory. A fundamental tool to study them is Fourier series, and I will start with a reminder about them. Taking inspiration from this setting, I will move to a more general context and explain how spectral analysis can be used to decompose the so-called right regular representation of some algebraic groups and give rise to automorphic forms.
We will introduce the concept of moving frames and where to use them, along with examples (and animations!). The notion of a quantum ribbon with the definition of the corresponding quantum Hamiltonian on such a strip will follow. Spectral results for different kinds of ribbons will be presented, and more importantly, explained. Last but not least, the spectrum of the Möbius strip will be tackled in three different models both analytically and numerically, with comparisons of the results.
This talk will be about the jamming state of granular materials and will introduce the random close packing problem. Granular materials which are composed of macroscopic grains such as sand, sugar, bearing balls etc. are ubiquitous in our everyday experience. Nevertheless, a fundamental description of both statistical and dynamic properties of granular matter is challenging. For example, it is not clear whether jamming transition (a fluid-to-solid phase transition) of granular materials is governed by a variational principle of an associated thermodynamic quantity like the free energy in equilibrium systems. Jamming transitions not only occur in granular media, but also in soft materials such as colloidal suspensions, compressed emulsions, foams, glasses and biological materials such as cells, DNA and protein packing. And you will see that this jamming transition is related with problem of identifying the densest packing of particles. Esma will try to explain the central role of the shape of the particles for this transition phase.