Past

"Show that there is a function $f$ such that for any sequence $(x_1, x_2, \dots)$ we have $x_n = f(x_{n + 1}, x_{n + 2}, \dots)$ for all but finitely many $n$."

Fred Galvin. Problem 5348. The American Mathematical Monthly, 72(10):p. 1135, 1965.\\

This quote is one of the earliest examples of an infinite hat problem, although it's not phrased this way. A hat problem is a non-empty set of colours together with a directed graph, where the nodes correspond to "agents" or "players" and the edges determine what the players "see". The goal is to find a collective strategy for the players which ensures that no matter what "hats" (= colours) are placed on their heads, they will ensure that a "sufficient" amount guess correctly.\\

In this talk I will discuss hat problems on countable sets and show that in a non-transitive setting, the relationship between existence of infinitely-correct strategies and Ramsey properties of the graph breakdown, in the particular case of the parity game. I will then introduce some small cardinals (uncountable cardinals no larger than continuum) that will be useful in analysing the parity game. Finally, I will present some new results on the sigma-ideal of meagre sets of reals that arise from this analysis.

Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G

is called p-nilpotent if S has a normal complement N in G, that is, G is

the semidirect product between S and N. The notion of p-nilpotency plays

an important role in finite group theory. For instance, Thompson's

criterion for p-nilpotency leads to the important structural result that

finite groups with fixed-point-free automorphisms are nilpotent.

By a classical result of Tate one can detect p-nilpotency using mod p

cohomology in dimension 1: the group G is p-nilpotent if and only if the

restriction map in cohomology from G to S is an isomorphism in dimension

1. In this talk we will discuss cohomological criteria for p-nilpotency by

Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the

1960s and 1970s. Finally, we will discuss how one can extend Tate's

result to study p-solvable and more general finite groups.

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.


In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges. Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.


Based on joint work with Oliver Riordan.

Ever wondered how the log function in your code is computed? This talk, which was prepared for the 400th anniversary of Napier's development of logarithms, discusses the computation of reciprocals, exponentials and logs, and also my own work on some special functions which are important in Monte Carlo simulation.

The talk will give a definition of matrix geometries, which are

particular types of finite real spectral triple that are useful for

approximating manifolds. Examples include fuzzy spheres and also the

internal space of the standard model. If time permits, the relation of

matrix geometries with 2d state sum models will also be sketched.

One of the holy grails of material science is a complete characterization of ground states of material energies. Some materials have periodic ground states, others have quasi-periodic states, and yet others form amorphic, random structures. Knowing this structure is essential to determine the macroscopic material properties of the material. In theory the energy contains all the information needed to determine the structure of ground states, but in practice it is extremely hard to extract this information.

In this talk I will describe a model for which we recently managed to characterize the ground state in a very complete way. The energy describes the behaviour of diblock copolymers, polymers that consist of two parts that repel each other. At low temperature such polymers organize themselves in complex microstructures at microscopic scales.

We concentrate on a regime in which the two parts are of strongly different sizes. In this regime we can completely characterize ground states, and even show stability of the ground state to small energy perturbations.

This is work with David Bourne and Florian Theil.

We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.

We study infinite (random) systems of interacting particles living in a Euclidean space X and possessing internal parameter (spin) in R¹. Such systems are described by Gibbs measures on the space Γ(X,R¹) of marked configurations in X (with marks in R¹). For a class of pair interactions, we show the occurrence of phase transition, i.e. non-uniqueness of the corresponding Gibbs measure, in both 'quenched' and 'annealed' counterparts of the model.

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections. After summarizing the evidence for these conclusions, I will delve into our efforts to create mathematical models that connect these patterns to the developmental mechanism leading to a beak. It turns out that there are simple (but precise) constraints on any mathematical model that leads to the observed phenomenology, leading to explicit predictions for the time dynamics of beak development in song birds. Experiments testing these predictions for the development of zebra finch beaks will be presented.

Based on the following papers:

http://www.pnas.org/content/107/8/3356.short

http://www.nature.com/ncomms/2014/140416/ncomms4700/full/ncomms4700.html

Ice sheets are among the key controls on global climate and sea-level change. A detailed understanding of ice sheet dynamics is crucial so to make accurate predictions of their mass balance into the future. Ice streams are the dominant negative component in this balance, accounting for up to 90$\%$ of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete.

A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localization feedback between ice flow, basal melting and water pressure in the underlying sediments. I will present a model of subglacial water flow below ice sheets, and particularly below ice streams. This hydrologic model is coupled to a model for ice flow. I show that under some conditions this coupled system gives rise to ice streams by instability of the internal dynamics.

We are dwelling in the Big Data age. The diversity of the uses of Big Data unleashes limitless possibilities. Many people are talking about ways to use Big Data to track the collective human behaviours, monitor electoral popularity, and predict financial fluctuations in stock markets, etc. Big Data reveals both challenges and opportunities, which are not only related to technology but also to human itself. This talk will cover various current topics and trends in Big Data research. The speaker will share his relevant experiences on how to use analytics tools to obtain key metrics on online social networks, as well as present the challenges of Big Data analytics.

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet Institute. His research is driven by a deep interest in analysing a wide range of sociotechnical problems by exploiting Big Data approaches, with the hope that this work could contribute to the intersection of social behavior and computational systems.

An important military task in a high-technology environment is to understand the set of radars present in it, since the radars will be, to a greater or lesser extent, indicative of the ships, aircraft and other military units which are present.

The transmissions of the different radars typically overlap in most of the dimensions which characterise then, such as frequency and bearing, and their pulses are interleaved in time. If, however, we are able to separate the individual pulse trains which are present then not only does this allow us to know how many different radars are present, but the characteristics of the pulse train are indicative of the type of the radar.

The problem of recognising the pulse trains is not trivial, because many radars 'jitter' their transmissions and pulses may be missing or two pulses may occur together, causing the characteristics of the pulse to be 'garbled.' The jittering may be used as a way to reject mutual interference between the radars, to resolve ambiguities in measurements of range or velocity or to make it harder to jam the radar.

The problems caused by pulses overlapping are likely to become more severe in the future because the pulses of the individual radars are becoming longer.

Although solutions currently exist which can cope, to at least some extent, with most of these issues, the purpose of bringing this topic to the seminar is to allow a fresh look at the problem from first principles.

The most valuable asset that people in a sovereign state can have is good, sustainable governance. Setting up a system of good, sustainable governance is not easy. The big and well-known problem is time inconsistency of optimal policies. A mechanism that has proven valuable in mitigating the time inconsistency problem is rule by law. The too-big-to-fail problem in banking is the result of the time inconsistency problem. In this lecture I will argue there is an alternative financial system that is not subject to the too-big-to-fail problem. The alternative arrangement I propose is a pure transaction banking system. Transaction banks are required to hold 100$\%$ interest bearing reserves and can pay tax-free interest on demand deposits. With this system, there cannot be a bank run as there is no place to run to. Mutual arrangements would finance all business investment, which is not currently the case.

I will give a short introduction to geometric stability theory and independence relations, focussing on the tree properties. I will then introduce one of the main examples for general measureable structures, the two sorted structure of a vector space over a field with a bilinear form. I will state some results for this structure, and give some open questions. This is joint work with William Anscombe.

We give an overview of Kitaev's lattice model in the setting of an arbitrary finite group G (where $G = Z_{2}$ is the famous Toric Code). We also exhibit the connection this model has with so-called 123-TQFTs (topological quantum field theories), making use of ideas coming from higher gauge theory and Hopf algebra representations.