Past

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.

Charles Babbage (1791–1871) was Lucasian Professor of mathematics in Cambridge from 1828–1839. He displayed a fertile curiosity that led him to study many contemporary processes and problems in a way which emphasised an analytic, data driven view of life.

In popular culture Babbage has been celebrated as an anachronistic Victorian engineer. In reality, Babbage is best understood as a figure rooted in the enlightenment, who had substantially completed his core investigations into 'mechanisation of thought' by the mid 1830s: he is thus an anachronistic Georgian: the construction of his first difference engine design is contemporary with the earliest public railways in Britain.

A fundamental question that must strike anybody who examines Babbage's precocious designs is: how could one individual working alone have synthesised a workable computer design, designing an object whose complexity of behaviour so far exceeded that of contemporary machines that it would not be matched for over a hundred years?

We shall explore the extent to which the answer lies in the techniques Babbage developed to reason about complex systems. His Notation which shows the geometry, timing, causal chains and the abstract components of his machines, has a direct parallel in the Hardware Description Languages developed since 1975 to aid the design of large scale electronics. In this presentation, we shall provide a basic tutorial on Babbage's notation showing how his concepts of 'pieces' and 'working points' effectively build a graph in which both parts and their interactions are represented by nodes, with edges between part-nodes and interaction-nodes denoting ownership, and edges between interaction-nodes denoting the transmission of forces between individual assemblies within a machine. We shall give examples from Babbage's Difference Engine 2 for which a complete set of notations was drawn in 1849, and compare them to a design of similar complexity specified in 1987 using the Inmos HDL.

L-functions have become a vital part of modern number theory over the past century, allowing comparisons between arithmetic objects with seemingly very different properties. In the first part of this talk, I will give an overview of where they arise, their properties, and the mathematics that has developed in order to understand them. In the second part, I will give a sketch of the beautiful result of Herbrand-Ribet concerning the arithmetic interpretations of certain special values of the Riemann zeta function, the prototypical example of an L-function.

The AO-model describes crystalline solids in the presence of defects like dislocation lines. We demonstrate that the model supports low-energy structures like grains and determine for simple geometries the grain boundary energy density. At small misorientation angles we recover the well-known Read-Shockley law. Due to the atomistic nature of the model it is possible to consider the the Boltzmann-Gibbs distribution at non-zero temperature. Using ideas by Froehlich and Spencer we prove rigorously the presence of long-range order if the temperature is sufficiently small.
 

We consider the coordinate-iterated-integral as an algebraic product on the shuffle algebra, called the (right) half-shuffle product. Its anti-symmetrization defines the biproduct  area(.,.), interpretable as the signed-area between two real-valued coordinate paths. We consider specific sets of binary, rooted trees known as Hall sets. These set have a complex combinatorial structure, which can be almost entirely circumvented by introducing the equivalent notion of Lazard sets. Using analytic results from dynamical systems and algebraic results from the theory of Lie algebras, we show that shuffle-polynomials in areas-of-areas on Hall trees generate the shuffle algebra.

We construct a cellular decomposition of the
Axelrod-Singer-Fulton-MacPherson compactification of the configuration
spaces in the plane, that is compatible with the operad composition.
Cells are indexed by trees with bi-coloured edges, and vertices are labelled by 
cells of the cacti operad. This answers positively a conjecture stated in 
2000 by Kontsevich and Soibelman.

This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.

The aim of this talk is to describe joint work with Gergely Berczi using a recent extension to non-reductive actions of geometric invariant theory, and its links with moment maps in symplectic geometry, to study hyperbolicity of generic hypersurfaces in a projective space. Using intersection theory for non-reductive GIT quotients applied to  compactifications of bundles of invariant jet differentials over complex manifolds leads to a proof of the Green-Griffiths-Lang conjecture for a generic projective hypersurface of dimension n whose degree is greater than n^6. A recent result of Riedl and Yang then implies the Kobayashi conjecture for generic hypersurfaces of degree greater than (2n-1)^6.

We consider CFTs with large gap in the spectrum of operators and a large number of degrees of freedom (large central charge). We analytically study a Heavy-Heavy-Light-Light correlation function, where Heavy, refers to an operator with conformal dimension which scales like the central charge and Light, refers to an operator whose dimension is of order unity in the large central charge limit. In certain regimes, the correlation function can be examined analytically leading to very simple and suggestive expressions.

Dominic Vella will talk about writing grants, Anna Seigal will talk about writing research fellow applications and Jason Lotay will talk about his experience and tips for applying for faculty positions. 

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the  workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

In the first part of the talk, I will describe surface colonization strategies of the motile bacteria Pseudomonas aeruginosa. During early stages of biofilm formation, the majority of cells that land on a surface eventually detach. After a prolonged lag time, cells begin to cover the surface rapidly. Reversible attachments provide cells and their descendants with multigenerational memory of the surface that primes the planktonic population for colonization. Two different strains use different surface sensing machinery and show different colonization strategies. We use theoretical modelling to investigate how the hydrodynamics of type IV pili and flagella activity lead to increased detachment rates and show that the contribution from this hydrodynamic effect plays a role in the different colonization strategies observed in the two strains.

In the second part of the talk, I will show that when cells migrate through constricting pores, there is an increase in DNA damage and mutations. Experimental observations show that this breakage is not due to mechanical stress. I present an elastic-fluid model of the cell nucleus, coupled to kinetics of DNA breakage and repair proposing a mechanism by which nuclear deformation can lead to DNA damage. I show that segregation of soluble repair factors from the chromatin during migration leads to a decrease in the repair rate and an accumulation of damage that is sufficient to account for the extent of DNA damage observed experimentally.

We present our works on two problems in global analysis (i.e.,analysis on manifolds): One concerns the compactness of the space of smooth $d$-dimensional immersed hypersurfaces with uniformly $L^d$-bounded second fundamental forms, and the other concerns the validity of W^{2,p}$-elliptic estimates for the Laplace--Beltrami operator on open manifolds. We construct explicit counterexamples to both problems. The onstructions involve rapid oscillations and wild spirals, with motivations derived from physical phenomena.

Background

The RON test is an engine test that is used to measure the research octane number (RON) of a gasoline. It is a parameter that is set in fuels specifications and is an indicator of a fuel to partially explode during burning rather than burn smoothly.

The efficiency of a gasoline engine is limited by the RON value of the fuel that it is using. As the world moves towards lower carbon, predicting the RON of a fuel will become more important.

Typical market gasolines are blended from several hundred hydrocarbon components plus alcohols and ethers. Each component has a RON value and therefore, if the composition is known then the RON can be calculated. Unfortunately, components can have antagonistic or complimentary effects on each other and therefore this needs to be taken into account in the calculation.

Several models have been produced over the years (the RON test has been around for over 60 years) but the accuracy of the models is variable. The existing models are empirically based rather than taking into account the causal links between fuel component properties and RON performance.

Opportunity

BP has developed intellectual property regarding the causal links and we need to know if these can be used to build a functional based model. There is also an opportunity to build a better empirically based model using data on individual fuel components (previous models have grouped similar components to lessen the computing effort)

A big problem in Riemannian geometry is the search for a "best possible" Riemannian metric on a given compact smooth manifold. When the manifold is complex, one very nice metric we could look for is a Kahler-Einstein metric. For compact Kahler manifolds with non-positive first chern class, these were proven to always exist by Aubin and Yau in the 70's. However, the case of positive first chern class is much more delicate, and there are non-trivial obstructions to existence. It wasn't until this decade that a complete abstract characterisation of Kahler-Einstein metrics became available, in the form of K-stability. This is a purely algebro-geometric stability condition, whose equivalence to the existence of a Kahler-Einstein metric in the Fano case is analogous to the Hitchin-Kobayashi correspondence for vector bundles. In this talk, I will cover the definition of K-stability, its relation to Kahler-Einstein metrics, and (time permitting) give some examples of how K-stability is verified or disproved in practice.

The 2007-2008 financial crisis forced governments to choose between the unattractive alternatives of either bailing out a systemically important bank (SIBs) or having it fail in a disruptive manner. Bail-in has been put forward as the primary tool to resolve a failing bank, which would end the too-big-to-fail problem by letting stakeholders shoulder the losses, while minimising the calamitous systemic impact of a bank failure. Though the aptness of bail-in has been evinced in relatively minor idiosyncratic bank failures, its efficacy in maintaining stability in cases of large bank failures and episodes of system-wide crises remains to be practically tested. This paper investigates the financial stability implications of the bail-in design, in all these cases. We develop a multi-layered network model of the European financial system that captures the prevailing endogenous-amplification mechanisms: exposure loss contagion, overlapping portfolio contagion, funding contagion, bail-inable debt revaluations, and bail-inable debt runs. Our results reveal that financial stability hinges on a set of `primary' and `secondary' bail-in parameters, including the failure threshold, recapitalisation target, debt-to-equity conversion rate, loss absorption requirements, debt exclusions and bail-in-design certainty – and we uncover how. We also demonstrate that the systemic footprint of the bail-in design is not properly understood without the inclusion of multiple contagion mechanisms and non-banks. Our evidence fortunately suggests that the pivot for stability is in the hands of policymakers. It also suggests, however, that the current bail-in design might be in the regime of instability.

My talk will have two protagonists: (1) Automorphic representations which -- let's be honest -- are very complicated and mysterious, but also (2) Involutions  (=automorphisms of order at most 2) of connected reductive groups -- these are very concrete and can often be represented by diagonal matrices with entries 1,-1 or i, -i. The goal is to explain how difficult questions about (1) can be reduced to relatively easy, concrete questions about (2).
Automorphic representations are representation-theoretic generalizations of modular forms. Like modular forms, automorphic representations are initially defined analytically. But unlike modular forms -- where we have a reinterpretation in terms of algebraic geometry -- for most automorphic representations we currently only have a (real) analytic definition. The Langlands Program predicts that a wide class of automorphic representations admit the same algebraic properties which have been known to hold for modular forms since the 1960's and 70's. In particular, certain complex numbers "Hecke eigenvalues" attached to these automorphic representations are conjectured to be algebraic numbers. This remains open in many cases (especially those cases of interest in number theory and algebraic geometry), in particular for Maass forms -- functions on the upper half-plane which are a non-holomorphic variant of modular forms.
I will explain how elementary structure theory of reductive groups over the complex numbers provides new insight into the above algebraicity conjectures; in particular we deduce that the Hecke eigenvalues are algebraic for an infinite class of examples where this was not previously known. 
After applying a bunch of "big, old theorems" (in particular Langlands' own archimedean correspondence), it all comes down to studying how involutions of a connected, reductive group vary under group homomorphisms. Here I will write down the key examples explicitly using matrices.

Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients method are derived, the necessary quantities occurring in the theoretical bounds estimated and a new practical algorithm derived. Numerical experiments suggest that the new method has significant potential, including in the steadily more important context of multi-precision computations.