There has been a great deal of attention paid to "Big Data" over the last few years. However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data. Even very small data sets can exhibit substantial complexity. There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models. The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets. On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions. In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces. In this talk, we will discuss what has been
done in this direction and what the future might hold, with numerous examples.
As the fundamental unit of life, the biological cell is a natural focus for computational simulations of growing cell population and tissues. However, models developed at the cellular scale can also be integrated into more complex multiscale models in order to examine complex biological and physical process that scan scales from the molecule to the organ.
This seminar will present a selection of the cellular scale agent-based modelling that has taken place at the University of Sheffield (where one software agent represents one biological cell) and how such models can be used to examine collective behaviour in cellular systems. Finally some of the issues in extending to multiscale models and the theoretical and computational methodologies being developed in Sheffield and by the wider community in this area will be presented.
G’s Growers supply salad and vegetable crops throughout the UK and Europe; primarily as a direct supplier to supermarkets. We are currently working on a project to improve the availability of Iceberg Lettuce throughout the year as this has historically been a very volatile crop. It is also by far the highest volume crop that we produce with typical weekly sales in the summer season being about 3m heads per week.
In order to continue to grow our business we must maintain continuous supply to the supermarkets. Our current method for achieving this is to grow more crop than we will actually harvest. We then aim to use the wholesale markets to sell the extra crop that is grown rather than ploughing it back in and then we reduce availability to these markets when the availability is tight.
We currently use a relatively simple computer Heat Unit model to help predict availability however we know that this is not the full picture. In order to try to help improve our position we have started the IceCAM project (Iceberg Crop Adaptive Model) which has 3 aims.
We believe that statistical mathematics can help us to solve these problems!!
Accurate simulation of coastal and hydraulic structures is challenging due to a range of complex processes such as turbulent air-water flow and breaking waves. Many engineering studies are based on scale models in laboratory flumes, which are often expensive and insufficient for fully exploring these complex processes. To extend the physical laboratory facility, the US Army Engineer Research and Development Center has developed a computational flume capability for this class of problems. I will discuss the turbulent air-water flow model equations, which govern the computational flume, and the order-independent, unstructured finite element discretization on which our implementation is based. Results from our air-water verification and validation test set, which is being developed along with the computational flume, demonstrate the ability of the computational flume to predict the target phenomena, but the test results and our experience developing the computational flume suggest that significant improvements in accuracy, efficiency, and robustness may be obtained by incorporating recent improvements in numerical methods.
Key Words:
Multiphase flow, Navier-Stokes, level set methods, finite element methods, water waves
In 1937 Quine introduced an interesting, rather unusual, set theory called New Foundations - NF for short. Since then the consistency of NF has been a problem that remains open today. But there has been considerable progress in our understanding of the problem. In particular NF was shown, by Specker in 1962, to be equiconsistent with a certain theory, TST^+ of simple types. Moreover Randall Holmes, who has been a long-term investigator of the problem, claims to have solved the problem by showing that TST^+ is indeed consistent. But the working manuscripts available on his web page that describe his possible proofs are not easy to understand - at least not by me.
Abstract: The simplest solutions of integrable systems are special functions that have been known since the time of Newton, Gauss and Euler. These functions satisfy not only differential equations as functions of their independent variable but also difference equations as functions of their parameter(s). We show how the inverse scattering transform method, which was invented to solve the Korteweg-de Vries equation, can be extended to its discrete version.
S.Butler and N.Joshi, An inverse scattering transform for the lattice potential KdV equation, Inverse Problems 26 (2010) 115012 (28pp)
We revisit the optimal exit problem by adding a moral hazard problem where a firm owner contracts out with an agent to run a project. We analyse the optimal contracting problem between the owner and the agent in a Brownian framework, when the latter modifies the project cash-flows with an hidden action. The analysis leads to the resolution of a constrained optimal stopping problem that we solve explicitly.
The talk will discuss the mean value theorem and Wooley's breakthrough with his "efficent congruencing" method.
Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.
Law in mathematics and mathematics in law: Probability theory and the fair price in contracts in England and France 1700–1850
From the middle of the eighteenth century, references to mathematicians such as Edmond Halley and Abraham De Moivre begin to appear in judgments in English courts on the law of contract and French mathematicians such as Antoine Deparcieux and Emmanuel-Etienne Duvillard de Durand are mentioned in French treatises on contract law in the first half of the nineteenth century. In books on the then nascent subject of probability at the beginning of the eighteenth century, discussions of legal problems and principally contracts, are especially prominent. Nicolas Bernoulli’s thesis at Basle in 1705 on The Use of the Art of Conjecturing in Law was aptly called a Dissertatio Inauguralis Matematico-Juridica. In England, twenty years later, De Moivre dedicated one of his books on probability to the Lord Chancellor, Lord Macclesfield and expressly referred to its significance for contract law.
The objective of this paper is to highlight this textual interaction between law and mathematics and consider its significance for both disciplines but primarily for law. Probability was an applied science before it became theoretical. Legal problems, particularly those raised by the law of contract, were one of the most frequent applications and as such played an essential role in the development of this subject from its inception. In law, probability was particularly important in contracts. The idea that exchanges must be fair, that what one receives must be the just price for what one gives, has had a significant influence on European contract law since the Middle Ages. Probability theory allowed, for the first time, such an idea to be applied to the sale of interests which began or terminated on the death of certain people. These interests, particularly reversionary interests in land and personal property in English law and rentes viagères in French law were very common in practice at this time. This paper will consider the surprising and very different practical effects of these mathematical texts on English and French contract law especially during their formative period in the late eighteenth and nineteenth centuries.
When formulated appropriately, the broad families of sequential quadratic programming, augmented Lagrangian and interior-point methods all require the solution of symmetric saddle-point linear systems. When regularization is employed, the systems become symmetric and quasi definite. The latter are
indefinite but their rich structure and strong relationships with definite systems enable specialized linear algebra, and make them prime candidates for matrix-free implementations of optimization methods. In this talk, I explore various formulations of the step equations in optimization and corresponding
iterative methods that exploit their structure.
I will discuss a puzzling theorem about smooth, periodic, real-valued functions on the real line. After introducing the classical Hardy-Littlewood maximal function (which just takes averages over intervals centered at a point), we will prove that if a function has the property that the computation of the maximal function is simple (in the sense that it's enough to check two intervals), then the function is already sin(x) (up to symmetries). I do not know what maximal local averages have to do with the trigonometric function. Differentiation does not help either: the statement equivalently says that a delay differential equation with a solution space of size comparable to C^1(0,1) has only the trigonometric function as periodic solutions.
In this talk, we will introduce the notions of systolic and residual girth growth for finitely generated groups. We will explore the relationship between these types of growth and the usual word growth for finitely generated groups.
I will report on recent progress towards understanding the growth of the torsion of the homology of subgroups of finite index in a given residually finite group G.
The cases I will consider are when G is amenable (joint work with P, Kropholler and A. Kar) and when G is right angled (joint work with M. Abert and T. Gelander).
Phylogenies, or evolutionary histories, play a central role in modern biology, illustrating the interrelationships between species, and also aiding the prediction of structural, physiological, and biochemical properties. The reconstruction of the underlying evolutionary history from a set of morphological characters or biomolecular sequences is difficult since the optimality criteria favored by biologists are NP-hard, and the space of possible answers is huge. Phylogenies are often modeled by trees with n leaves, and the number of possible phylogenetic trees is $(2n-5)!!$. Due to the hardness and the large number of possible answers, clever searching techniques and heuristics are used to estimate the underlying tree.
We explore the combinatorial structure of the underlying space of trees, under different metrics, in particular the nearest-neighbor-interchange (NNI), subtree- prune-and-regraft (SPR), tree-bisection-and-reconnection (TBR), and Robinson-Foulds (RF) distances. Further, we examine the interplay between the metric chosen and the difficulty of the search for the optimal tree.
In 1995, celebrated Russian mathematician V.I. Arnold conjectured that, contrary to common belief, convex, homogeneous solids with just two static balance points ("weebles without a bottom weight") may exist. Ten years later, based on a constructive proof, the first such object, dubbed "Gömböc", was built. In the process leading to the discovery, several curious properties of the shape emerged and evidently some tropical turtles had evolved similar shells for the purpose of self-righting.
This Public Lecture will describe those properties as well as explain the journey of discovery, the mathematics behind the journey, the parallels with molecular biology and the latest Gömböc thinking, most notably Arnold's second major conjecture, namely that the Gömböc in Nature is not the origin, rather the ultimate goal of shape evolution.
Please email @email to register.
How many $H$-free graphs are there on $n$ vertices? What is the typical structure of such a graph $G$? And how do these answers change if we restrict the number of edges of $G$? In this talk I will describe some recent progress on these basic and classical questions, focusing on the cases $H=K_{r+1}$ and $H=C_{2k}$. The key tools are the hypergraph container method, the Janson inequalities, and some new "balanced" supersaturation results. The techniques are quite general, and can be used to study similar questions about objects such sum-free sets, antichains and metric spaces.
I will mention joint work with a number of different coauthors, including Jozsi Balogh, Wojciech Samotij, David Saxton, Lutz Warnke and Mauricio Collares Neto.
In this talk I will give an overview of the algorithms used by Chebfun to numerically compute polynomial and trigonometric minimax approximations of continuous functions. I'll also present Chebfun's capabilities to compute best approximations on compact subsets of an interval and how these methods can be used to design digital filters.
The study of infrared singularities, due to the emission of “soft” (low momentum) gauge bosons, remains a highly active research area in a variety of quantum field theories. After motivating both phenomenological and formal reasons as to why we should care about IR singularities, this talk will review their structure in QED, QCD and quantum gravity, examining the similarities and differences between these three contexts. The role of Wilson lines will be examined, which provide a useful unifying language. Finally, I will examine recent work on moving beyond the soft approximation, and why this might be useful.
This'll be a nice and slow paced introduction to topological quantum field theory in general, and 1-2-3 dimensional theories in particular. If time permits I will explain the spin version of these and their connection to physics. There will be lots of pictures.
I will introduce the general idea of p-adic Hodge theory from the view point of a beginner. Also, I will give a sketch of the proof of the crystalline comparison theorem in the case of good reduction using 'almost mathematics'.
I will discuss some properties of Hermitian metrics on compact complex manifolds, having constant Chern scalar curvature, focusing on the existence problem in fixed Hermitian conformal classes (the "Chern-Yamabe problem"). This is joint work with Daniele Angella and Simone Calamai.
In this talk, gauged quiver quantum mechanics will be analysed for BPS state counting. Despite the wall-crossing phenomenon of those countings, an invariant quantity of quiver itself, dubbed quiver invariant, will be carefully defined for a certain class of abelian quiver theories. After that, to get a handle on nonabelian theories, I will overview the abelianisation and the mutation methods, and will illustrate some of their interesting features through a couple of simple examples.
A recommendation system for multi-modal journey planning could be useful to travellers in making their journeys more efficient and pleasant, and to transport operators in encouraging travellers to make more effective use of infrastructure capacity.
Journeys will have multiple quantifiable attributes (e.g. time, cost, likelihood of getting a seat) and other attributes that we might infer indirectly (e.g. a pleasant view). Individual travellers will have different preferences that will affect the most appropriate recommendations. The recommendation system might build profiles for travellers, quantifying their preferences. These could be inferred indirectly, based on the information they provide, choices they make and feedback they give. These profiles might then be used to compare and rank different travel options.
I will explain the framework of quasiminimal structures and quasiminimal classes, and give some basic examples and open questions. Then I will explain some joint work with Martin Bays in which we have constructed variants of the pseudo-exponential fields (originally due to Boris Zilber) which are quasimininal and discuss progress towards the problem of showing that complex exponentiation is quasiminimal. I will also discuss some joint work with Adam Harris in which we try to build a pseudo-j-function.
I plan to discuss finite time singularities for the harmonic map heat flow and describe a beautiful example of winding behaviour due to Peter Topping.
I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.
Security Constrained Optimal Power Flow is an increasingly important problem for power systems operation both in its own right and as a subproblem for more complex problems such as transmission switching or
unit commitment.
The structure of the problem resembles stochastic programming problems in that one aims to find a cost optimal operation schedule that is feasible for all possible equipment outage scenarios
(contingencies). Due to the presence of power flow constraints (in their "DC" or "AC" version), the resulting problem is a large scale linear or nonlinear programming problem.
However it is known that only a small subset of the contingencies is active at the solution. We show how Interior Point methods can exploit this structure both by simplifying the linear algebra operations as
well as generating necessary contingencies on the fly and integrating them into the algorithm using IPM warmstarting techniques. The final problem solved by this scheme is significantly smaller than the full
contingency constrained problem, resulting in substantial speed gains.
Numerical and theoretical results of our algorithm will be presented.
Associahedra are polytopes introduced by Stasheff to encode topological semigroups in which associativity holds up to coherent homotopy. These polytopes naturally form a topological operad that gives a resolution of the associative operad. Muro and Tonks recently introduced an operad which encodes $A_\infty$ algebras with homotopy coherent unit.
The material in this talk will be fairly basic. I will cover operads and their algebras, give the construction of the $A_\infty$ operad using the Boardman-Vogt resolution, and of the unital associahedra introduced by Muro and Tonks.
Depending on time and interest of the audience I will define unital $A_\infty$ differential graded algebras and explain how they are precisely the algebras over the cellular chains of the operad constructed by Muro and Tonks.
Artist Antoni Malinowski has been commissioned to produce a major wall painting in the foyer of the new Mathematical Institute in Oxford, the Andrew Wiles Building. To celebrate and introduce that work Antoni and a series of distinguished speakers will demonstrate the different impacts and perceptions of colour produced by the micro-structure of the pigments, from an explanation of the pigments themselves to an examination of how the brain perceives colour.
Speakers:
Jo Volley, Gary Woodley and Malina Busch, the Pigment Timeline Project, Slade School of Fine Art, University College London
‘Pigment Timeline’
Dr. Ruth Siddall - Senior Lecturer in Earth Sciences, University College London
‘Pigments: microstructure and origins?’
Antoni Malinowski
‘Spectrum Materialised’
Prof. Hannah Smithson Associate Professor, Experimental Psychology, University of Oxford and Tutorial Fellow, Pembroke College
‘Colour Perception‘
11.30am, Lecture Theatre 1
Mathematical Institute, University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
No booking required
In the classification of surfaces, K3 surfaces hold a place not dissimilar to that of elliptic curves within the classification of curves by genus. In recent years there has been a lot of activity on the problem of rational points on K3 surfaces. I will discuss the problem of finding the Picard group of a K3 surface, and how this relates to finding counterexamples to the Hasse principle on K3 surfaces.
Let X be a smooth projective curve (of genus greater than or equal to 2) over C and M the moduli space of vector bundles over X, of rank 2 and with fixed determinant of degree 1.Then the Fourier-Mukai functor from the bounded derived category of coherent sheaves on X to that of M, given by the normalised Poincare bundle, is fully faithful, except (possibly) for hyperelliptic curves of genus 3,4,and 5
This result is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space M.
Results on the deformation and inversion of Picard bundles (already known) follow from the full faithfulness of the F-M functor
In this talk we will explore the convergence of Krylov methods when used to solve $Lu = f$ where $L$ is an unbounded linear operator. We will show that for certain problems, methods like Conjugate Gradients and GMRES still converge even though the spectrum of $L$ is unbounded. A theoretical justification for this behavior is given in terms of polynomial approximation on unbounded domains.
Let $G(n,d)$ be a random $d$-regular graph on $n$ vertices. In 2004 Kim and Vu showed that if $d$ grows faster than $\log n$ as $n$ tends to infinity, then one can define a joint distribution of $G(n,d)$ and two binomial random graphs $G(n,p_1)$ and $G(n,p_2)$ -- both of which have asymptotic expected degree $d$ -- such that with high probability $G(n,d)$ is a supergraph of $G(n,p_1)$ and a subgraph of $G(n,p_2)$. The motivation for such a coupling is to deduce monotone properties (like Hamiltonicity) of $G(n,d)$ from the simpler model $G(n,p)$. We present our work with A. Dudek, A. Frieze and A. Rucinski on the Kim-Vu conjecture and its hypergraph counterpart.
Sparse matrices occur in numerical simulations throughout science and engineering. In particular, it is often desirable to solve systems of the form Ax=b, where A is a sparse matrix with 100,000+ rows and columns. The order that the rows and columns occur in can have a dramatic effect on the viability of a direct solver e.g., the time taken to find x, the amount of memory needed, the quality of x,... We shall consider symmetric matrices and, with the help of playdough, explore how best to order the rows/columns using a nested dissection strategy. Starting with a straightforward strategy, we will discover the pitfalls and develop an adaptive strategy with the aim of coping with a large variety of sparse matrix structures.
Some of the talk will involve the audience playing with playdough, so bring your inner child along with you!