This is our Department Colloquium, with distinguished speakers giving an overview of a topic of general mathematical interest. The talks are aimed at a level to be accessible to all postgraduate students and staff in the department. Advanced undergraduates and members of other departments are also welcome to attend.
Talks last a full hour and, unless otherwise stated, take place in the Wolfson Theatre (4W1.7), beginning at 4:15pm. Tea is available from 30 mins before the talk in the foyer outside the lecture theatre.
| Date | Speaker | Title/Abstract |
|---|---|---|
| 17 Feb 2012 | Oliver Riordan University of Oxford |
Explosive percolation? Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology). One of their most interesting features, both mathematically and in terms of applications, is the "phase transition": as the ratio of the number of edges to vertices increases past a certain critical point, the global structure changes radically, from only small components to a single macroscopic ("giant") component plus small ones. Recent work on Achlioptas processes (a natural type of evolving random graph model) suggested that in some such processes the transition is particularly radical: more or less as soon as the macroscopic component appears, it is already extremely large. The talk will include general background on Achlioptas and other random graph processes, plus a discussion of joint work with Lutz Warnke on the explosive percolation phenomenon. |
| 2 Mar 2012 | Bálint Tóth Technical University Budapest (Hungary) |
Brownian motion and "Brownian motion" The physical phenomenon called Brownian motion is the apparently random motion of a particle suspended in a fluid, driven by collisions and interactions with the molecules of the fluid which are in permanent thermal agitation. One of the idealised mathematical models of this random drifting is the stochastic process called commonly "Brownian motion", or Wiener process. A dynamical theory of Brownian motion should link these two: derive in mathematically satisfactory way -- as a kind of macroscopic scaling limit -- the idealised mathematical description from physical principles. The first attempt was made by Einstein in his celebrated 1905 paper and we are still very far from the end of this endeavour. I will survey some attempts. |
| 16 Mar 2012 | Angus Macintyre Queen Mary, University of London |
Logic and the exponential functions of analysis I will give an overview of how a model-theoretic analysis of the real and complex exponential functions has assisted in the solution of purely analytic problems, and the formulation of novel ones. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 14 Oct 2011 | Mark Chaplain University of Dundee |
Multiscale mathematical modelling of cancer growth Cancer growth is a complicated phenomenon involving many inter-related processes across a wide range of spatial and temporal scales, and as such presents the mathematical modeller with a correspondingly complex set of problems to solve. In this talk we will present multiscale mathematical models for the growth and spread of cancer and will focus on three main scales of interest: the sub-cellular, cellular and macroscopic. The sub-cellular scale refers to activities that take place within the cell or at the cell membrane, e.g. DNA synthesis, gene expression, cell cycle mechanisms, absorption of vital nutrients, activation or inactivation of receptors, transduction of chemical signals. The cellular scale refers to the main activities of the cells, e.g. statistical description of the progression and activation state of the cells, interactions among tumour cells and the other types of cells present in the body (such as endothelial cells, macrophages, lymphocytes), proliferative and destructive interactions, aggregation and disaggregation properties. The macroscopic scale refers to those phenomena which are typical of continuum systems, e.g. cell migration, diffusion and transport of nutrients and chemical factors, mechanical responses, interactions between different tissues, tissue remodelling. The models presented will be either systems of nonlinear partial differential equations or individual-forced based models and, in addition to presenting our computational simulation results, we will discuss some analytical and numerical results and issues. Finally, we will present an overview of recent analytical results in the area concerning a new notion of multiscale convergence, called "three-scale convergence". |
| 28 Oct 2011 | Yves Balasko University of York |
Some mathematical aspects of economic theory Abstract in PDF |
| 11 Nov 2011 | Terry Lyons University of Oxford |
Expected signature and all that... |
| 25 Nov 2011 | Angus Macintyre Queen Mary, University of London |
Rescheduled for 16 Mar 2012. |
| 9 Dec 2011 | Mark Lewis University of Alberta (Canada) |
The Mathematics Behind Biological Invasion Processes Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change. In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 4 Mar 2011 | Roger Heath-Brown University of Oxford |
Counting Solutions of Diophantine Equations Given a Diophantine equation f(x_1,...,x_n)=0, where f is an integer polynomial, let N(B) be the number of integral solutions in which |x_i| is at most B for each index i. The talk will be about the growth of N(B) as B tends to infinity. Why should one be interested in this? What phenomena affect the growth rate? How does one prove anything about N(B)? |
| 18 Mar 2011 | Gero Friesecke Technical University of Munich (Germany) |
How good is the quantum mechanical explanation of the periodic table? Abstract in PDF |
| 1 Apr 2011 | Elmer Rees University of Bristol |
Real Linear Algebra and Topology I will review the history of the construction of real division algebras starting with Hamilton, Graves and Cayley. I will also discuss some of the attempts to prove that there are none of dimension more than eight. The proof of the final non-existence theorem needs methods from topology. These methods can also be used to give restrictions on the spaces of real matrices of given rank. One of the first published results of this kind was the Adams, Lax, Philips theorem motivated by the construction of hypo-elliptic operators. Most of these results could easily be stated in a first course on Linear Algebra but the only known proofs for several of them use topology. |
| 15 Apr 2011 | Andrew Stuart University of Warwick |
Bayes Theorem and Inverse Problems Inverse problems arise in many areas of science and technology, and are interesting for mathematicians because they are typically ill-posed. The Bayesian approach to inverse problems is useful in many of these arenas for several reasons, in particular because it provides a clean and transparent approach to regularization and because it allows for the quantification of uncertainty. I will introduce the Bayesian approach to inverse problems via the simple (to state) problem of predicting the next number in a sequence. I will then develop these ideas for more complex PDE-based models arising in the physical sciences. Finally I will point to open questions and new directions in the field. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 14 Jan 2011 | Mark Peletier |
Special Landscape Seminar - Leverhulme Lecture Gradient flows, optimal transport, and fresh bread In 1997, Jordan, Kinderlehrer, and Otto pioneered a new way of looking at age-old equations for diffusion, thus giving an exact mathematical description of the sense in which `diffusion is driven by entropy'. This sense revolves around the concept of optimal transport. Introduced by Monge in 1781, this theory focuses on optimal ways to transport given quantities from A to B. Its development took off after Kantorovich improved the formulation in 1942, and in recent years the theory has exploded, with applications in differential geometry, probability theory, functional analysis, analysis on non-smooth spaces, and many more. In this talk I will revisit the original connection between the diffusive problems on one hand and the theory of optimal transport on the other. I will show how the two are connected, discuss many consequences of this, and describe recent insight into the deeper meaning of this connection. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 15 Oct 2010 | Dorothy Buck Imperial College |
The Topology of DNA-Protein Interactions The central axis of the famous DNA double helix is often constrained or even circular. The shape of the axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology - for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as by-product of their interaction with DNA. |
| 29 Oct 2010 | Timothy J. Hollowood Swansea University |
Solitons and Integrability in Quantum Field Theory Since John Scott Russel observed a soliton-like wave in a canal in 1834, the theory of solitons and integrable systems has been a fruitful area of research in mathematical physics. In the area of Quantum Field Theory and Statistical Mechanics, integrability has proved to be a very powerful idea and often such models can be solved exactly leading to insights that take us beyond the perturbative regime. The classic example is the sine-Gordon theory whose exact spectrum and scattering matrix were written down over 30 years ago. More recent work has shown that the sine-Gordon theory is the simplest model of a class associated to (pseudo-) Riemannian Symmetric Spaces whose integrability is linked to underlying mathematical structures involving affine Lie algebras and their deformations known as quantum groups. These generalized sine-Gordon theories also plan a key role in the integrability underlying strong theory and quantum gravity. |
| 12 Nov 2010 | Richard Gill Leiden University (The Netherlands) |
Murder by numbers In March 2003, Dutch nurse Lucia de Berk was sentenced to life imprisonment by a court in The Hague for 5 murders and 2 murder attempts of patients in her care at a number of hospitals where she had worked in the Hague between 1996 and 2001. The only hard evidence against her was a statistical analysis resulting in a p-value of 1 in 342 million which purported to show that it could not be chance that so many incidents and deaths occured on her ward while she was on duty. On appeal in 2003 the life sentence was confirmed, this time for 7 murders and 3 attempts. This time, no statistical evidence was used at all: all the deaths were proven to be unnatural and Lucia shown to have caused them using scientific medical evidence only. However, after growing media attention and pressure by concerned scientists, including many statisticians, new forensic investigations were made which showed that the conviction was unsafe. After a new trial, Lucia was spectacularly and completely exhonerated in 2010. I'll discuss the statistical evidence and show how it became converted into incontrovertible medical-scientific proof in order to secure the second, and as far as the Dutch legal system was concerned, definitive conviction. I'll also show how statisticians were instrumental in convincing the legal establishment that Lucia should be given a completely new trial. The history of Lucia de Berk brought a number of deficiencies to light in the way in which scientific evidence is evaluated in criminal courts. Similar cases to that of Lucia occur regularly all over the world. The question of how that kind of data should be statistically analyzed is still problematic. I believe that there are also important lessens to be learnt by the medical world, however, the Dutch medical community, where most people still believe Lucia is a terrible serial killer, is resisting all attempts to uncover what really happened. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 12 Feb 2010 | Brian Conrey American Institute of Mathematics (USA) |
The Riemann Hypothesis 150 years ago B. Riemann discovered a pathway to understanding the prime numbers. But today we still have not completed his vision. I will give an introduction to Riemann's Hypothesis, one of the most compelling mathematics problems of all time, and describe some of its colorful history. |
| 5 Mar 2010 | Richard Thomas Imperial College |
Counting curves in algebraic geometry One can study "complex manifolds" or "algebraic varieties" via invariants that "count holomorphic curves in them". Without assuming prior knowledge of geometry (just holomorphic functions), this talk will explain the notions in inverted commas. In particular, there are at least four different ways to define curve counting. |
| 19 Mar 2010 | Endre Süli University of Oxford |
Mathematical challenges in kinetic models of dilute polymers We shall review recent developments concerning the existence of global weak solutions to coupled Navier-Stokes-Fokker-Planck systems of partial differential equations that arise in kinetic models for dilute polymers. We shall also survey some recent developments concerning the numerical analysis of high-dimensional Fokker-Planck equations with unbounded drift terms featuring in these models. |
| 30 Apr 2010 | James Davenport University of Bath |
The mathematics of the internet There is a lot of mathematics underpinning daily use of the Internet, be it Google or Internet shopping. In general, but not always, the mathematics has come first, and the application later. In this talk, we will sketch some of the applications, and describe, as one non-specialist to others, part of the mathematics. As a side-effect, we will explain why Google Scholar is much more reliable than "Impact Factors". |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 30 Oct 2009 | Sir John Kingman University of Bristol |
Forbidden transitions: continuous time versus discrete space Markov processes are the stochastic analogues of ordinary dynamical systems governed by first order differential equations. Almost all the theory of such processes deals with the autonomous case, in which the equations do not explicitly involve time, but this is an unnatural restriction, for instance in applications to biology or OR. One of the few known results about the general case came from an ingenious argument of former Bath Professor David Williams, and has led to a complete analysis for processes taking only finitely many values. Much deeper problems arise when there is a countable infinity of possible states, as often occurs in applications. |
| 13 Nov 2009 | Raphaël Rouquier University of Oxford |
Dunkl operators: from analysis to algebra and back We will introduce deformations of partial derivatives (Dunkl operators) and discuss their actions on polynomial functions. This brings in special functions and gives rise to Calogero-Moser integrable systems. Various flavours of Hecke algebras control these structures. The space of families of points in the plane provides a rich background. Eventually, the Dunkl operators can be studied via microlocal methods. |
| 27 Nov 2009 | Claude Le Bris Ecole Nationale des Ponts et Chaussées (Paris, France) |
Random media in computational material science The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization, for the modelling of random materials. In particular, some variants of the classical theory will be introduced. The relation between stochastic homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. |
| 11 Dec 2009 | Reidun Twarock University of York |
Viruses and geometry Viruses have protein containers that encapsulate, and hence provide protection for, their genomic material. For a significant number of viruses these containers are organised according to icosahedral symmetry, which allows us to model their structural organisation via group theory. We show here that a wide spectrum of distinct viral features can be predicted in striking detail via a classification of affine extensions of the icosahedral group. Examples discussed in this talk include the sizes and shapes of the protein building blocks of the containers, the double-shelled genomic RNA structure in MS2, the dodecahedral RNA cage in Pariacoto virus, and the heterogeneity in the genomic organisation of Picornaviridae. Some of the implications of this fundamental geometric principle of virus architecture for virus assembly and evolution are also discussed. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 27 Feb 2009 | Nick Evans University of Southampton |
The Large Hadron Collider and the Search for the Higgs Boson The Large Hadron Collider project is underway at CERN, Geneva - this 27km round ring will collide protons with ten times higher energy than has been achieved before. The goal is to probe the structure of matter at scales of 10-18 m. I will review the mathematical structure of the Standard Model of particle physics and the crucial unconfirmed mechanism for symmetry breaking, the Higgs mechanism. LHC discovery signals for the Higgs boson will be discussed. The simplest model is fine tuned though and we believe the Higgs is liable to be found in conjunction with other new physics from compositeness, to supersymmetry, to extra dimensions of space. |
| 20 Mar 2009 | Vincent Beffara Ecole Normale Supérieure de Lyon (France) |
Isotropic embeddings of planar lattices In recent years, huge progress has been made in the understanding of critical 2D models of statistical physics, and especially about their scaling limits (through the use of SLE processes). However, one question remains widely open, and that is the reason for universality, i.e. the belief that similar system on different lattices, even though they have different critical points, nevertheless converge to the same scaling limit as the lattice mesh goes to 0. It seems that a key question along the way to understanding it is, given the combinatorics of a lattice, how to embed it in the plane in order to obtain a conformally invariant scaling limit - and a surprising fact is that the "right" embedding depends not only on the lattice but also on the model. I will present the few results I was recently able to obtain in this direction. |
| 1 May 2009 | Ulrike Tillmann University of Oxford |
Topological Field Theory, via configuration spaces of points and
moduli spaces of surfaces In this introductory talk, I will give an outline of some of the ideas that have led to some remarkable theorems describing the topology of moduli spaces of Riemann surfaces (Madsen-Weiss) and a very recent classification theorem for Topological Field Theories (Hopkins-Lurie). |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 10 Oct 2008 | Heinz Engl University of Vienna |
Regularization of Inverse Problems: Convergence Analysis,
New Applications and Challenges We illustrate, via a parameter identification problem from systems biology, the numerical difficulties arising when solving inverse problems and give an overview over problem areas where inverse problems appear and over regularization methods for their stable solution. Recent emphasis has been on nonlinear problems in a non-Hilbert-space setting, e.g., in the connection with TV-regularization and sparsity-enforcing regularization. We illustrate the importance of the latter via an inverse bifurcation problem from systems biology. Finally, we mention some inverse problems that appear in finance and illustrate the effect of regularization. |
| 7 Nov 2008 | Martin Bridson University of Oxford |
Dimension, rigidity and fixed-point theorems The first half of the lecture will recall three strands of thought from 20th century geometry/topology: (super)rigidity, which constrains the way in which lattices in certain Lie groups can act, e.g. SL(n,Z); Smith theory, wherein one studies the fixed-points sets of finite groups acting on spheres and contractible spaces; and finally Helly's theorem, which characterises the dimension of a Euclidean space in terms of the possible intersection patterns of convex subsets. In the second half will describe in outline some recent theorems that draw inspiration from the foregoing ideas. For example, I shall sketch a proof that SL(n,Z) admits no non-trivial actions by homeomorphisms on a sphere of dimension less than n-1 or a contractible manifold of dimension less than n. I shall also motivate and explain a fixed point theorem that settles a long-standing question of Kropholler. |
| 21 Nov 2008 | John Ball University of Oxford |
Mathematical Problems of Liquid Crystals Most mathematical work on nematic liquid crystals has been in the context of the Oseen-Frank theory, which models the mean orientation of the constituent rod-like molecules by means of a director field consisting of unit vectors. However nowadays most physicists use the Landau - de Gennes theory, whose basic variable is a tensor-valued order parameter. Unlike the Oseen-Frank theory, that of Landau - de Gennes does not assign an unphysical orientation to the director field. The lecture will describe the two theories and the relationship between them, as well as other interesting mathematical problems related to the Landau-de Gennes theory. |
| Date | Speaker | Title/Abstract |
|---|---|---|
| 8 Feb 2008 | Alexander Bobenko Technical University Berlin (Germany) |
Discrete differential geometry: from organizing principles to applications Discrete differential geometry aims at the development and application of discrete equivalents of the geometric notions and methods of differential geometry. The latter appears then as a limit of refinements of the discretization. Current progress in this field is to a large extent stimulated by its relevance for applications (computer graphics etc). Concrete examples considered in the talk include discrete curvature line parametrizations, discrete Willmore energy, and applications to architecture. |
| 22 Feb 2008 | Stephen Senn University of Glasgow |
Why I hate minimisation If publicly funded collaborative trials are compared to those run in the pharmaceutical industry it will be found that the former generally use more complex allocation algorithms but simpler approaches to analysis. For example a popular design and analysis combination is to control covariate imbalance using an approach known as 'minimisation' but then ignore the covariates altogether in the analysis. In this talk I will explain why minimisation is not even a sensible approach to controlling covariate imbalance and also explain why sophistication in analysis can make a bigger contribution to improving the efficiency of inferences than complication in allocation. |
| 7 Mar 2008 | Julian Besag University of Bath |
Geographical analysis and ethical dilemmas in the study of
childhood leukemias in Great Britain The causes of childhood leukaemias are largely unknown but it is often claimed that cases occur in geographical clusters. This has led to various possible explanations, including the existence of a virus and the effects of radiation from nearby nuclear installations. It is also true that apparent clusters may occur purely by chance, even if there is no underlying cause, especially if the database is extensive. Thus, "cluster busting" has been highly controversial. This talk will describe a very simple method (Besag and Newell, 1991, "The detection of clusters in rare diseases", Journal of the Royal Statistical Society, A 154, 143-155) designed specifically to trawl a database of more than 100,000 census enumeration districts (ED's) in Great Britain for evidence of clusters in childhood leukaemias. Results for a five-year period centred on a census date will be given in detail, together with reasons for not publishing them or releasing them to the press. The talk is intended to be accessible to anyone familiar with the Poisson distribution. |
| 11 Apr 2008 | Kenneth Falconer University of St Andrews |
Symmetry and enumeration of self-similar fractals We describe a general method for enumerating the distinct self-similar sets that arise as attractors of certain families of iterated function systems, using a little group theory to analyse the symmetries of the attractors. The talk will be illustrated by a range of pictorial examples and is suitable for a broad audience. |
| 25 Apr 2008 | Rob Stevenson University of Amsterdam (The Netherlands) |
Adaptive wavelet methods for solving high dimensional PDEs We discuss a non-standard numerical method for solving (linear) PDE's. Such equations can be written in the form Lu=f, where L is a boundedly invertible operator between Hilbert spaces. By equipping these spaces with (wavelet) bases, an equivalent infinite matrix vector equation is obtained. We will show that the adaptive wavelet scheme produces a sequence of approximations to the solution vector that converges with the best possible rate, where the cost of producing these approximations is proportional to their length. Finally, we discuss the application of this scheme to problems on product domains, where we will obtain rates that are independent of the space dimension. |
| 2 May 2008 | Peter Bickel University of California, Berkeley (USA) |
Low dimensional features which enable high dimensional inference Theoretical analysis seems to suggest that standard problems such as estimating a function of high dimensional variables with noisy data (regression or classification) should be impossible without detailed knowledge or absurdly large amounts of data. Nevertheless, algorithms to perform classification of images or other high dimensional objects are remarkably successful. The generally held explanation is the presence of sparsity/low dimensional structure. I'll discuss analytically and with examples why this may be right. |
| 7 May 2008 | Jerry L. Bona University of Illinois at Chicago (USA) |
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| 9 May 2008 | Trevor Wooley University of Bristol |
Tales from the wild Diophantine west: integral solutions of
polynomial equations Diophantine equations (polynomial equations to be solved in integers) with few variables have attracted the enthusiastic attention of number theorists for millenia, and indeed the work of Wiles concerning Fermat's Last Theorem even attracted the attention of the mass media. In contrast, the solubility of diophantine equations in many variables is a wild frontier with, for the most part, only sketchy knowledge and speculative conjectures. We will provide an overview of the latter area, illustrating our discussion with an exotic tale - the story of the little known race half a century ago to solve cubic equations in many variables. The ideas underlying this story continue to inspire modern developments. |