Mathematics Colloquium
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Spring Term 2009/10Organisers: Volker Betz (V.M.Betz@warwick.ac.uk) and Brian Bowditch (B.H.Bowditch@warwick.ac.uk)
Additional colloquia, titles and abstracts will be added as details become available. (
Colloquia take place on Friday afternoons at 4.00pm, Lecture Room B3.02 in the Mathematics Institute, Zeeman Building. They are directed towards a general mathematical audience. In particular, one the functions of these Colloquia is to inform non-specialists and graduate students about recent trends, ideas and results in some area of mathematics, or a closely related field.
Abstract: Variational principles and the exploitation of goal functions for characterizing the dynamics of a system are ubiquitous in mathematical physics. Their origin is somewhere with Maupertuis' and Fermat's principle, becoming a crucial tool in analytical mechanics. We will discuss what becomes of these principles when dissipation is added, and how analysis meets probability theory in the characterization of complex behaviour.
Abstract: Wigner matrices are N by N (hermitian or real symmetric) matrices, whose entries are, up to the symmetry constraints, independent and identically distributed random variables. In this talk, I will present recent results concerning the spectral properties of Wigner matrices, in the limit of large N. In particular, I am going to discuss a proof of the universality of the local eigenvalue statistics.
Abstract: Two great puzzles in solar astrophysics concern the source of coronal heating and the distribution of solar flares. The atmosphere of the sun is heated to one million degrees or more, possibly by swarms of tiny flares. These tiny flares could be consequences of the braiding of magnetic field lines. Reconnection between braided threads of magnetic flux can release energy stored in the braid. The larger flares exhibit a power law energy distribution. Several authors have suggested that a self-organization process in the solar magnetic field could lead to such a distribution. Here we show how reconnection of braided lines can organize the small scale structure of the field, leading to power law energy release. An application of braids to mixing theory will also be discussed. Magnetic Helicity Magnetic helicity measures geometric and topological properties of a field, such as twist, writhe, shear, linking, and braiding. I will discuss the relation between helicity and field structure, and show how magnetic helicity can be calculated directly from field line data. The flow of helicity into and out of the solar corona can be observed, and gives information on both the solar dynamo and solar activity.
Abstract: In this talk, I will discuss the third generation approach to the classification of the finite simple groups. I will particularly focus on what might be called the end game: the point where the groups are actually named.
Abstract: It is a classical fact that the isometries of the hyperbolic space correspond to the Moebius maps of its boundary. In the talk we will discuss the Moebius geometry on the boundary of other symmetric spaces, in particular of the complex hyperbolic space, and we will give characterizations of this structure in terms of metric Moebius geometry.
Abstract: The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe. This is joint work with Apala Majumdar and Arghir Zarnescu.
Abstract: In recent years a surprising number of significant insights and results in noncommutative algebra have been obtained by using the global techniques of projective algebraic geometry. In particular the classification of noncommutative analogues of curves and surfaces has lead to some startling examples and constructions that have had wide-ranging applications. This talk will survey some of these results and applications.
Abstract: A binomial is a polynomial with two terms. Algebraic varieties cut out by binomials have a rich combinatorial structure. In the first part ot the talk we shall highlight basic -yet not so known- facts about binomial systems as basic blocks in the study of general polynomial systems. In the second part, we shall concentrate on two occurrences of binomials in a differential setting: in the beautiful formulation by Gelfand, Kapranov and Zelevinsky of multivariate hypergeometric systems of linear PDE´s, and in the reversible mass action kinetics chemical reaction systems of non linear ODE´s.
Abstract: Some mathematicians have a problem with a smile -- a mathematical one. The talk will present results of a joint project with surgeons in the field of cranio-maxillofacial surgery. In this kind of surgery, faces are operatively changed to remove unpleasant face distortions. Bone from the upper or lower jaws is removed or shifted on a scale of cm's. The questions treated by mathematicians are: (1) operation planning within teleconferences of ZIB and the clinic, based on a detailed geometric model of skull and facial tissue, (2) prediction of the facial appearance on the basis of numerical simulation of the patients soft tissue, based on fast adaptive multilevel finite element methods for the elastomechanic partial differential equations, either the linear Navier-Lamé equations or nonlinear enrichments like geometric nonlinearity or nonlinear material laws of Ogden type. The second problem type leads to nonconvex optimization. Extensions of affine conjugate Newton methods for convex optimization problems to the nonconvex case here (more precisely: polyconvex case) are given. During the talk, a lot of results for real patients are displayed, including the effect of such operations on the smile.
Abstract: (Topological) branched coverings of the sphere, modulo a natural ("isotopy") relation, are interesting combinatorial objects; and a result by Thurston explains, at least theoretically, when such a covering is equivalent to a rational map. I will explain how such coverings can be conveniently encoded in group theory, and how that language can be used to answer a long-standing open problem by Douady and Hubbard, the "Twisted rabbit problem". I will then discuss visualizations of "matings" of polynomials (the topological branched covering obtained from gluing together two polynomials at infinity) through the same method. This is joint work with Volodya Nekrashevych.
Autumn Term 2009/10Organisers: Volker Betz (V.M.Betz@warwick.ac.uk) and Brian Bowditch (
Additional colloquia, titles and abstracts will be added as details become available. (
Colloquia take place on Friday afternoons at 4.00pm, Lecture Room B3.02 in the Mathematics Institute, Zeeman Building. They are directed towards a general mathematical audience. In particular, one the functions of these Colloquia is to inform non-specialists and graduate students about recent trends, ideas and results in some area of mathematics, or a closely related field.
Abstract: The talk describes model reduction in the multi-particle time-dependent Schrodinger equation via the Dirac-Frenkel variational approximation principle and then turns to the multi-configuration time-dependent Hartree method (MCTDH) as an important, practically very successful example. This approach can be viewed as a dynamical low-rank approximation. The MCTDH nonlinear equations of motion and their numerical integration are discussed. The talk closes with an analysis of the modelling error in the MCTDH method, showing the mechanisms that may lead to convergence or failure.
Abstract: Many algebraic geometers know the story of an example, published by the young Theodor Vahlen in 1891, which showed that an algebraic curve in 3-space could not in general be described by less than 4 equations. It held its own for a full fifty years until Oskar Perron destroyed it in 1941 with perfectly elementary and transparent arguments. A few algebraic geometers know that, due to the differing attitudes of Vahlen and Perron to the Nazi regime, Perron's paper triggered a few politically oriented mathematical publications (!), which only after the war gave way to a discussion of "set- theoretic" versus "ideal-theoretic (complete) intersections" in Algebraic Geometry. Almost no algebraic geometer has looked at Vahlen's original paper, though. We will recall the story and take it a step further. This will bring up the question how mathematics contrives to be a cumulative science. Note. No knowledge of Algebraic Geometry is required to follow the plot.
Abstract: We will discuss swimming strategies for microscopic swimmers and recipes to optimize their strokes. The talk will review the fundamentals of biological fluid dynamics, applications to the engineering of micro-robots, the geometric structure underlying the mathematics of low Reynolds number swimming, and numerical algorithms for optimal control.
Abstract: The world of holomorphic correspondences on the Riemann sphere - multivalued functions defined by algebraic equations - is an unfamiliar mathematical environment in which Kleinian groups can 'morph' into rational maps, but where the classic tools of conformal dynamics, like the Measurable Riemann Mapping Theorem, still deliver results. We describe the behaviour of a family of examples in which the modular group $PSL_2(Z)$ may be deformed to the quadratic map $z^2+1/4$, and discuss some questions that arise.
Abstract: JSJ decompositions originated in 3-manifold theory and are the starting point for the classification of 3-manifolds. They have been generalized since in other contexts. I will give a review of JSJ-type results in graphs, groups and metric spaces, focusing more in JSJ theories for hyperbolic groups, CAT(0) groups and finitely presented groups.
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