# Mathematics Colloquium

## Summer Term 2009/10

Organisers: 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.

• Friday 30 April 2010 Juan Luis Vázquez (Universidad Autónoma de Madrid) Porous Medium Flow With Fractional Diffusion

Abstract: We study a model for flow in porous media including nonlocal (long-range) diffusion effects. It is based on Darcys law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed, which is unexpected in fractional diffusion models.
The model has also the very interesting property that mass preserving selfsimilar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that the asymptotic behaviour is described after renormalization by these solutions which play the role of the Barenblatt profiles of the standard porous medium model.
This is a joint project with Luis Caffarelli. Other authors are involved.
References:
arXiv:1001.0410v1 [math.AP], Caffarelli-Vazquez;
arXiv: 1004.1096v1 [math.AP] Caffarelli-Vazquez;
arXiv:1001.2383v1 [math.AP], with de Pablo et al.

• Friday 7 May2010 Terry Gannon (Alberta) The Search for the Exotic
• Friday 14 May 2010 John Hunton (Leicester) Aperiodic Order

Abstract: Aperiodic order is a term that has developed to refer to the study of structures such as tilings or Delone sets in euclidian space that have high degrees of regularity, but no global translational symmetry. Although such objects have appeared in various branches of mathematics for centuries, the recent interest in them began in the 1960's with a problem in theoretical computer science; it developed as a topic in combinatorial and discrete geometry. By the 1970's we had well known examples such as the Penrose tilings, but from the 1980's the topic has taken a new twist with the discovery of such patterns "in nature", first as the atomic structures of so-called Quasicrystals, and more recently as possible models for virus capsids. Over the years, the study of aperiodically ordered patterns has used ideas from a variety of mathematical disciplines, including dynamics, topology and various branches of geometry as well as ideas from mathematical physics. This talk will introduce the topic and describe some recent work in trying to understand aperiodic order, presented mainly from a topological perspective.

• Friday 21 May 2010 Peter Deuflhard (ZIB Berlin) The smile of the mathematicians: mathematical modelling and simulation in cranio-maxillofacial surgery

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.

• Friday 28 May 2010 Gabriel Paternain (Cambridge) Symplectic Topology of Mane's Critical Values

Abstract: Consider a closed Riemannian manifold $M$ and let $\sigma$ be a closed 2-form whose pull-back to the universal covering of $M$ is exact. I will discuss the changes in the symplectic topology of a hypersurface $|p|^2=2k$ in the twisted cotangent bundle determined by $\sigma$ as $k$ makes its transition from high energies to low energies. It has been known for some time (Aubry-Mather theory) that drastic changes in the dynamical properties of the hypersurface take place at the Ma\~n'e critical values.

I will try to relate these phase transitions to symplectic properties like displacement, stability and vanishing of the Rabinowitz Floer homology.

• Friday 11 June 2010 Karen Vogtmann (Cornell) Outer Spaces

Abstract: An “outer space” for a group G is a contractible space with a proper action of the group Out(G) of outer automorphisms of G. Classical examples include homogeneous spaces and Teichmuller spaces. For a free group F of finite rank, an outer space was introduced in the mid-1980s. The basic idea is to think of an automorphism of a free group topologically, as a homotopy equivalence of a finite graph. In this talk, I will describe this outer space and indicate how it is used to obtain algebraic information about Out(F). I will then show how theses ideas have recently led to the construction of outer spaces for other types of groups.

• Friday 18 June 2010 Yuri Prokhorov (Moscow) Finite Subgroups of the Cremona Group and a Question of Serre
• Friday 25 June 2010 Nicholas Ercolani (University of Arizona) Random Matrices, Caustics and Counting Graphs
Abstract: This talk will report on some recent developments in the study of (Hermitian) random matrices that brings together several areas of mathematics including statistical mechanics, combinatorics, classical function theory, integrable systems theory, and asymptotic analysis. In particular, we will focus on describing the asymptotic behavior of the free energy of the random matrix ensemble as the matrix size becomes arbitrarily large. This leads to some significant progress on a graphical enumeration program initiated by Tutte in the 1960's and, as a consequence, some surprising new insights into a 20 year old conjecture related to the so-called double scaling limit of 2D quantum gravity.

## Spring Term 2009/10

Organisers: 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.

• Friday 15 January 2010 Christian Maes (Leuven) Goal functions and Variational principles, from analytical mechanics to irreversable thermodynamics

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.

• Friday 22 January 2010 Benjamin Schlein (Cambridge) Bulk Universality for Wigner Matrices

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.

• Friday 29 January 2010 Mitchell Berger (Exeter) Applications of braid theory

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.

• Friday 5 February 2010 Chris Parker (Birmingham) Identifying Groups

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.

• Friday 12 February 2010 Viktor Schroeder (Zürich) Moebius geometry on the boundary of symmetric spaces

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.

• Friday 19 February 2010 John Ball (Oxford) The Q-tensor theory of liquid crystals

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.

• Friday 26 February 2010 Toby Stafford (Manchester) Noncommutative Projective Surfaces

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.

• Friday 5 March 2010 Alicia Dickenstein (Buenos Aires) Binomials, hypergeometric functions and mass action kinetics chemical reaction network

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.

• Friday 12 March 2010 Peter Deuflhard (ZIB Berlin) The smile of the mathematicians: mathematical modelling and simulation in cranio-maxillofacial surgery

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.

• Friday 19 March 2010 Laurent Bartholdi (Göttingen) Insanely twisted rabbits
1. 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 2009/10

Organisers: 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.

• Friday 9 October 2009 Christian Lubich (Tübingen) Variational approximations in quantum dynamics and the MCTDH method

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.

• Friday 16 October 2009 Norbert Schappacher (Strasbourg) Political Space Curves

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.

• Friday 23 October 2009 Peter Cox (Exeter) Climate change: feedbacks, tipping points and radical solutions

• Friday 30 October 2009 Antonio DeSimone (Trieste) Excellent Swimmers

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.

• Friday 6 November 2009 Gero Friesecke (Warwick/Munich) How good is the quantum mechanical explanation of the periodic table?

• Friday 13 November 2009 Michel Crucifix (Louvain) Reduced order models of palaeoclimates : techniques, challenges and promises

• Friday 20 November 2009 Shaun Bullett (Queen Mary) Holomorphic correspondences: how to deform the modular group to a quadratic map

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.

• Friday 27 November 2009 Panos Papasoglu (Athens) JSJ decompositions for graphs, groups and metric spaces

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.

• Friday 4 December 2009 Burkard Wilking (Münster) Sharp estimates for the Ricci flow

• Friday 11 December 2009 Chris Budd (Bath) Optimal transport methods for mesh generation with applications to meteorology