# Mathematics Colloquia

## Spring Term 2008/2009

## Abstracts

Friday 9 January 2009 **Nigel Hitchin** (Oxford)

*The space of icosahedra*

Friday 16 January 2009 **Alfio Quarteroni** (Milan / Lausanne)

*Mathematical modeling and the Galileo legacy*

Mathematical modeling aims to describe the diﬀerent aspects of the real world, their interaction, and their dynamics through mathematics. It constitutes the third pillar of science and engineering, achieving the fulﬁllment of the two more traditional disciplines, which are theoretical analysis and experimentation. Nowadays, mathematical modeling has a key role also in ﬁelds such as the environment and industry, while its potential contribution in many other areas is becoming more and more evident. One of the reasons for this growing success is deﬁnitely due to the impetuous progress of scientiﬁc computation; this discipline allows the translation of a mathematical model—which can be explicitly solved only occasionally—into algorithms that can be treated and solved by ever more powerful computers. They can explore new solutions in a very short time period, thus allowing the speed of innovation cycles to be increased. In this lecture we will discuss the role of mathematical modeling and of scientiﬁc computation in applied sciences; the way modern mathematical modeling is influenced by Galileian intuition on the description of natural phenomena and the Galileo scientific method; the importance of mathematical modeling in simulating, analyzing, and decision making. We will show some results and underline the perspectives in diﬀerent ﬁelds such as industry, environment, life sciences, and sports. The interplay between analytical models, numerical methods, and advanced algorithms for scientific computing will be highlighted.

Friday 23 January 2009 **Bill Meeks** (University of Massachusetts Amherst)

*The classical theory of minimal and constant mean curvature surface theory with an emphasis on the uniqueness of examples found by Euler, Delaunay, Scherk and Riemann.*

Part of the classical theory of minimal surfaces in three-dimensional Euclidean space deals with the asymptotic properties of complete embedded classical examples M. This theory then lends itself to obtain classification results of the surfaces subject to the constraint that the surfaces satisfy some geometric or topological constraint. For example, if M is simply-connected, then recent work of Colding-Minicozzi and of Meeks-Rosenberg demonstrates that the only examples are the plane and the helicoid. Very recently, Meeks and Tinaglia have extended this result to show that a complete, simply-connected, embedded of non-zero constant mean curvature M must be a sphere, thereby completing a more general classification question. My talk will be for a general audience and undergraduates should enjoy the computer graphics images and the history presented.

Friday 30 January 2009 **Robert Kerr** (Warwick)

*Circulation collapse in 3D Euler: Numerical bounds.*

Friday 6 February 2009 **Nigel Boston** (Madison)

*Random Groups in Number Theory and Topology".
*

In a recent Inventiones paper Dunfield and Thurston compared fundamental groups of random 3-mainfolds and random discrete groups. Analogously, we develop a theory comparing certain random Galois groups and random p-groups. This leads to some mysterious mass formulae, some infinite groups that arise with nonzero probability, and applications to number theory.

Friday 13 February 2009 **Michael Berry** (Bristol)

*Three recent results on asymptotics of oscillations*

The results are separate, and apparently paradoxical, and have implications for physics. First, when two exponentials compete, their interference can be dominated by the contribution with smaller exponent. Second, repeated differentiation of almost all functions in a wide class generates trigonometric oscillations ('almost all functions tend to cosx'). Third, it is possible to find band-limited functions that oscillate arbitrarily faster than their fastest Fourier component ('superoscillations').

Friday 20 February 2009 **Joe Chuang** (City University London)

*TBA*

Friday 27 February 2009 **Nick Higham** (Manchester)

*How and How Not to Compute the Exponential of a Matrix*

Friday 6 March 2009 **John Ratcliffe**

*Some examples of aspherical homology 4-spheres*

In this talk the construction of infinitely many examples of aspherical Riemannian 4-manifolds that are homology 4-spheres will be described. These manifolds are constructed by performing Dehn surgery on a complete, open, hyperbolic 4-manifold of finite volume which can be realized as the complement of five disjoint tori in the 4-sphere. Infinitely many of these homology 4-spheres admit an Einstein metric of negative scalar curvature. The existence of aspherical homology 4-spheres answers an old question of William Thurston and solves Problem 4.17 on Kirby's 1977 low-dimensional topology problem list. This work is joint with Steven Tschantz of Vanderbilt University.

Friday 13 March 2009: TBA

## Autumn Term 2008/2009

## Abstracts

Friday 3 October 2008 **Andreas Dress** (Shanghai Institute for Computational Biology)

Some mathematical aspects of molecular evolution

Friday 10 October 2008 **Chris Jones** (Warwick)

*Data and models: A marriage made in mathematics*

The technology-enabled increase in data acquisition is matched by the greater amounts of available information from computational modeling. The subject of Data Assimilation (DA) addresses the issue of making optimal use of this profusion of observational and computational data. I will argue that DA can contribute to enhancing both the predictive capacity of models and the physical understanding of the underlying system. A mathematical framework for DA is provided by Bayes’ Theorem which in turn opens up challenging mathematical areas. From the perspective of Lagrangian DA, which attempts to assimilate data from ocean instruments that flow with the fluid, I will discuss some open issues and suggest a guiding principle.

Friday 31 October 2008 **Erhard Scholz** (Wuppertal)

*Transitions from Weyl's early gauge geometry to physics and differential geometry*

Hermann Weyl formulated his gauge geometrical generalization of Riemannian geometry in 1918 ( purely infnitesimal geometry ), trying to unify gravity with electromagnetism and to find a field-theoretic explanation of matter. Although his original goal remained without success, the broader concept of gauge structures had a tremendous and varied influence on subsequent developments in physics and mathematics, and even on the philosophy of space (although less than in the other fields). Surprisingly enough, even Weyl's original concept of scale ( length ) gauge of 1918 had a kind of revival in the 1970s in foundational studies of general relativity and field physics. This talk will follow some of these developments,although necessarily highly selectively.

Friday 21 November 2008 **Stefan Friedl** (Warwick)**
**When does a knot bound a disk?

A knot is called slice if it bounds a disk in the 4-ball. We will give some explicit examples and we will explain how the theory

of slice knots ties in with the seminal work of Simon Donaldson and Mike Freedman.