## Reading Distinguished Colloquium

Once or twice a term, the Department of Mathematics and Statistics invites a distinguished mathematician or statistician to give a special colloquium. Unless stated otherwise, the colloquia take place on a Tuesday at 4pm, in the Ditchburn Theatre (JJ Thompson Building) and are followed by a reception hosted by the department. For further information please contact Brigitte Calderon or Michael Levitin.

### Colloquium 16 October 2012

**4pm - Room 314 Mathematics Building**

**Thanasis Fokas (Cambridge)**

A new method for analyzing boundary value problems (BVPs) for linear and integrable nonlinear PDEs, extending ideas of the inverse scattering transform method, was introduced in [1] and further developed by several researchers. First, this method will be introduced by using linear evolution PDEs in the half-line as illustrative examples. Then, regarding the nonlinear analogues of these problems it will be emphasised that: (i)For linearizable BVPs, the new method is as effective as the usual inverse scattering transform method. (ii) For general BVPs, with either decaying, or t-periodic boundary conditions, it yields effective long time asymptotic formulae.

The related development of the emergence of a new analytical approach to inverting integrals with applications in medical imaging, will be mentioned.

References [1] A.S. Fokas, A Unified Transform Method for Solving Linear and Certain Nonlinear PDEs, Proc. R. Soc. Lond. A 453, 1411-1443 (1997).

### Past Colloquia:

**Colloquium 22 May 2012**

**4pm - Room 113 Mathematics Building**

**Mikhail Feldman (Wisconsin)**

*Shock Reflection, Free Boundary Problems and Degenerate Elliptic Equations*

**Abstract:**

We discuss shock reflection problem for compressible gas dynamics, and von Neumann

conjectures on transition between regular and Mach reflections. Then we will talk about

some recent results on existence, regularity and geometric properties of regular reflection

solutions for potential flow equation. The approach is to reduce the shock reflection

problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic

type. Open problems will also be discussed. The talk will be based on the joint work

with Gui-Qiang Chen, and with Myoungjean Bae.

### Colloquium 13th March 2012

**4pm - Room 113 Mathematics Building**

**Lloyd N. Trefethen (University of Oxford)**

*Robust PadÃ© Approximation via SVD*

**Abstract:**

Approximating functions or data by polynomials is an everyday tool, starting with Taylor series. Approximating by rational functions can be much more powerful, but also much more

troublesome. In different contexts rational approximations may fail to exist, fail to be unique, or depend discontinuously on the data. Some approximations show forests of seemingly meaningless pole-zero pairs or "Froissart doublets", and when these artifacts should not be there in theory, they often appear in practice because of rounding errors on the computer. Yet for some applications, like extrapolation of sequences and series, rational approximations are indispensable.

In joint work with Pedro Gonnet and Stefan Guettel we have developed an elementary method to get around many of these problems for the computation of rational Pade approximants,

based on the singular value decomposition. The talk will include many examples.

* *

### Colloquium 7th February 2012

**4pm - Room 113 Mathematics Building**

**Dmitri Vassiliev (University College London)**

**Problems in the spectral theory of differential operators**

**Abstract:**

The first part of the talk will deal with spectral problems for scalar partial differential operators and, in particular, with the so-called Weyl Conjecture (existence of a two-term asymptotic formula for the counting function). This will be, essentially, a popular overview of the subject, charting its development from the non-rigorous work of physicists to the eventual rigorous proof of the Weyl Conjecture. This part of the talk will be based on the Safarov & Vassiliev book [1].

The second part of the talk will deal with spectral problems for systems.

The general consensus has always been that spectral analysis of systems is similar to that of scalar problems, only somewhat more complicated technically. I will show that systems are fundamentally different in that as soon as you are working with a pair of unknown functions instead of one, you get a very rich geometric structure emerging from spectral analysis. In particular, I will show that geometric concepts such as metric, teleparallel connection, torsion, spinor, Dirac Lagrangian and electromagnetic covector potential appear naturally as a result of spectral analysis of first order systems. This gives a new perspective on the origins of the main equations of theoretical physics in which all geometric concepts are, traditionally, introduced in an axiomatic fashion.

[1] Yu.Safarov and D.Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1997 (hardcover), 1998 (softcover).

### Colloquium 15th June 2011

(Wednesday, 4pm)

**Gilbert Strang (Massachusetts Institute of Technology)**

*Factoring Banded Matrices and Matrix Polynomials*

**Abstract:**

It seems fair to call these the four great factorizations of linear algebra:

A = LU and QR and QDQ' and USV'. I will speak about those and two others (minor by comparison). Banded matrices with banded inverses form an interesting group: they can be factored into tridiagonal matrices with tridiagonal inverses. These matrices are rare but useful - wavelet matrices and "CMV matrices" are leading examples.

When those matrices are block Toeplitz, with submatrices repeating down each (block) diagonal, all the information is in the matrix polynomial with those submatrices as coefficients. Suppose its determinant is 1 (constant!).

Then we look for linear factors with det = 1. This is a start on doubly infinite matrices, and I will look at the ordinary A = LU (or A = LPU) factorization when the usual elimination process has no reasonable place to start.

### Colloquium 18th February 2011

(Friday, 4pm)

**Wolfgang Wendland (Stuttgart)**

**Boundary integral equations, Trefftz elements, Levi functions**

### Colloquium 31st January 2011

**Jon Keating, FRS (Bristol)**

*Random Matrices and Number Theory*

**Abstract: **

Over the last 35 years, evidence has accumulated hinting at profound connections between random matrices, which are important in many areas of Mathematics, Science and Engineering, and the theory of the Riemann zeta function. Recently, a general understanding has developed which sets this in a much wider context, conjecturally linking a range of fundamental problems in number theory to properties of random matrices.

This talk will be a survey of some of the key ideas and developments.

Colloquium 15th November 2010

**Angus Macintyre (QMUL)President of the London Mathematical Society**

*Schanuel's Conjecture, and its implications for complex analysis*

**Abstract**: Schanuel's Conjecture, formulated around 1960, is about the algebraic independence of the values of the complex exponential function. Roughly speaking, it says that there are no algebraic relations between values of the exponential function which do not follow formally from the functional equation. Rather surprisingly, it has recently emerged, largely due to work in model theory, that this conjecture has implications for some old and very difficult questions in complex analysis.I will sketch the connections, and, if time permits, consider implications in the real domain.

Time and venue: 4pm - Physics Ditchburn Lecture Theatre, Whiteknights Campus, Reading