Past

Countable Borel equivalence relations arise naturally as orbit equivalence

relations for countable groups. For each countable Borel equivalence relation E

there is an infinitary sentence such that E is equivalent to the isomorphism

relation on countable models of that sentence. For first order theories the

question is open.

The development and quantitative validation of complex nonlinear differential equation models is a difficult task that requires the support by numerical methods for sensitivity analysis, parameter estimation, and the optimal design of experiments. The talk first presents particularly efficient "simultaneous" boundary value problems methods for parameter estimation in nonlinear differential algebraic equations, which are based on constrained Gauss-Newton-type methods and a time domain decomposition by multiple shooting. They include a numerical analysis of the well-posedness of the problem and an assessment of the error of the resulting parameter estimates. Based on these approaches, efficient optimal control methods for the determination of one, or several complementary, optimal experiments are developed, which maximize the information gain subject to constraints such as experimental costs and feasibility, the range of model validity, or further technical constraints.

Special emphasis is placed on issues of robustness, i.e. how to reduce the sensitivity of the problem solutions with respect to uncertainties - such as outliers in the measurements for parameter estimation, and in particular the dependence of optimum experimental designs on the largely unknown values of the model parameters. New numerical methods will be presented, and applications will be discussed that arise in satellite orbit determination, chemical reaction kinetics, enzyme kinetics and robotics. They indicate a wide scope of applicability of the methods, and an enormous potential for reducing the experimental effort and improving the statistical quality of the models.

(Based on joint work with H. G. Bock, S. Koerkel, and J. P. Schloeder.)

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

I will talk about some ongoing work, motivated by a long standing problem in

the theory of dynamical systems. In particular, I will explain how p-adic

methods lead to the construction of elements in SL_n(Z) whose eigenvalues e_1,

., e_n generate a free abelian subgroup of rank n-1 in the multiplicative group

of positive real numbers. This is a special instance of a more general theorem,

asserting the existence of strongly hyperbolic elements in arithmetic subgroups

of SL_n(R).