This "Invitation to Mathematics" consists of 14 contributions, many from the world's leading mathematicians, which introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active mathematicians, but together they show mathematics as a rich and lively field of research. The contributions are written for interested students at the age of transition between high school and university who know high school mathematics and perhaps competition mathematics and who want to find out what current research mathematics is about. We hope that it will also be of interest to teachers or more advanced mathematicians who would like to learn about exciting aspects of mathematics outside of their own work or specialization. Together with a team of young 'test readers', editors and authors have taken great care, through a substantial 'active editing' process, to make the contributions understandable by the intended readership.

Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.

What is the correct concept behind measures of inflation? Does money cause business activity or is it the other way around? Shall we stimulate growth by raising aggregate demand or rather by lowering taxes and thereby providing incentives to produce? Policy-relevant questions such as these are of immediate and obvious importance to the welfare of societies. The standard approach in dealing with them is to build a model, based on economic theory, answer the question for the model world and then apply the results to economic phenomena outside. Data come in, if at all, only in testing a limited number of the model's consequences. Despite some critical voices, economic methodology too has by and large subscribed to a "theory first" approach to applied economics. Error in Economics systematically develops an alternative to the theory-based orthodoxy. It places the methodical study of evidence at the centre of the scientific enterprise and thus provides a foundation for a methodology of evidence-based economics. But the book does not stop at the truism that claims should be based on the best available evidence. Rather, detailed studies in the areas of measurement, causal inference and policy analysis show what it means for a claim to be evidence-based in the context of a concrete case. The examples discussed concern topics as diverse as consumer price indices, radio spectrum auctions, the transmission mechanism, natural experiments on minimum wages and the evaluation of counterfactuals for policy. Error in Economics is essential reading for economic methodologists, philosophers of science and anyone interested in how claims about socio-economic matters are validated.