There is a widely held conception that progress in science and technology is our salvation, and the more of it, the better. This, however, is an oversimplified and even dangerous attitude. While the future will certainly offer huge changes due to such progress, it is far from certain that all of these changes will be for the better. The unprecedented rate of technological development that the 20th century witnessed has made our lives today vastly different from those in 1900. No slowdown is in sight, and the 21st century will most likely see even more revolutionary changes than the 20th, due to advances in science, technology and medicine. Particular areas where extraordinary and perhaps disruptive advances can be expected include biotechnology, nanotechnology, and machine intelligence. We may also look forward various ways to enhance human cognitive and other abilities using, e.g., pharmaceuticals, genetic engineering or machine-brain interfaces - perhaps to the extent of changing human nature beyond what we currently think of as human, and into a posthuman era. The potential benefits of all these technologies are enormous, but so are the risks, including the possibility of human extinction. This book is a passionate plea for doing our best to map the territories ahead of us, and for acting with foresight, so as to maximize our chances of reaping the benefits of the new technologies while avoiding the dangers.
This is a guest editorial for a so-called virtual illue of the journal. Its first paragraph reads as follows.
This virtual issue of Physica Scripta grew out of the one-day meeting Emerging technologies and the future of humanity that took place on 17 March 2014 at the Royal Swedish Academy of Sciences. The backdrop of the discussions at the meeting (as well as those in this virtual issue) is the realization that technologies developed in the next few decades can have profound influence on the future of humanity.
Oded Schramm (1961--2008) influenced greatly the development of percolation theory beyond the usual $\Z^d$ setting, in particular the case of nonamenable lattices. Here we review some of his work in this field.
This paper provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on $\mathbb{Z}^d$. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on $\mathbb{Z}^d$, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.
This paper begins with a review of the collection 18 Unconventional Essays on the Nature of Mathematics edited by Hersh (2006). Inspired especially by the contribution by Thurston to that collection, I then go on to discuss, by means of a couple of thought experiments involving computer "oracles", the nature of mathematics as a human activity, hopefully providing some balance to to the simplified view (sometimes held by research mathematicians such as myself) of the discipline as purely a quest for objective truth.
By means of a series of examples, taken from classic contributions to probability theory as well as from the author's own practice, an attempt is made to convince the reader that problem solving is often a matter of cooking up an appropriate Markov chain. Topics touched upon along the way include coupling, correlation inequalities, and percolation.
This paper surveys my debunking in the paper "Intelligent Design and the NFL Theorems" of anti-Darwinist William Dembski's mathematical smokescreens. (In Swedish.)
This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on $\Z^d$ and more generally on transitive graphs. For iid percolation on $\Z^d$, uniqueness of the infinite clusters is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
This paper gives an elementary discussion of the role of randomness in the theory of evolution, and a heuristic explanation of how advanced creatures such as ourselves can come about "by chance".
The book is now (as of May 30, 2002) available in the bookstores! Click here to order (or here if you are in North America). For copyright reasons - and to encourage you to buy the book - I no longer keep a postscript file publically available.
From the back cover:
Based on a lecture course given at Chalmers University of Technology, this book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomized algorithms with important applications in optimization and other problems in computing. Amongst the algorithms covered are the Markov chain Monte Carlo method, simulated annealing, and the recent Propp-Wilson algorithm. This book will appeal not only to mathematicians, but also to students of statistics and computer science. The subject matter is introduced in a clear and concise fashion and the numerous exercises included will help students to deepen their understanding.
From the ISI Short Book Reviews:
``Has climbing up onto the MCMC juggernaut seemed to require just too much effort? This delightful little monograph provides an effortless way in. The chapters are bite-sized with helpful, doable exercises (by virtue of strategically placed hints) that complement the text.''
This is a popular discussion about the law of large numbers, and an attempt to rectify some of the most common misunderstandings about it. (In Swedish.)
This is a large survey paper about applications of percolation theory in equilibrium statistical mechanics. The chapters are entitled as follows:
1. Introduction.
2. Equilibrium phases.
3. Some
models.
4. Coupling and stochastic
domination.
5. Percolation.
6. Random-cluster
representations.
7. Uniqueness and exponential mixing from
non-percolation.
8. Phase transition and percolation.
9. Random
interactions.
10. Continuum models.
This is a highly nontechnical discussion of the relation between random walks and electrical networks. (In Swedish.)
This is a set of lecture notes (90 pages) meant for an advanced undergraduate course in probability theory, with emphasis on concepts of convergence of random variables, transform methods, and a first gentle exposure to measure theory. (In Swedish.)