Mathematics flourishes when ideas from one area of mathematics can be used in another area.  For a long time (certainly since Descartes), algebra has been a great asset to geometry.  In return, geometry has been helpful to algebra, for instance through representation theory.  Here are two very fine reviews of papers that bring algebra and geometry to the question of computational complexity.  Both reviews are by Peter Bürgisser. Continue reading →

Ron Solomon has been quite involved in the study of finite simple groups and their classification.  He is one of the coauthors of the monumental book series on the subject, which starts with The Classification of the Finite Simple Groups.  Now that we have a good classification of these groups, it is possible to look more carefully at their structure.  Solomon wrote a valuable review of a paper by Salarian and Stroth where they investigate the $p$-local structure of finite simple groups.  The review discusses the work that leads up to the present paper: earlier results that make the problem interesting and earlier results that make the present paper possible.  By laying out the context and how the present work fits into that context, Solomon gives us a look into how group theorists are obtaining increasingly refined information about the finite simple groups, revealing them to be rich objects of study.  Continue reading →

The problem of bad science has been in the news recently.  See, for instance this article in the New York Times, the website Retraction Watch, or this retracted article in Science.  In any of the sciences, researchers can get a sense that something is amiss with a published paper just by reading it.  Perhaps some circumstantial fact seems unusual.  A statistical analysis can provide some evidence that there really is a problem, either with the data or with the methods.  But in many cases, a researcher would have to try to recreate the experiment and the data to test the results – a daunting task.  Not so in mathematics.  In a mathematical article, everything is there on the page, open for scrutiny.  If there is a problem, the expert reader need only examine the arguments to discover it.  Oscar Zariski does exactly that in his review of an early attempt to prove resolution of singularities (in characteristic zero).  Zariski was famous for the way he increased the rigor in algebraic geometry by increasing the algebra in algebraic geometry.  In his analysis of Derwidué’s paper, he points out early on that the approach is too geometric to work.  Rather than stopping there, Zariski forges on, zeroing in on the weak points of the argument and giving full explanations of their weaknesses.

The real proof of the resolution of singularities in characteristic zero came thirteen years later, in the famous work of Hironaka.  Continue reading →