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Next week, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence. This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, and the maximum principle, but there will be more of an emphasis on rigour, generalisation and abstraction, and connections with other parts of mathematics. If time permits I may also cover topics such as factorisation theorems, harmonic functions, conformal mapping, and/or applications to analytic number theory. The main text I will be using for this course is Stein-Shakarchi (with Ahlfors as a secondary text), but as usual I will also be writing notes for the course on this blog.
For the past few months, Cambridge University Press (in consultation with a number of mathematicians, including Tim Gowers and myself) has been preparing to launch a new open access journal (or more precisely, a complex of journals – see below) in mathematics, under the title “Forum of Mathematics“, as an experiment in moving away from the traditional library subscription based model of mathematical academic publishing. (The initial planning for this journal happened to precede the Cost of Knowledge boycott, but the philosophy behind the journal is certainly aligned with that of the boycott, which I believe is further evidence that the time has come for mathematical journal reform.) The journal will formally begin accepting submissions on October 1st, but it has already been officially announced by Cambridge University Press, with an editorial board (with Rob Kirby as managing editor, thirteen other editors including Tim and myself, and a board of associate editors that is still in the process of being assembled) and FAQ already in place.
In many respects, Forum of Mathematics functions as a regular mathematics journal, in that papers are submitted by the authors, sent out to referees by the editors, and (if accepted) published by the Forum. There are however a couple of important features that distinguish the Forum from traditional mathematics journals. The first is the open access, publication-charge based publishing model (sometimes known as “gold open access”). Namely, all articles will be freely available without subscription charges, but authors, upon their paper being accepted, be asked to pay a publication charge to cover costs. The publication charges will be set at zero for the first three years, and then raised to somewhere around £500 GBP or $750 USD after the initial three-year period (with fee waivers available for authors from developing countries). (One reason that the publication charges are not entirely fixed at this point is that there is the possibility of obtaining additional funding sources for this journal, for instance from philanthropic organisations, which may allow for fee reductions or additional waivers.) This is of course a non-trivial sum of money, but it is significantly lower than the charges for most other gold open access journals. Also, editorial decisions will not be influenced by the author’s ability to pay for the charges, which only come into effect in the event that the paper is accepted for publication.
(One way in which Cambridge University Press is keeping costs low, by the way, is to keep the journal purely electronic, with physical issues available on a print-on-demand basis only. A side benefit of this choice is that there is no hard constraint on how many or how few pages will be published each year, so that acceptance decisions will not be influenced by artificial constraints such as the size of the journal backlog.)
Another distinctive feature of Forum of Mathematics lies in its scope and structure. It is not exactly a single journal, but is instead a complex consisting of a generalist flagship journal (officially known as Forum of Mathematics, Pi) and a specialist journal (Forum of Mathematics, Sigma) which is in turn loosely organised for editorial purposes into “clusters” for each of the major subfields of mathematics (analysis, topology, algebra, discrete mathematics, etc.). As a first approximation, Pi is intended as a top-tier journal (on the level of, say, the Journal of the American Mathematical Society, Inventiones, or Annals of Mathematics) that only accepts significant papers of interest to a wide audience of mathematicians, while Sigma resembles a collection of specialist journals, one for each major subfield of mathematics. However, the journals will be using the same editorial interface, and so it will be possible to easily transfer a submission between journals or clusters (while retaining all the referee reports and other editorial data). This is meant to help address a common issue in traditional mathematical journals, namely that if the editorial board decides that a submission falls too far outside the scope of the journal, or is not quite at the desired level of quality, then the authors have to start all over again with a new journal (and new referee reports). Of course, it is still possible (and perhaps even fairly common) that a submission to Forum of Mathematics will be deemed unsuitable for either Pi or Sigma, and thus rejected entirely; but the structure of the Forum should give some additional flexibility, to reduce the frequency that papers are rejected for artificial reasons such as being out of scope. (Of course, we would still expect authors to aim their submission at the most appropriate location to begin with, in order to reduce the time and effort expended on processing the paper by everyone involved.)
Further discussion of this journal can be found at Tim Gowers’ blog. It should be fully operational in a few months (barring last-minute hitches, we should be open for submissions on 1 October 2012). Of course, a single journal is not going to resolve all the extant concerns about the need for journal publishing reform, such as those raised in the Cost of Knowledge boycott; but I feel that it is important to have some experimentation with different publishing models, to see what alternatives to the status quo are possible.
In a few weeks (and more precisely, starting Friday, September 24), I will begin teaching Math 245A, which is an introductory first year graduate course in real analysis. (A few years ago, I taught the followup courses to this course, 245B and 245C.) The material will focus primarily on the foundations of measure theory and integration theory, which are used throughout analysis. In particular, we will cover
- Abstract theory of -algebras, measure spaces, measures, and integrals;
- Construction of Lebesgue measure and the Lebesgue integral, and connections with the classical Riemann integral;
- The fundamental convergence theorems of the Lebesgue integral (which are a large part of the reason why we bother moving from the Riemann integral to the Lebesgue integral in the first place): Fatou’s lemma, monotone convergence theorem, and the dominated convergence theorem;
- Product measures and the Fubini-Tonelli theorem;
- The Lebesgue differentiation theorem, absolute continuity, and the fundamental theorem of calculus for the Lebesgue integral. (The closely related topic of the Lebesgue–Radon-Nikodym theorem is likely to be deferred to the next quarter.)
See also this preliminary 245B post for a summary of the material to be covered in 245A.
Some of this material will overlap with that seen in an advanced undergraduate real analysis class, and indeed we will be revisiting some of this undergraduate material in this class. However, the emphasis in this graduate-level class will not only be on the rigorous proofs and on the mathematical intuition, but also on the bigger picture. For instance, measure theory is not only a suitable foundation for rigorously quantifying concepts such as the area of a two-dimensional body, or the volume of a three-dimensional one, but also for defining the probability of an event, or the portion of a manifold (or even a fractal) that is occupied by a subset, the amount of mass contained inside a domain, and so forth. Also, there will be more emphasis on the subtleties involved when dealing with such objects as unbounded sets or functions, discontinuities, or sequences of functions that converge in one sense but not another. Being able to handle these sorts of subtleties correctly is important in many applications of analysis, for instance to partial differential equations in which the functions one is working with are not always a priori guaranteed to be “nice”.
Ben Green, Tamar Ziegler, and I have just uploaded to the arXiv the note “An inverse theorem for the Gowers norm (announcement)“, not intended for publication. This is an announcement of our forthcoming solution of the inverse conjecture for the Gowers norm, which roughly speaking asserts that norm of a bounded function is large if and only if that function correlates with an -step nilsequence of bounded complexity.
The full argument is quite lengthy (our most recent draft is about 90 pages long), but this is in large part due to the presence of various technical details which are necessary in order to make the argument fully rigorous. In this 20-page announcement, we instead sketch a heuristic proof of the conjecture, relying in a number of “cheats” to avoid the above-mentioned technical details. In particular:
- In the announcement, we rely on somewhat vaguely defined terms such as “bounded complexity” or “linearly independent with respect to bounded linear combinations” or “equivalent modulo lower step errors” without specifying them rigorously. In the full paper we will use the machinery of nonstandard analysis to rigorously and precisely define these concepts.
- In the announcement, we deal with the traditional linear nilsequences rather than the polynomial nilsequences that turn out to be better suited for finitary equidistribution theory, but require more notation and machinery in order to use.
- In a similar vein, we restrict attention to scalar-valued nilsequences in the announcement, though due to topological obstructions arising from the twisted nature of the torus bundles used to build nilmanifolds, we will have to deal instead with vector-valued nilsequences in the main paper.
- In the announcement, we pretend that nilsequences can be described by bracket polynomial phases, at least for the sake of making examples, although strictly speaking bracket polynomial phases only give examples of piecewise Lipschitz nilsequences rather than genuinely Lipschitz nilsequences.
With these cheats, it becomes possible to shorten the length of the argument substantially. Also, it becomes clearer that the main task is a cohomological one; in order to inductively deduce the inverse conjecture for a given step from the conjecture for the preceding step , the basic problem is to show that a certain (quasi-)cocycle is necessarily a (quasi-)coboundary. This in turn requires a detailed analysis of the top order and second-to-top order terms in the cocycle, which requires a certain amount of nilsequence equidistribution theory and additive combinatorics, as well as a “sunflower decomposition” to arrange the various nilsequences one encounters into a usable “normal form”.
It is often the case in modern mathematics that the informal heuristic way to explain an argument looks quite different (and is significantly shorter) than the way one would formally present the argument with all the details. This seems to be particularly true in this case; at a superficial level, the full paper has a very different set of notation than the announcement, and a lot of space is invested in setting up additional machinery that one can quickly gloss over in the announcement. We hope though that the announcement can provide a “road map” to help navigate the much longer paper to come.