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Emissary Mathematical Sciences Research Institute Spring/Summer 2010 Emissary Mathematical Sciences Research Institute www.msri.org An Eventful Spring Free Boundary Problems A word from Director Robert Bryant Henrik Shahgholian The spring semester here at MSRI has been an especially active one. In addition to our two scientific programs, Arithmetic Statis- Many problems in physics, industry, finance, biology, and other ar- tics and Free Boundary Problems, which have attracted two en- eas can be described by partial differential equations that exhibit a thusiastic groups of mathematicians to keep the building busy and priori unknown sets of positive codimension, such as phase bound- which have had numerous well-attended workshops, we’ve had a aries, optimal stopping strategies, contact sets or shocks, as the variety of other exciting activities and workshops. case may be. (continued on page 2) The study of such sets, known as free boundaries, has a central role in such problems. The geometry and analysis of the under- lying mathematical problems are the qualitative features that have provided the greatest challenges for mathematicians for decades and still pose many open problems. Paradigmatic examples are the classical Stefan problem and more general models of phase transitions, where the free boundary is the moving interface between phases. Other examples come from problems in surface science, plastic molding and glass rolling, filtration through porous media, where free boundaries occur as fronts between saturated and unsaturated regions, and others from reaction-diffusion, fluid dynamics, mathematical finance, biology, and water wave theory. Many of these applications are divided into one-, two-, and multi- phase problems, depending on the number of regions in which the underlying field equations are to be solved. FBPs is an area of mathematics and its application that remains theoretically vibrant while giving unique practical insight and it will surely stay on the mathematical map forever. (continued on page 3) Contents Arithmetic Statistics (cont’d) 8 Free Boundary Problems (cont’d) 3 Puzzles Column 7 Math Circles 3 Director’s Letter (cont’d) 2 Honors 5 Fredrik Strömberg Call for Applications 11 What does this stunningly beautiful pattern have to do with Staff Roster 12 counting primes? See article starting on page 8. 1 What is Arithmetic Statistics? Brian Conrey, Barry Mazur, Michael Rubinstein, and Nina Snaith Number Theory has its share of conjecture and heuristics that thrive Manjul Bhargava together with his students and co-authors have on—if not depend on—the accumulation of aggregates of in- been• developing extremely precise methods for counting appropri- stances, aggregates of numerical data. ate orbits of certain arithmetic groups acting on integral points on certain lattices. This approach follows and significantly refines the To see that numerical data related to numbers themselves is also classical Methods in the Geometry of Numbers (as had pursued by at the very heart of the pleasure of number theory, and is a major Gauss, Minkowski, Siegel, and others). A major application of this reason for the very theory itself, consider this letter of Gauss to one work of Bhargava and co-authors is to establish counts of important of his students (italics mine): ingredients of the arithmetic of elliptic curves. Among their appli- Even before I had begun my more detailed investigations cations is the result of Bhargava–Shankar that the average rank of into higher arithmetic, one of my first projects was to turn the Mordell–Weil group of elliptic curves over Q — when they are my attention to the decreasing frequency of primes, to ordered in any of the standard ways—is less than 1:5. which end I counted the primes in several chiliads [groups of 1000] and recorded the results on the attached white This result is related to their study of the average size of the 2- pages. I soon recognized that behind all of its fluctuations, Selmer rank of elliptic curves (again over Q — and when they are this frequency is on the average inversely proportional to ordered in any of the standard ways). They show that the average the logarithm, so that the number of primes below a given size is three.1 For any prime number p the reduced p-Selmer rank 2 bound n is approximately equal to R dn= log n [...]. Later of an elliptic curve over a number field has this important prop- on, when I became acquainted with the list in Vega’s tables erty: it is finite (!), computable (!) (at least in principle), and is an (1796) going up to 400031, I extended my computation fur- upper bound for the rank of the Mordell–Weil group of the ellip- ther, confirming that estimate. In 1811, the appearance of tic curve over the number field. If the Shafarevich–Tate conjecture Chernau’s sieve gave me much pleasure and I have fre- holds, then for all but finitely many primes p, the reduced p-Selmer quently (since I lack the patience for a continuous count) rank would be equal to that Mordell–Weil rank. So it is natural, as spent an idle quarter of an hour to count another chiliad in the results of Bhargava and co-authors alluded to above, to ex- here and there. pect that the statistics of p-Selmer ranks (e.g., even when restricted to p = 2) contribute to our understanding of Mordell–Weil ranks. Often, in modern number theory, to actually sample a sufficient In the course of our program we have been learning the most recent quantity of data that might allow you to guess even approximate advances in this direction (3-Selmer, 5-Selmer). qualitative behavior of the issue you are studying, you may have to go out to very high numbers. For example, there are basic ques- The heuristics predicting “average sizes” of quite a few impor- tions about elliptic curves (e.g., what is the frequency of those pos- tant• arithmetic objects have also been the focus of our program. We sessing two independent rational points of infinite order) where if are fortunate to have both Henri Cohen and Hendrik Lenstra among 8 you only look at curves of conductor < 10 , you might be tempted us. They are the co-originators of the Cohen–Lenstra heuris- to make guesses that are not only wrong, but qualitatively wrong. tics that guides conjectures regarding average sizes of ideal class groups and other important invariants in number theory. The latest Our program Arithmetic Statistics then stands for those aspects development in the formidable toolbox of heuristics is due to Bjorn of number theory—be it theory or computation—that connect Poonen and Eric Rains and has a somewhat different slant; it gives closely with concrete (important) numerical data related to num- one precise guesses for the probabilities of reduced p-Selmer ranks bers themselves. for elliptic curves over a given number field (when these curves Many people in our program are engaged in the theoretical side of are ordered in the usual way). This too has been one of the fo- our subject, and many in the computational side. Much “theoreti- cuses of our program. A few years ago, Peter Swinnerton-Dyer, cal” modern number theory bears on, and sometimes has vital need extending earlier results of Heath-Brown, studied the probabilities of large scale computing projects and large data-bases. And both of reduced 2-Selmer ranks of families of elliptic curves that are the computational and theoretical facets connect to some of the fa- quadratic twists of some very specific types of elliptic curves over mous heuristics in our subject: Cohen–Lenstra heuristics (average Q. One grand (and enticing) feature of Swinnerton-Dyer’s study expected size of various finite abelian groups that appear in our is that the probabilities arise as if they were the product of a spe- subject); and Random matrix heuristics. The computational and cific Markov process; another curious feature — a drawback, per- theoretical facets of our subject form one interlocking unity. haps—is that the statistics are dependent upon ordering the elliptic curves in the twist family not in the standard way but in terms of We next discuss a few examples of recent work that have been the number of primes dividing the discriminant. All the issues that themes of our program. are brought up by this work are focuses of current research in our 1 Of course no 2-Selmer group can have such a size: these 2-Selmer groups are then all either above or below average. 2 This is the dimension of the so-called p-Selmer group minus the rank of rational p-torsion of the elliptic curve over the number field. 8 program. Specifically, Dan Kane’s work in the program is towards Now, with their asymptotic version of the large sieve, CIS have 1 relating such Swinnerton-Dyer statistics dependent upon different studied zeros on the 2 -line, not only of Dirichlet L-functions, but orderings of the collection of elliptic curves being sampled, while of other families as well: twists of a fixed L-function of degree 2 by 1 Karl Rubin, Zev Klagsbrun, and Barry Mazur are developing an Dirichlet characters (at least 36% of their zeros are on the 2 -line) approach (which has a “Markov process feel”) to unconditionally and twists of L-functions of degree 3 (at least 0.5% of their zeros 1 prove the expected statistics for reduced 2-Selmer ranks over an are on the 2 -line). In addition, CIS have been able to confirm a arbitrary number field for all quadratic twists families of many el- prediction from random matrix theory about the sixth moment of 1 liptic curves (the elliptic curves in any of these families are ordered Dirichlet L-functions at the point 2 , averaged over characters with in a certain not entirely unnatural, but again nonstandard, way).
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