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 underlying 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). In moduli up to Q. They prove a formula which includes all of the separate work, Jonathan Hanke is collaborating with Bhargava and main terms and has an error term which is a power of Q smaller Shankar to obtain the asymptotics for the 2-part of the class group than the main terms The main terms are expressed in terms of sim- of n-monogenic orders in cubic fields. ple factors multiplied by a ninth degree polynomial in log Q. The leading coefficient of the polynomial is 42 and the lower terms are Dirichlet L-functions are the simplest generalizations of the Rie- • given explicitly in terms of complicated arithemetic and geometric mann zeta function. They have been used to prove an asymptotic factors. The theorem exactly matches the predictions arising from formula for the number of primes up to a quantity X in a given random matrix theory, and provides excellent confirmation of the arithmetic progression modulo q. Like the Riemann zeta function RMT models for L-functions. We are very fortunate to have as each Dirichlet L-function can be expressed as Dirichlet series (the participants in our program all five authors of the paper in which Riemann zeta function has Dirichlet series coefficients 1;1;1;::: the predictions, now confirmed, were first detailed: Brian Conrey, and the first Dirichlet L-function has coefficients periodic modulo David Farmer, Jon Keating, Michael Rubinstein, and Nina Snaith. 3, namely 1; -1;0;1; -1;0;::: ), has a functional equation and Eu- 1 ler product, and is conjectured to have its zeros on the 2 -line; the Several researchers are examining statistics for curves over finite latter assertion is sometimes called the generalized Riemann hy- •fields. The zeros of the zeta function are the inverses of the eigen- pothesis. It can be proven that each individual Dirichlet L-function values of the Frobenius endomorphism. The work of Katz and 1 has at least 40% of its zeros on the 2 -line. Conrey, Iwaniec, and Sarnak indicates that when the genus g is fixed and the characteris- Soundararajan (CIS) have now shown that when all of the zeros of tic q tends to infinity, the normalized zeros are distributed like the these Dirichlet L-functions are taken together at least 55% of these eigenvalues of matrices in a group of random matrices determined 1 zeros are on the 2 -line. To be specific, take a large number Q and by the monodromy group of the moduli space of the curves. But consider all of the L-functions associated with a primitive character the related question of studying statistics as q remains fixed and the modulo q where q Q. Now consider for all these L-functions genus g grows to infinity is still largely unknown, though recent ≤ all the zeros located in the rectangle of complex numbers with real progress has been made in computing the distribution of the trace parts between 0 and 1 and imaginary parts between 0 and log Q. of the Frobenius endomorphism for various families by Kurberg– CIS can prove that at least 55% of the zeros in this rectangle have Rudnick, Bucur–David–Feigon–Lalín and Bucur–Kedlaya. 1 real part 2 . 14 The technique used by CIS is called the• asymptotic large sieve. It can be used to give an asymptotic formula for a quan- 12 tity that would have previously been es- timated by the large sieve inequality. The latter has been a staple of number theorists 10 for over 40 years. One spectacular application of the large sieve inequality is to prove the Bombieri–Vinogradov theorem, 8 which asserts that when counting primes up to X in arithmetic progressions with moduli up to Q, the error terms behave, 6 on average, as well as could be expected, that is, as well as could be proved assum- 4 ing the Generalized Riemann Hypothesis. This is a deep and famous result; indeed, Enrico Bombieri won the Fields medal in 2 1974 for this work. A few years ago Gold- ston, Pintz, and Yildirim used the BV theorem to prove their celebrated theorem to 0 -10000 -5000 0 5000 10000 the effect that the smallest gaps between Michael Rubinstein consecutive prime numbers are an order of The imaginary parts of the zeros of quadratic Dirichlet L-functions of discriminant magnitude smaller than the average gaps. with absolute value less than 10000.

9 A plot of the zeta function near the 1032nd zero, by Jonathan Bober and Ghaith Hiary.

The broader question of computing the global distribution of the make it more suitable for computations proving finiteness theo- zeros in the g limit remains. This is a nontrivial modeling job, rems, and making tables of quadratic forms in 3 and 4 variables since the global obstruction imposes an infinite, but discrete, set (over Q and some small number fields) together with Robert Miller. of conditions that the matrix model should satisfy. Such a model needs to exhibit both discrete and continuous features in order to In the study of elliptic curves over totally real number fields capture the global phenomenon. Bucur and Feigon, together with like• Q(√5) (recent work of William Stein) one is naturally led to their collaborators David and Lalín, will be working in this direc- Hilbert modular forms. Work of Shimura and recently of Ikeda in tion while at MSRI. Japan indicates that there is a similar connection between modular forms of half integral weight and modular forms of integral weight Computation and experimentation have been playing a large over number fields as it is well-known for Q. However, as it is role• in our program. For example, postdocs Jonathan Bober and known from the theory over Q, there are several advantages to re- Ghaith Hiary have been implementing Hiary’s world’s fastest al- place in such a theory the modular froms of half integral weight gorithms for the Riemann zeta function, computing zeros of “(s) by Jacobi forms. The Fourier coefficients of these Jacobi forms with =s near 1036, and using their data to test some conjectures correspond (in the theory over Q) to the central value in the critical about the behavior of the zeta function. Rubinstein is developing strip of the twisted L-series of the associated Hilbert modular forms general purpose algorithms for computing L-functions and is also or elliptic curve over Q. Skoruppa and his student Hatice Boylan gathering extensive numerical evidence in favor of the generalized are preparing a longer article to set up such a theory over arbitrary Riemann hypothesis. William Stein has been tabulating ellip- number fields based on results of Boylan’s thesis. In particular, tic curves over Q(p(5)), and verifying the Birch–Swinnerton- they want, in joint work with Fredrik Strömberg, to compute suffi- Dyer conjecture. John Cremona is working on his programs to ciently many examples of Jacobi forms over Q(√5) which should systematically find curves of a given conductor over Q, with a complement the computations of William Stein on elliptic curves view to doubling the range of his tables and verifying (or other- over Q(√5). wise) that there is no curve of rank 4 and conductor less than 234446. Nathan Ryan, Nils Skoruppa, and Gon- zalo Tornaría, in collaboration with Martin Raum, have been studying methods for computing with Siegel modular forms which have degree 4 L-functions associated to them. Duc Khiem Huynh is attempting to develop probabilistic models for these L-functions with the goal of testing the predictions using data provided by Ryan, Skoruppa, Tornaría, and Raum. Skoruppa is working on a new algorithm for computing modular forms of half integral weight directly from the periods of the associated modular forms of integral weight. This will make it possible to tabulate half integral modular forms of very high level without the need of computing complete (and then very high dimensional) spaces as is required by the currently known algorithms. David Farmer, Stefan Lemurell, and Sally Koutsoliotas are developing methods for finding Maass forms for higher rank groups and testing conjectures regarding their Fourier coefficients and associated L-functions. Jonathan Hanke is working with Tornaría, and also collaborator Will Jagy, on classi- fying regular and spinor regular ternary quadratic forms, Contour plot of a Maass form, by Fredrik Strömberg. Another example improving the the modular symbols code in SAGE to can be seen on page 1.

10 In the case of Siegel modular forms a conjecture of Böcherer— the Arithmetic Statistics program, which most of the Connections originally stated in the case of the forms for the full symplectic participants stayed on to enjoy. group—relates sums of Fourier coefficientsof a form to the central Three other workshops form a part of our program. Our intro- values of its twisted spinor L-series, generalizing the formulas for ductory workshop was held from January 31 to February 4 and the coefficients of Jacobi forms (over Q) mentioned above. Similar featured talks to help define the direction of our program. Talks formulas for the case of paramodular forms are being investigated were given, in order of appearance, by: Henri Cohen, Karl Ru- by Ryan and Tornaría. This case is of particular interest due to the bin, Manjul Bhargava, Michael Rubinstein, Nina Snaith, Melanie so-called Paramodular Conjecture which proposes a bridge to ge- Wood, Brian Conrey, Andrew Sutherland, Jordan Ellenberg, David ometry by relating spinor L-series attached to paramodular forms Farmer, John Voight, Henryk Iwaniec, Akshay Venkatesh, John with Hasse–Weil L-functions attached to rational abelian surfaces Cremona, Bjorn Poonen, William Stein, Kannan Soundararajan, (analogue to the Modularity Theorem of Wiles et al). Ryan and Chantal David, and Frank Thorne. Tornaría are trying to find algorithms to compute Fourier coefficients of these paramodular forms in a large scale. This would not Several of the partipants in our program are also involved in a large only provide more evidence for both the Paramodular Conjecture scale NSF funded collborative Focused Research Group project to and the paramodular extension of Böcherer Conjecture, but would develop methods for computing with L-functions and associated also allow a large scale computation of central values of twists for automorphic forms, as well as verify many of the important con- degree 4 L-series, useful for testing and refining random matrix jectures in this area. In order to help diffuse the large mount of heuristics for degree 4 L-series. data being generated by the project, an archive with a user friendly front end for browsing and searching the data is being developed, How the Program Unfolded and a workshop involving fifteen participants was held at MSRI from February 21 to 25 to continue developing the archive. Learning seminars, whereby our participants meet weekly to teach The last workshop for our program was held from April 11 to 15 each other and discuss material relevant to our research, form an on the theme of Arithmetic Statistics. It highlighted some of the important part of our program. The Bhargava–Shankar group has work being carried out at the MSRI during our program. been meeting to learn material related to the work of Bhargava and Shankar on ranks of elliptic curves. The explicit formula group is studying the problem of ranks from an analytic perspective. The MSRI invites membership applications for the 2012-2013 aca- low lying zeros seminar has been looking at papers related to the demic year in these positions: distribution of zeros in families of L-functions. Quadratic twists of elliptic curves meets to discuss the problem of ranks of ellip- Research Professors: by October 1, 2011 tic curves in fmailies of quadratic twists. Earlier, a group was Research Members: by December 1, 2011 meeting to study the Cohen–Lenstra heuristics and its extension Postdoctoral Fellows:by December 1, 2011 to Tate–Shafarevich groups by Christophe Delaunay. Lastly, a few researchers are holding a seminar to study paramodular forms. In the academic year 2012–2013, the research programs are: The first workshop to take place as part of the Arithmetic Statistics Cluster Algebras, fall semester program was the two-day Connections for Women event. This tar- Organized bya˘ Sergey Fomin (University of Michigan), Bern- geted female mathematicians in fields related to the program, but hard Keller (Université Paris VII), Bernard Leclerc (Université we were pleased to see that all aspects of the workshop were well- de Caen), Alexander Vainshtein (University of Haifa; chair- attended by the program’s participants, which lead to a very even man), Lauren Williams (University of California, Berkeley). mix of male and female researchers. The Connections for Women workshop was a very agreeable mixture of excellent talks, a buzz Commutative Algebra, fall and spring semesters Organized of mathematical discussion and a chance to meet new people; ev- bya˘ David Eisenbud (University of California, Berkeley; chair- ery math workshop should be like this! The audience enjoyed six man), Srikanth Iyengar (University of Nebraska), Ezra Miller superb talks by leading women in the area, ranging from the num- (Duke University), Anurag Singh (University of Utah), and ber theory involved in cryptography to several of the questions of Karen Smith (University of Michigan).a˘ counting (ranks, points on curves, number fields) that are themes Commutative Algebra Noncommutative Algebraic Geome- of the rest of the program. try and Representation Theory, spring semester. The discussion session on pursuing a career in mathematics saw Organized bya˘ Mike Artin (MIT), Victor Ginzburg (Univer- senior mathematicians giving advice on how to apply for first jobs sity of Chicago), Catharina Stroppel (Universitat Bonn), Toby and postdoctoral positions, some anecdotes about how dual-career Stafford (University of Manchester; chairman), Michel Van den couples have found posts in the same institution, and strategies for Bergh (University of Hassalt), Effim Zelmanov (University of departments keen to increase the number of women in their fac- California, San Diego).a˘a˘ ulty. With participants covering the spectrum from undergraduates to those with a long career behind them, the discussion was lively MSRI uses MathJobs to process applications for its positions. and productive. Interested candidates must apply online at www.mathjobs.org after August 1, 2011. These two days then lead into the main Introductory workshop for