A handout was given out with background info on Somos sequences. It contains a typo; "Your job: find it!" We are currently meeting in the 4th floor math lounge at 3 on Tuesdays and Thursdays, moving to room 411 at 4. We are looking for a new location. The people present at the beginning of the meeting introduced themselves: Jim Propp: Head of the group, Professor at Harvard, interested in Combinatorics, Probability and Algebra. Trevor Bass, Seth Kleinerman, Dennis Clark, David Speyer, Gabriel Carroll, Dan Abramson, Gregg Musiker and Jason Burns. Jason is an MIT grad student Propp has hired to help run the group, he will also be doing research. Some other people arrived later, I don't have their names. Propp talked a little about relations to physics, CS, number theory and linguistics. We will be studying something called the Hirota equation, which also has physical connections, and many of our combinatorial problems arise from or connect to models from statistical mechanics. In CS, we will be studying huge combinatorial objects and polynomials, so there is a major need for efficient code; there are also connections to the theory of formal languages. In number theory, the Somos sequences we will be studying have connections to values of elliptic and theta functions, which also relate to elliptic curves and abelian varieties, no one seems to have really worked this out in detail. In Linguistics, we will often be given lists of monomials or tilings that are admissible and have to determine the "grammer". The program has two goals. The first is to further Propp's research. The second is to teach us (the students) how to do math research, communicate it to others and collaborate on it. The goal of REACH this term will be to understand connections between multi-dimensional combinatorics problems and polynomial recurrences. As a trivial example, consider the number of ways to tile a 1xN strip with 1x1 and 1x2 rectangles; this is F_N, the Nth Fibonacci. F_N obeys the linear recurrence F_N=F_{N-1}+F_{N-2}. When we move to higher dimensional problems, linear recurrences don't work. This is obvious to tell because the data grow too fast. E.g. consider the number of tilings of a (2n)x(2n) rectangle with 1x2 and 2x1 rectangles. This is clearly bounded below by 2^{n^2}, because we can divide it into n^2 2x2 squares, each of which can be tiled with two dominoes in two different ways. Solutions to linear recurrences can only grow like c^n or c^{n log n}, never like c^{n^2}. However, quadratic recurrences often do the job. For example, consider what is known as the Aztec diamond of order n. As an example, the Aztec diamond of order 4 is given below XX XXXX XXXXXX XXXXXXXX (Each "X" represents a 1-by-2 square.) XXXXXXXX XXXXXX XXXX XX We again ask how many ways this can be tiled by dominos. There is a simple answer: 2^{n(n+1)/2}. A better way to understand this, though, is to say that these numbers obey the recurrence A_{n+1}=(2*A_n^2)/A_{n-1}. Not that, if we had not known that it arose from a combinatorial problem, it would not be clear that this recurrence gives integers. There are many polynomial recurrences of this sort, including the Somos sequences described on the handout, for which it is known that they give integer values, but no combinatorial reason is known. These current proofs use a technique known as "cluster algebras". Moreover, in these cases, it is conjectured but not proven that, in a certain sense, the terms are not only integral but positive. This would follow immediately from a combinatorial proof, but is immune to cluster algebras. Essentially, there are generalizations of these recurrences that give sequences fo polynomials instead of integers, it is conjectured that all the coefficients of these polynomials are positive. More logistical issues: we will meet on Tuesdays and Thursdays, 3-5. Let Prof. Propp know if you're going to miss a session. If you have a regular conflict but still want to participate, tlak to Propp, but it may not be possible. In general, attendance is important so that collaboration occurs. Each day should have a notetaker. The notes should be sent to Propp the next morning after the meeting; ASCII, PostScript and HTML are acceptable formats. The minutes will be archived. Dennis will take notes at the next meeting (2/14). The REACH webpage is the one sent out by Propp in his recent e-mail. Remember that this is meant to be secret, so do not link to it or websites will index it. It isn't really secure, so don't post national secruity matters to it. There is also a public website, www.math.harvard.edu/~propp/reach . This site needs work, especially in the way of posting documentation of our work. In general, we need more documentation, in order to help bring new members into the group and in order to show to grant organizations. There will be a group T-shirt, so think about designs throughout the term. Aztec diamonds exhibit highly interesting statistical behavior. (Random domino tilings of large Aztec diamonds don't look random!) This term, one of our goals will be to determine what Somos-4 counts. Some goals after that might be to find an algorithm to sample uniformly from whatever that class is that is counted and to determine statistical properties of a typical member. We have an e-mail list, reach@math.harvard.edu . It will be archived, we need someone to volunteer to maintain the archive. Some scripts for this purpose already exist. Do not send attachments or long files to the list; instead, post this online and send the list links to them. Also, sign your messages. Propp moderates two forums: domino@math.wisc.edu and bilinear@math.wisc.edu. domino discusses tilings by "domino like objects", bilinear discusses quadratic recurrences. You should probably ask Propp's advice before posting. REACH participants will earn $10 an hour. If you are not a US citizen or not an undergraduate, there may be complications, talk to Propp now. Keep track of the hours you have worked. Work means time spent thinking seriously about REACH problems. Thinking while shaving doesn't count, thinking on a bus does, thinking in the shower might. Pay is capped at $1,000. Send Propp weekly reports on hours worked that week and how, total cumulative hours and future plans. Going to REACH meetings is billable time, so is going to Propp's talk at MIT. Veterans who haven't yet shold write summaries of their work last term. If you want to purchase references for your work with REACH, REACH can reimburse you. At the end of the term, REACH will keep the books it has paid for. A problem: define a sequence of rational functions f_i by f_0=x f_1=y f_n=(f_{n-1}^2+z^n)/f_{n-2} Compute the first 10 f_i, make conjectures and prove them. Jim hopes to write two omnibus articles this term, surveying reciprocity (last term's work) and Laurentness. It would be great to come up with some succint open problems in Laurentness phenomena. Future reading topics will be Somos sequences and Alternating Sign Matrices. It would also be good to learn about the connection between Somos sequences and theta functions.