REACH meeting notes for 2/4/03 Jim said: "Welcome to REACH! I'm really excited to see so many of you here. My goal is to see you doing math, learning new things, and having fun as soon as possible." The notetaker for next time is Ananda. The notetaker is the referee: gets to say "Could you please spell that?" or "Could you send me that by email so that I can include it?" The notetaker makes sure to get everything down asking questions and pausing the meeting if needed. When in doubt, the note-taker should ask me or others "Am I supposed to be getting this down?" The notetaker should remind Jim to pick a notetaker for the next meeting (if Jim forgets). The notetaker also records commitments made by members of the group, e.g. Jim's commitment to updating the REACH web-page (see next item). Jim will update the REACH webpage since it is a new semester. The meeting begins with people stating their names and a mathematics related quote. The new members of reach this semester are Neil, Ian, Julian (not present), Cecilia, and Alex. Jim explained the group was getting bigger, making it more difficult than before to manage such a group. In the past it worked fairly well, though with large variance. However, he still wishes to improve the overall experience for all groups. "Teach what you know, learn what you don't, document everything." February plans - it's everyone's job to make sure that everyone is contributing. Give feedback as well. People last semester were often silent, and hung with projects they were not as enthusiastic about. Be friendly to newcomers and get them involved with the group. Jim mentioned http://www.math.harvard.edu/~propp/reach/minutes/min.02.09.24.txt as a good place to turn to for more basic info about REACH. The private REACH website: security through obscurity. New members: submit applications. Managing Groups Jim posed the question to the group of how to shuffle the groups for this semester. Perhaps all the new members should be put in a new group? He also posed the question of how to bring new members up to speed. The Math 192 course videos were an option, but it would take a long time and not all of the material was relevant to the current project. A 10 minute session was opened to the group to discuss this. Gabriel suggested that new members find projects that they find interesting and be brought up to speed by the experienced members. He was not sure how much of the body of knowledge from previous semesters of REACH was available online. Rui noted that doing the homework problems assigned at the beginning of REACH last semester were useful in getting a feel for the math. Help on Computer Issues Jim inquired if anyone needed computer help, especially in the areas of LaTeX, developing pictures, and programming help. Much of the work done in REACH is very geometric so it helps to have diagrams, pictures, and illustrations. Gabriel was not sure which mathematics program people preferred, but said he was willing to give people a session on mathematica. Alex is familiar with Matlab. Jim commented that java applets would be nice as it would make visible the mathematics we do on the web. T-shirt for REACH 2002/3 Anyone is welcome to contribute to the production of the shirts. The two main jobs are the design of the t-shirt (other than saying REACH on it semewhere, it could be pretty much anything) and the logistics of ordering and picking them up. A Math Problem: Spanning trees of bow-ties. Jim isn't sure whether this is a warm-up exercise or the point of entry for a whole new article that could be written. A spanning tree connects the vertices of a graph such that it is possible to get from one vertex to any other vertex in exactly one way. Ex. / has one spanning tree / Ex. /\ /\ / \ / \ = / \ , / , \ ----- / \ /----- ------\ The triangle has 3 spanning trees. Ex. I won't write it out, but /------/ / \ / / \ / ------\/ has 8 spanning trees. What about the graph below? /------/\ / \ / \ / \ / \ ------\/------ It turns out to have 21 spanning trees. The pattern is 1,3,8,21. Every other fibonnaci number. The question now is how to extend this sequence backwards? If so, we'll get .... -21, -8, -3, -1, 0, 1, 3, 8, 21 .... New Idea: Use lots of variables to create polynomials that correspond to spanning trees of these graphs. They will satisfy a similar recurrence relation as the fibonnacci's. We can associate polynomials to the set of all spanning trees of a graph by using a monomial term to represent a specific spanning tree. This involves labelling edges of the graphs with the variable x_1, x_2, ... /\ /\ / \ x_1 / \ x_3 = / \ , / , \ = x_1x_3+x_1x_2+x_2x_3 ----- / \ /----- ------\ x_2 Jim predicts that after running the recurrence backwards and finding combinatorial objects that correspond to this recurrence, the coefficients of the polynomials will be negative which would explain the negative value for the 'number' of spanning trees. Note while these initial graphs let us find a recurrence relation in the polynomials, we can run the polynomial recurrence backward to get clues as to the combinatorial objects we're observing. Here's how to develop the recurrence relation. We have to look at 'special forests' and spanning trees of these graphs. F of /------/**\ / \ / \ / \ / \ ------\/-------** are all the forests such that the two vertices labeled ** are not joined or connected via a path, and such that there is a path from all the other vertices to one of the two ** vertices. T of /-------/**\ / \ / \ / \ / \ ------\/-------** are just all the spanning trees of this particular graph. In the end we get the following two relations: F ( /-- ---/**\ ) = T( /\------/ )+F( /\------/ ) / \ / \ / \ / / \ / / \ / \ / \ / / \ / -----\/--------** ------ \/ -------\/ T ( /--------/**\ ) = 2*T ( /\-------/ ) + F( /\------/ ) / \ / \ / \ / / \ / / \ / \ / \ / / \ / ------\/--------** -------\/ -------\/ Translated into the polynomials, we get F ( /-- ---/**\ ) = T ( /\------/ ) + x_6*F( /\------/ ) / \ / \ / \ / / \ / / \ / \ / \ / / \ / -----\/-------** -------\/ -------\/ T ( /-----/**\ ) = (x_6+x_7) T ( /\------/ ) + x_6*x_7*F( /\------/ ) / \ / \ / \ / / \ / / \ / \ / \ / / \ / ------\-------** -------\/ -------\/ Running it backwards will give negative indices, negative exponents (due to division that will occur) and negative coefficients (due to subtraction that will occur) , but the expressions we get will still be Laurent polynomials. Working out the details is homework for Tuesday. Groups and Teams Jim plans to turn back to the original way of running REACH with more related topics. He plans on studying recurrences. One last recurrence: s_{n+1} = frac{s_n +1}{s_{n-1}} or frac{s_n^4 + 1}{s_{n-1}} alternating whether n is even or odd. (pick an index). The first two terms are 1 and 1. Then the first few terms of this sequence are..... 1, 1, 2, 17, 9, 386, 43, 8857, 206, ... Is there a combinatorial model for this? It might be interesting, though not a required thing to look at. Back to big-picture stuff: What Jim wants: people learning, teaching, reading, writing, creating pictures, creating models, creating software, running experiments, enjoying themselves. What Jim also wants: articles to get written (with or without his name on them) What Jim offers: good problems; an overview of group-members and the ability to connect people with each other; some individual support. David and Kyle will be augmenting the amount of support Jim can offer (in addition to participating in specific groups) What Jim also wants: logistical support for his own work (e.g., he'll be giving a talk in Florida next month; he could use some help working out specific formulas, and he could use people to proof-read my slides) End of REACH meeting. Notes submitted by Andy Itsara, and revised by Jim Propp.