SSL Notes for meeting #1 (9/9) Martin is the notetaker for today Introductions, including outside interests Martin - Music, comics, computer science Hal - Did SSL before. Abby - Music, movies, fashion, Students for a Free Tibet Emily - Minnesota Twins, skiing, read, math Josh - Music (recording, synth), fables, job? Sam - Modern European and American literature, jazz, backgammon Paul - Golf, guitar/piano, music, reading Steven - Basketball (dislocated shoulder), sail Jeremy - sailing, snowboarding, hiphop music Jim - Cafes. Cambridge is home of the best cafe but at least Madison has wireless cafes (Muddy Waters, Electric Earth, Mother Fool's). Acapella/choral singing. Jeremy: EVP Coffee has wireless? But at least good coffee. This is the smallest group I've run in a long time (maybe 7 years!) This year, more of a focus on process (as well as outcome) One part of that is giving me feedback on how things are going for you, for others, or for the group as a whole (even more than usual). Another part is, there may be some formal assessment (via questionnaires). Distribute contracts Student commitments ------------------- "Attend weekly group meetings" "Help create a comfortable intellectual atmosphere": We're here to have fun at the frontier of knowledge Attending to content and process; how we say what we say Two goals: * advance my research agenda (things I *really* want to know the answer to) * advance your educations In particular, the note taker records people's statements of what they intend to do. As in when I say: "By this weekend, I'll make sure that you've all been added to the email lists for the domino and bilinear forums". Expectation: Note taker should post draft of minutes on her/his SSL site later that evening or the next morning; Jim will send his corrections Note-taker should anticipate problems. "Wendy, I'm not going to be able to copy that all down. Can you email it to me for inclusion in the notes?" Spend 6 hours per week on SSL outside of meetings. Some weeks more, some weeks less. You'll get more out of it if you put in more time. This room will be available 2 hours a day. In mid-October, a web-accessible cluster of computers can let you run Maple wherever you are. For now, less on computers and more on math background. The reading will be on articles by other undergraduates. Hal's article? Hal will send the link to the list when it is up (will wait for the test message). Last year, lots of students wrote papers but not enough read and critiqued each other's papers. WHen you find a passage you don't understand, can be fixed. "Spend 6 hours per week on SSL outside of meetings": You'll get more out of it if you put in more time Everything counts: background reading; infrastructure (email, web-sites), even "homework" Suggest you keep a research notebook Get Mathematica or Maple NOW (salary cap: $700 for students who buy Maple/Mathematica/whatever; $600 for students who don't) Related: What to do This room (B107 w/ Maple computers) will be available 2 hours a day In mid-October, a web-accessible cluster of computers can let you run Maple wherever you are For now, less on computers and more on math background The reading will be on articles by other undergraduates. Hal's? - Hal will send the link to the list when it is up Last year, lots of students wrote papers but not enough read and critiqued each other's papers. "Create a term project (document or software)" Could even be a t-shirt! "Write an end-of-term report" Only a few paragraphs, maybe an hour of time max. Helps me get more grants (e.g., if we need extra money for the spring!) Helps me remember what you did when you're looking for a letter of recommendation "Stay in contact with Supervisor and other group-members": "Call me Jim or Prof. Propp" send email to ssl@math.wisc.edu (?) (don't send big files on it, give people a webpage URL instead!) "Create and maintain a personal SSL web-page": avoid unintentional duplication of effort update every couple of weeks keep track of hours (no particular format, but should be clear) What does Wisconsin provide? SIT (Student Information Technology). Notoriously bad service - losing password means everything is gone? Everything is automated. www.sit.wisc.edu. CS and engineering have similar systems. Hal can help out if necessary. "Report on hours and activities every week": keep track! ("What counts?") weekly one-paragraph report sent on Friday, Saturday, or Sunday starting THIS WEEK [give them examples] To be posted on your private web-page (or, if something is private, via email) Primary missions (every week): What you've done (e.g., computations, summary of what you've read) How many hours you spent (hours spent at meetings count!) (Don't assume I've taken attendance; include all hours.) Secondary missions (every two or three weeks): What you're planning to do How you're feeling about your involvement with SSL (overwhelmed? understimulated? bored with the problems? left out because you don't know other people in the group?) Suggestions for how SSL could better serve your needs Billable time: Attending meetings counts (also Weds.) Avoid blind alleys: send me an email every 10 hours of research time "Communicate and cooperate with other students working on the same project": buddy system? importance of listening, debugging, proof-reading "Read email regularly and respond promptly": read every day (preferably) "Follow through on tasks" "Exercise initiative": e.g., tee-shirts; if you want some meeting-time to be devoted to something or other, propose it to me in email! Jim's commitments ----------------- "Give students interesting topics to think about" What I offer that most REU programs don't is a network of interconnected problems "Help students develop skills in solving problem and inventing new ones" "Help students obtain publishable results" I need to get better at this; lowered teach load may help "Maintain a central clearinghouse for exchange of information": jamespropp.org/SSL/ (public stuff) (minutes of meetings; software links) jamespropp.org/... (keep this private) Minutes: I can edit and return; it gets sent out about 24 hours after the meeting Other sources of information are www.math.wisc/~propp/tiling/, jamespropp.org/reach/, the domino forum and the bilinear forums. (Bilinear stuff is related to the talk that was on 9/8) (Don't be disturbed if you don't follow the postings!) Privacy issues (security through obscurity) Library can grow (manuals). Money from the grant can also be spent on supplies. Want to learn C? Maple? Mathematica? We can buy a book that you can borrow. (Catch: it ends up belonging to the SSL Library, not to you.) Maybe buy software too: send me email if you need something. "Pay salary (or submit grades) promptly": $10/hour; salary cap currently $600/term or $700/term. Where the money is coming from: VIGRE, (NSF encouraging vertical integration). "Write letters of recommendation for students": your final report comes in handy! A room for food in the break time will hopefully be reserved for next time. We take turns bringing refreshment. In the second half of the meeting, Jim showed the group: www.math.harvard.edu/~propp/reach/ www.math.harvard.edu/~propp/192/ "In the next few weeks, I'll do something like this for SSL." Students helped each other set up web pages, get access to Maple or Mathematica, etc. What I want: people learning, teaching, reading, writing, creating pictures, creating models, creating software, running experiments, enjoying themselves What I also want: articles to get written (with or without my name on them) What I offer: good problems; an overview of group-members and the ability to connect people with each other; some individual support Stephen will be augmenting the amount of support I can offer (in addition to participating in specific groups) What I also want: logistical support for my own work (e.g., if I'm giving a talk, I could use people to proof-read my slides) (Talk about puzzles and their relation to tilings) For Thursday: An understanding of snake graphs, which are a bunch of boxes. Each box is to the right of or above the last box. Good procedures for counting perfect matchings of these snake graphs. A two- box snake graph: .__.__. | | | .__.__. A perfect matching: Each vertex is paired up with exactly one other vertex by selecting edges in the graph. Here's an example: . . . | | | . . . If we select the first vertical, there's another way: . .__. | . .__. Or we could pair the upper-left vertex with the vertex to its right, in which case the other choices are forced: .__. . | .__. . Work on: Various small snakes, counting matchings other than brute force elimination. Systematic. Interested in both solid ways "you can count on" to count these, as well as conjectured ways that seem to work better but you haven't proven it yet. Jim feels that he knows how to do this efficiently. Here is the pattern: .__.__.__. | | | | .__.__.__. Again, for the first (upper left) vertex we have two choices, down and right. After having matched first vertex down, what remains is the other graph we looked at. So 3 perfect matchings of that type: . . . . | | | | . . . . . . .__. | | . . .__. . .__. . | | . .__. . Or, we can pair the first vertext to the right. If we do this, there are two more possibilities: .__. .__. .__. .__. and .__. . . | | .__. . . .__.__.__.__. | | | | | has 5+3=8 perfect matchings .__.__.__.__. Fibonacci numbers: (1,1),2,3,5,8,13,21... Or we can alternate: 1, 2, 5, 13, ... vs 1, 3, 8, 13, ... 1+5 is thrice 2. 2+13 is thrice 5. 1+8 is thrice 3. 3+21 is thrice 8. F_{n-2}+F_{n+2}=3F_n Find a combinatorial proof. (Let O^k be the snake graph of k segments, e.g. OOOO is the graph with 8 perfect matchings shown above.) M(graph) = # of matchings of the graph. . U is disjoint union. . . . M(OOOOOO) U M(OO) = M(OOOO) U M(OOOO) U M(OOOO) Since disjoint sets, the union has cardinality equal to the product of their respective sizes. How do we process the left side to generate the right side? A bijection between the sets. A bijective proof you already know. Examine . M(OOOOOO) = M(OOOOO) U M(OOOO). 21 is the number of pefect matchings of OOOOOO. We've already given this. Either the first vertex is matched below or to the right. Cardinality of the straight snake follows the fibonacci recurrence and begins like fibonacci, therefore is equal to the fibonacci sequence (aside from the 1's). Not only a multiplicative but an additive relation between c and c'. It's like 3 copies of the same set. Disjoint union: Sums of the cardinalities, ignoring the intersection. "Nothing as complicated" as taking the cardinality of the multiset. . {-1,0} U {0,1} = {-1,0_1,0_2,1} . #(A U B) = #(A) + #(B) Ignore the metaphysical issue of taking 2 copies of the element 0. Want to see a variety of answers. Get started for Thursday. Want to know what our steps have been. More interested in examining matchings of small snakes. (Straight snakes are simple. Zigzag snakes are going to be harder. Winding snakes may be the next step.) If you had only 5 minutes, how would you work out this snake: OOOO O What about only 30 seconds? OO Jim will post his Markoff notes from the talk yesterday. Next notetaker is Paul. Notetaker should remind Jim to ask for the next notetaker at the end of the meeting. Notetaker should also remind Jim to ask for the next snackbringer. Abby is bringing the snack.