Math 496 F1H Home Page

This is the home page for Math 496F1H, "Introduction to Mathematical Research". This class meets for the Spring 2018 semester, MWF 2:00-2:50 in 343 Altgeld (or 242 Armory for the duration of the GEO strike.)

Class Diary, blogger style

F 3/2 -- We are in 242 Armory now for the duration of the strike. TN talked about random graphs, one of the innovations of Paul Erdos. I started to talk about finite differences.

M 2/26, W 2/28 -- Class cancelled in deference to extra classes at the end of the semester.

F 2/23 -- I talked about Beatty sequences and will eventually find a good link for them.

W 2/21 -- DT talked about "Euclidean rhythms" and illustrated them with the best drumming in 20 years of Math 496. I meant to distribute a paper I co-wrote with Doug West in the 1980s, on Tverberg's theorem. I'll give it out on Friday. Class will be cancelled for M 2/26 and two other days, to allow for extra class meetings at the end of the semester.

M 2/19 -- MF talked about a topic in algebraic combinatorics on the pattern of the parity (mod 2) of the size of sets and their intersections, and a connection with quadratic forms and rank. I mentioned Tverberg's theorem, and then talked about that crazy theorem every math major should know.

F 2/16 -- No student speakers. I gave a generalization of Zeilberger's problem, in which we summed the squares of the powers minus 1. This gave a review of second semester calculus. I told some stories, gave out a useful guide to LaTex, and then talked about the Chebyshev polynomials. Lots of sources are available; here's Wikipedia.

W 2/14 -- No student speakers. I gave a variety of proofs of the centroid triangle problem; and distributed a couple of pages of a 1986 paper of mine on lattice point simplices. I also gave Doron Zeilberger's brilliant tricky proof (handout for Friday) of a problem from Goldbach.

M 2/12 -- JH talked about his Math 428 project from last fall on the Stern sequence. I gave out a copy of Keith Devlin's essay on tools for solving "real-world" mathematical problems, and talked a little bit about barycentric coordinates and Caratheodory's Theorem. More on W 2/14.

F 2/9 -- TN talked about Turan's theorem and maximal graphs without certain subgraphs. Here is a handout involving computations with symmetric polynomials. I also gave a couple of proofs of the "one point triangle theorem."

W 2/7 -- I spoke the whole time, on Pick's Theorem and various related matters. More to follow, as time allows. Links here are to a handout on Pick's Theorem, and to the GIMPS project for finding new Mersenne Primes.

M 2/5 -- PG talked about a generalization of the Chinese Remainder Theorem to more general rings. MF talked about Young Tableaux and their generalizations. I distributed a few pages from our recommended text on the Art Gallery problem and the LAS page honoring David Blackwell in the Gallery of Excellence and his New York Times obituary. I also talked a bit about what I'd like to see in talks, including "topic sentences" and more explanation of what you are doing in your presentation. The time is now available for more presentations. The next "required one" will be after you choose your semester project.

F 2/2 -- XL_2 talked about the "Art Gallery Problem", taken in part from the recommended text, "Proofs from the Book". I distributed a few pages of an article on Maria Agnesi's famous calculus textbook which we will see later this semester in the UIUC Rare Book and Manuscript Library. I also talked about David Blackwell: The LAS page honoring David Blackwell in the Gallery of Excellence and his New York Times obituary and read excerpts from his biographical interview.

W 1/31 -- ZC talked about Dirichlet's theorem on primes in arithmetic progression, a few instances in which it can be proved by elementary means. (I added a bit more on the full theorem and the Prime Number Theorem.) XL_1 talked about "denesting", by which is meant the exact evaluation of certain infinitely nested square roots. The original results were by the great mathematician Ramanujan. This link will give a .pdf of the source material for the talk. (Some of the identities on p.2 have been repurposed in the past as problems for the Mauritius math contest.)

M 1/29 -- CDS talked about differentiation and integration in the complex plane, especially as they apply to the exponential function. I talked a bit more about the x^y = y^{2x}. (Error in 1/26 web entry will stay there.) I passed out the Monthly article, and by comparison, here is the preprint.

M 1/29 (addition to W 1/24 entry): DW's sources were: were the video websites Scam school and 3Blue1Brown.

F 1/26 -- JB talked about the Principle of Inclusion and Exclusion and the Derangement problem, and TN talked about bipartite graphs and odd cycles with some generalizations to planar graphs. This took most of the class, but I mentioned briefly some history of the three utilities problem. (This link might only work from UI machines.) I hinted at the next math I'll talk about, which involves the equation x^y = y^x and x^{2y} = y^x for positive reals and positive rationals.

W 1/24 -- DW talked about the famous "three utilities" problem, which is equivalent to the planarity of the complete bipartite graph K_{3,3}. The handouts were: the wikipedia page on the MSC classification, plus the search page at MathSciNet, the research classifications page on the department website, the FAQ for the arXiv, and some puzzle problems from the OEIS.

M 1/22 -- JH talked about some graph theoretical questions, including the maximum number of edges in a planar graph. Two handouts: Scott Kim's What is a puzzle? and the Welcome to the essential OEIS. I finished saying what I wanted to say about the Fibonacci numbers.

F 1/19 -- You will start talking on Monday! Several more handouts: Two old sets of notes on the Fibonacci numbers, and linear algebra. Prof. Francis Edward Su's guide to writing homework for math classes, Doug Shaw's tips for Making good talks into great ones, Joe Gallian, an expert in undergraduate research, on How to give a good talk,

W 1/17 -- First day. Several handouts: Course Organization, from George Polya, a problem-solving template , Class questionnaire and "One Introduction to Mathematical Research". This contains the homework assignment for Friday. This should be written up (no more than a page is needed):

  • Present your favorite theorem and proof or problem and solution.
  • Change your favorite in some way, and prove or solve it again.
  • Change your favorite in another way, so that you no longer know how to prove or solve it. Top of Page