Math 496 F1H Home Page

This is the home page for Math 496F1H, "Introduction to Mathematical Research". This class meets for the Spring 2020 semester, MWF 2:00-2:50 in 343 Altgeld

Class Diary, blogger style

The mathematical links page is here.

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

M 3/9 -- Jet-lagged return. No handouts, various mathematical topics. You had to be there.

M 3/2, W 3/4, F 3/6 -- At Obwerwolfach. Photographic proof

F 2/28 -- JR spoke about cellular automata, including the Garden of Eden Theorem. I think I told some stories about mathematicians I knew, including John Horton Conway, who invented the brilliant cellular automaton called Conway's Game of Life; see the links at the bottom. If some of these are especially useful, please let me know and I'll put them in later.

W 2/26 -- Three handouts: two on creative blocking. The first was April in the 2014 Math Department Calendar, and the second was the report of a 2012 IGL Project. The third was a compact cribsheet on the Stern sequence. The math I talked about was mostly related to the Stern sequence. There is a huge amount of information on the Stern sequence at this page from an old summer graduate course.

M 2/24 I gave out a Wikipedia handout on Arnold's cat map, and some information about Oberwolfach, which is where I will be next week. SQ gave a talk about group theory and groups defined by generators and various Burnside problems. I talked a bit about an old IGL project of mine (handouts on W) and then about the Stern sequence, which I'll be returning to periodically for the rest of the semester.

F 2/21 -- I read a few more anecdotes from David Blackwell's oral history, linked on 2/10. Most of the time was spent on a presentation by YT on the famous Arnold Cat Map, which involves dynamical systems, linear algebra and, somehow, Fibonacci numbers. Lots of ideas.

W 2/19 -- Lots of odds and ends about topics from the previous days, plus, a handout on Pick's Theorem and one page from my paper on lattice point simplices. I also talked about Doron Zeilberger's brilliant tricky solution to Goldbach's problem. At the end of the class, we made a field trip to the stacks of the Mathematics Library. You can visit there yourself as often as you like.

M 2/17 -- I spoke about, and at the end of the day, gave a handout on Pick's Theorem. I also distributed two Wikipedia articles which were relevant to JY's presentation on F 2/14.

F 2/14 -- I made a few preliminary remarks (on the inappropriateness of Valentine's Day for STEM students and the quest to find "good problems") and I completed some of my 2/12 thoughts by giving the theorem which characterizes polynomials over C which are a sum of two *cubes*. I also distributed a proof of the Arithmetic-Geometric inequality. But the main part of the day was a long and surprisingly advanced presentation by JY on Monster Moonshine, which touched on simple groups and group representations theory and the complex torus and Ramanujan and the significance of 196883. It was a good harbinger of what you will see in your first week of classes in grad school!

W 2/12 -- A discussion of convex functions and inequalities and the arithmetic-geometric inequality, and how monomial substitutions into it will always give a positive polynomial, but one which might (or might not) be a sum of squares of polynomials. The instructor indulgently referred to his own research much of the time!

M 2/10 -- Handouts: Two old sets of notes on the Fibonacci numbers, and linear algebra, how to write for the Putnam, an obituary for David Blackwell and from 2/7, the FAQ of the arXiv. I also distributed a copy of the sort of experimentation I was doing on the Chebyshev polynomials on 2/7. I ended with some discussion of sums of two squares and sums of three squares.

F 2/7 -- I did all the talking, and rambled a bit, to be sure. I riffed on the Chebyshev polynomials -- see MathWorld, Wikipedia, Mactutor, and looked at the value of T_n(2), which gives a second-order linear recurrence, a cousin of the Fibonacci numbers. There will be old Fibonacci handouts on M 2/10.

W 2/5 -- Lots of handouts, not all of which will be linked. One is to the email I got accepting my article on the arXiv, but see the article. I also distributed information about the Student Assistance Center and what to do in case of an emergency; ie, Run > Hide > Fight. I distribute an old table containing the exact value of the sine of multiples of three degrees. Regarding MathSciNet, I distributed both the MathSciNet search page, and a Searching Tips link. Mathematically, I gave a proof of Niven's Theorem on rational values of trig functions at rational multiples of pi, which involved both the rational root theorem as well as Chebyshev Polynomials.

M 2/3 -- MN spoke on Ramanujan's proof of Bertrand's Postulate: for every integer n, there is a prime between n and 2n. SJ spoke about even and odd permutations and rotations of an n-gon. I talked a little bit about the symmetries of regular polytopes and the duality of a cube and an octahedron. Three handouts: for fun, The OREIS puzzle page, and the wikipedia page on the Mathematics Subject Classification and the way we break down research areas in this department: Urbana research areas.

F 1/31 -- YZ talked about Ramsey's theorem on two-coloring the edges of K_n and started to talk about three-coloring. JY talked about various difficult theorems on continuous vector fields on S^n, with reference to the Radon-Hurwitz-Hilbert Theorem. No handouts.

W 1/29 -- KS gave a problem about rational points on a circle in the plane whose center is irrational. NW talked about Mobius inversion and arithmetic functions and gave a generalization. I gave out Scott Kim's What is a puzzle?

M 1/27 -- YT spoke on Riemannian manifolds and their classification. JR talked about complexity problems in graph theory; in particular, the existence of Hamiltonian and Eulerian cycles. I distribute the Overleaf summary of n particular, here is a quick guide to LaTex and Francis Edward Su's guide to writing homework for math classes. Not much time to finish x^y = y^(mx).

F 1/24 -- Here is an NSF link to REU sites. SQ spoke on the Kakeya Needle Problem and variations. Handouts (each of which I spoke about a little): Joe Gallian, an expert in undergraduate research, on How to give a good talk, and Doug Shaw's tips for Making good talks into great ones, plus a printed version of Resources for Research , and my joint Monthly article on x^y = y^(mx)

W 1/22 -- -- First day. Several handouts: Course Organization; from George Polya, a problem-solving template; a Class questionnaire, and the instructor's "One Introduction to Mathematical Research". (That's the original version. The one I distributed in class is from JSTOR: "One Introduction to Mathematical Research". One thing you'll learn is that editors sometimes change things!
This contains the homework assignment for Monday 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.

    I gave an overview about how I've applied this homework to my own research, gave five possible topics and had a class vote. The winner was x^y = y^{2x}, which I'll start talking about on Friday after I cover some other things and give you a chance to talk.

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