Math 496 F1H Home Page

This is the home page for Math 496F1H, "Introduction to Mathematical Research". This class meets for the Spring 2019 semester, MWF 2:00-2:50 in 343 Altgeld

Class Diary, blogger style

The mathematical links page will be here. Suggest a new one!

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

Advice on your Project Proposals:

  • These are due before Spring Break.
  • I accept a variety of formats, it shouldn't be much longer than one page.
  • In practice, proposals are iterative: it's extremely common for me to return a proposal for a friendly revision; it's very rare that I tell somebody that what they want to do is not satisfactory.
  • Be as bold and imaginative as you want, but be as clear as possible.
  • For a pure math project, it could be a few paragraphs explaining the problem, what's known, what's not known, and what you want to do.
  • For an applied math project, you should also explain some of the underlying subject outside of mathematics.
  • Your second talk will be a presentation of your accepted proposal to the class.

    M 2/25 -- Two presentations. NQ gave a geometric problem which has many different solutions. JT talked about machine learning and a problem related to it which turns out to be equivalent to the Continuum Hypothesis. I talked about my failures in geometry on the Putnam and about my PhD dissertation committee, which included Paul J. Cohen, who proved the independence of the Continuum Hypothesis. I also gave a "crazy theorem that every math major should know". Handouts on W.

    Major catch-up: M 2/11 (I was sick), plus five classes on W 2/13, F 2/18, M 2/18, W 2/20, F 2/22. Two of you spoke: NQ gave a talk on Sudoku and SZ1 gave an introduction to Lebesgue integration. I gave out a couple of handouts: Scott Kim's What is a puzzle? and a handout on Pick's Theorem. I talked about numeration systems and generating functions and lattice point simplices and barycentric coordinates and non-measurable sets and functional equations involving x^2-2 and other things I've probably forgotten about. Remind me if you want me to put a link in!

    F 2/8 -- SZ1 talked about Borel sets, and what you can say about A+B if all you know about A and B is that they are closed. I distributed a handout which included Furstenberg's original paper, and Idris Mercer's exposition, plus Doron Zeilberger's brilliant tricky solution to Goldbach's problem, and, finally, a biography of Riemann. I talked about the weighting on a balance problem and how it relates to digital representation problems. Notes later, if there is interest.

    W 2/6 -- We're up to bonus speakers, entirely up to you. CB discussed Riemann and the memoir in which he introduced the Riemann Hypothesis. I talked a bit about Riemann and then moved on to Doron Zeilberger's remarkable elementary proof of a question of Goldbach's and then to Hillel Furstenberg's remarkable topological proof of the infinitude of primes. Lots of unplanned stories, etc. Handouts will appear on Friday.

    M 2/4 -- Final speaker for the first round: SZ2 gave an introduction to partitions and gave one of many interesting identities for the generating functions related to them. This inspired me to give a few other generating function examples and talk about partitions in more generality. I talked in boring detail about the timeline for the Cantor paper. I also distributed the finished notes on the Stirling approximation to n!.

    F 2/1 -- Two speakers: BMB talked about a model in biomathematics about stability of populations and BD talked about a parameterization to the solutions to the Diophantine equation x^2 + y^2 = z^2 over the integers. This inspired me to talk a lot more about Diophantine equations and the "point-slope" method. I also distributed my very recent paper on the Cantor set, written with Jayadev Athreya (who founded IGL) and Jeremy Tyson (currently department chair).

    W 1/30 -- Cold and wind day! Here are links to the handout I sent along with the email: Two old sets of notes on the Fibonacci numbers, and linear algebra.

    M 1/28 -- Two speakers: AH talked about elliptic curves and the group structure of points, and NL talked about the eversion of the sphere and showed a remarkable video. I talked more about elliptic curves, especially regarding the equation x^3 + y^3 = c, and continued the derivation of the Stirling approximation to n!

    F 1/25 -- Two speakers: CC on Euler summation with proofs, and JT on the Intermediate Value Theorem. Handouts: Joe Gallian, an expert in undergraduate research, on How to give a good talk, Francis Edward Su's guide to writing homework for math classes and Doug Shaw's tips for Making good talks into great ones. I talked some about Stirling's formulas. Hopefully, everyone will complete their first talk by the end of the week.

    W 1/23 -- One volunteer speaker: TT talked about the Halting Problem involving Turing machines in theoretical computer science. In addition to the LaTeX guide mentioned below, I passed out a printed version of Resources for Research . Mathematically, I talked a bit about a Putnam problem I wrote involving the function f(n) = n + [\sqrt{n}], and then a lot about the Stern sequence. I will pass out notes on Friday. We need people to volunteer to speak, or I'll have to start assigning dates!

    F 1/18 -- Two volunteer speakers. XC talked about "Fermat's Little Theorem" and XW gave a proof that there are infinitely many primes of the form 4k+1 using Quadratic Reciprocity. I talked a bit about Carmichael numbers and mentioned the website, which has many useful LaTeX tools. In particular, here is a quick guideLaTex, which I will pass out on Wednesday. I also started to talk about the Cantor set as the limiting set on iterating the set function F(S) = (1/3)S U ((1/3)S + (2/3)). More as we have time. As a reminder: your final project proposals are due before Spring Break, which is about two months away.

    W 1/16 -- Three volunteer speakers! CB discussed the Prime Number Theorem, which counts the number of primes less than N and mentioned some variations, including Primes in short intervals. SZ1 (sorry, there are two SZ's in the class) discussed topological notions, including Bolzano-Weierstrass, as they generalize from R to R^n to metric spaces. NQ gave a novel heuristic for factoring polynomials in Z[x], by plugging in x = 10 and factoring the resulting polynomials. I passed out two articles on Cantor set, and didn't talk about them.

    M 1/14 -- 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". 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.

    I gave an overview about how I've applied this homework to my own research, involving the equation x^y = y^x and the Cantor set, plus a proof of Fermat's Last Theorem for polynomials.

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