Math 496 F1H Home Page

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 overleaf.com, 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):

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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