For ease, I re-list here all the presentations we had throughout the term. I also include some of them. If you gave a presentation and would like your notes to be included, please email them to me and I’ll add them here.

Jeremy Elison, Wednesday, October 12: Georg Cantor and infinity.

Kevin Byrne, Wednesday, October 26: Alan Turing and Turing machines.

Keith Ward, Monday, November 7: Grigori Perelman and the Poincaré conjecture.

David Miller, Wednesday, November 16: Augustin Cauchy and Cauchy’s dispersion equation.

Taylor Mitchell, Friday, November 18: Lajos Pósa and Hamiltonian circuits.

Sheryl Tremble, Monday, November 28: Pythagoras and the Pythagorean theorem.

Blake Dietz, Wednesday, November 30: and the Happy End problem.

Here are Jeremy’s notes on his presentation. Here is the Wikipedia page on Cantor, and a link to Cantor’s Attic, a wiki-style page discussing the different (set theoretic) notions of infinity.

Here are a link to the official page for the Alan Turing year, and the Wikipedia page on Turing. If you have heard of Conway’s Game of Life, you may enjoy the following video showing how to simulate a Turing machine within the Game of Life; the Droste effect it refers to is best explained in by H. Lenstra in a talk given at Princeton on April 3, 2007, and available here.

Here is a link to the Wikipedia page on Perelman, and the Clay Institute’s description of the Poincaré conjecture. In 2006, The New Yorker published an interesting article on the unfortunate “controversy” on the priority of Perelman’s proof.

Here are David’s slides on his presentation, and the Wikipedia page on Cauchy.

Here is a link to Ross Honsberger’s article on Pósa (including the result on Hamiltonian circuits that Taylor showed during her presentation).

Here are Sheryl’s slides on Pythagoras and his theorem. In case the gif file does not play, here is a separate copy:

The Pythagorean theorem has many proofs, even one discovered by President Garfield!

Finally, here is the Wikipedia page on . Oakland University has a nice page on him, including information on the number; see also the page maintained by Peter Komjáth, and an online depository of most of papers.

This entry was posted on Tuesday, January 10th, 2012 at 5:26 pm and is filed under 187: Discrete mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

This is a nice problem. Here is what I know. (Below, I refer to the Handbook. This is the Handbook of Set Theory, Foreman, Kanamori, eds., Springer, 2010.) First of all, the consistency of the failure of diamond at a weakly compact cardinal seems open. Woodin has asked this explicitly, I do not know if the question itself is due to him. Of course, $\diamonds […]

I thought about this question a while ago, while teaching a topics course. Since one can easily check that $${}|{\mathbb R}|=|{\mathcal P}({\mathbb N})|$$ by a direct construction that does not involve diagonalization, the question can be restated as: Is there a proof of Cantor's theorem that ${}|X|

(I am replacing prior nonsense with a completely different suggestion. I am also turning this into CW so details can be added by somebody with time (which, sadly, most likely won't be me). Comments prior to Feb. 9, 2011, refer to said prior nonsense.) Start with $V=L$ and force to add a Mathias real $s$. Let $W$ be the resulting extension. Let $A$ be th […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is open whether the continuum hypothesis for an infinite set $E$ implies the well-orderability of $E$. Of course, if $CH(E)$ holds, then the assumption in your (first) statement holds. ($CH(E)$ is the statement that any subset $A$ of $\mathcal P(E)$, either $A$ injects into $E$, or else $A$ is in bijection with $\mathcal P(E)$.) This is a question that da […]

Both directions seem straightforward. If there is such a sequence, for any $Z\subseteq\omega$ there is an $\alpha$ such that $Z=A_\alpha$, since $Z=Z\cap\beta$ for any infinite $\beta$. This shows there is at most $\omega_1$ many subsets of $\omega$, that is, $\mathsf{CH}$ holds. On the other hand, if $\mathsf{CH}$ holds, we have that the set of bounded subs […]

For a different solution, pick any derivative $f$ that is not continuous, for instance, the derivative of $$g(x)=\left\{\begin{array}{cl}x^2\sin(1/x)&x\ne0,\\0&x=0.\end{array}\right.$$ We can always find a sequence of continuous functions converging to $f$ pointwise, but the convergence cannot be uniform, or $f$, being the uniform limit of continuous […]

The solution you suggest is correct: Recall that the complement of a closed set is an open set and that any open set in $\mathbb R$ is the union of disjoint open intervals. If $x$ is not in either closed set, then it is in one of these open intervals, whose endpoints belong to the closed sets. So you can indeed just use a straight line on the interval to ext […]

Here is a nice example I recently discussed in lecture. Recall that a function $ f:\mathbb R\to \mathbb R $ is Baire one iff it is the pointwise limit of a sequence of continuous functions. These functions do not need to be continuous, but Baire proved that they always have uncountably many points of continuity; in fact, the set of points of continuity of $ […]