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.

43.614000-116.202000

Like this:

LikeLoading...

Related

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.

The argument you are looking for is given in Kanamori's book, see Theorem 28.15. For the more nuanced version of the lemma, see section 7D in Moschovakis's descriptive set theory book (particularly 7.D.5-8), or section 3.1 in the Koellner-Woodin chapter of the Handbook.

This problem is very much open. Cheng Yong calls Harrington's $\star$ the assumption that there is a real $x$ such that all $x$-admissible ordinals are $L$-cardinals. From the work of Yong we know that Second- and even Third-order arithmetic do not suffice to prove that Harrington's $\star$ implies the existence of $0^\sharp$. Whether this was poss […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

What precisely do you mean by a standard model? An $\omega$-model? (That is, a model whose set of natural numbers is isomorphic to $\omega$.) Or a $\beta$-model? (That is, a model whose ordinals are well-ordered.) If the latter, the Mostowski collapse theorem tells us any such model is isomorphic in a unique way to a unique transitive model. If the former, t […]

This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR5,151d. (Anyway, it is probably easier to read a more modern presentation, […]

It is not possible to give a finitary proof of this fact. Consider, for instance, the statement that all Goodstein sequences terminate. This is an arithmetic claim, it is independent of $\mathsf{PA}$, and it is equivalent to the claim that some (very specific) computable non-increasing sequences of polynomial towers stabilize. Instead of Goodstein sequences, […]

Given any field automorphism of $\mathbb C$, the rational numbers are fixed. In fact, any number that is explicitly definable in $\mathbb C$ (in the first order language of fields) is fixed. (Actually, this means that we can only ensure that the rationals are fixed, I expand on this below.) Any construction of a wild automorphism uses the axiom of choice. Se […]

A proof can be found in Kanamori's book The higher infinite. The result follows from work of Solovay on the theory of uniform indiscernibles and work of Martin on projective scales. Look at sections 14, 28, and 30 of the book. A different approach (using the infinite partition properties of $\omega_1$ and $\omega_2$, themselves due to Solovay and Martin […]

Forcing with sufficiently homogeneous forcing that adds reals is enough to obtain the negation of $(*)$. The point is that if a formula $\phi$ defines a parameter-free well-ordering of $\mathbb R$, then for any ordinal $\alpha$, the statement "$x$ is the $\alpha$-th real in the well-ordering defined by $\phi$" uniquely characterizes $x$ in terms of […]

Assuming that $\gamma$ is finite, the argument is fairly simple: Suppose $\beta\to(\alpha)^\gamma_\delta$, and fix a bijection $f$ between $|\beta|$ and $\beta$. Consider a coloring $c:[|\beta|]^\gamma\to\delta$. Using $f$ , this gives us a coloring $c':[\beta]^\gamma\to\delta$ (here we used that $\gamma$ is finite). We want to argue that there is a $c$ […]