(This post is specifically for Math 287 students.)

Starting Monday, you guys should be organized in new groups. No group can have three members that are together in the current groups. When I arrive on Monday, the groups should already be formed. You guys should start working on the laboratory on Polyhedra, Chapter 7. Make sure to bring whatever materials may be needed for this.

There have been complaints about not everybody contributing their share to their respective groups. This is not acceptable, but it is sadly the main reason for the reorganization. So: If I receive two complaints about a member not contributing as required, and there are no reasonable extenuating circumstances, that person will be dropped out of their group (receiving a zero in their current project as a result). If the issue is not expected to be resolved for the next report, a new reorganization of groups will be triggered as a result.

Also, I am unhappy with the level of some of the reports. It seems the peer reviewing of other groups’ drafts is not being taken as seriously as needed. So: Now, on the dates drafts are due, each group should bring three copies of their current draft. When you review another group’s draft, write the members of your group on the copy you are reviewing. I’ll collect the copies, with your comments, copy them for my records and return. If I identify something that a group should have noticed and mentioned, but did not, that group will be penalized (since this means the group did not take their refereeing role seriously). Conversely, if a group mentions something that should be addressed, but I do not see the issue resolved in the final report, the group that failed to address the given comments will be penalized.

Finally, this being a mathematics course, I expect your projects to include proofs. If a project lacks proofs it will receive a failing grade.

Feel free to contact me be email if any of the above needs clarifying.

[…] de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit […]

This is a very interesting question (and I really want to see what other answers you receive). I do not know of any general metatheorems ensuring that what you ask (in particular, about consistency strength) is the case, at least under reasonable conditions. However, arguments establishing the proof theoretic ordinal of a theory $T$ usually entail this. You […]

This is false; take a look at https://en.wikipedia.org/wiki/Analytic_set for a quick introduction. For details, look at Kechris's book on Classical Descriptive Set Theory. There you will find also some information on the history of this result, how it was originally thought to be true, and how the discovery of counterexamples led to the creation of desc […]

This is open. In $L(\mathbb R)$ the answer is yes. Hugh has several proofs of this, and it remains one of the few unpublished results in the area. The latest version of the statement (that I know of) is the claim in your parenthetical remark at the end. This gives determinacy in $L(\mathbb R)$ using, for example, a reflection argument. (I mentioned this a wh […]

A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, […]

There is a new journal of the European Mathematical Society that seems perfect for these articles: EMS Surveys in Mathematical Sciences. The description at the link reads: The EMS Surveys in Mathematical Sciences is dedicated to publishing authoritative surveys and high-level expositions in all areas of mathematical sciences. It is a peer-reviewed periodical […]

You may be interested in the following paper: Lorenz Halbeisen, and Norbert Hungerbühler. The cardinality of Hamel bases of Banach spaces, East-West Journal of Mathematics, 2, (2000) 153-159. There, Lorenz and Norbert prove a few results about the size of Hamel bases of arbitrary infinite dimensional Banach spaces. In particular, they show: Lemma 3.4. If $K\ […]

You just need to show that $\sum_{\alpha\in F}\alpha^k=0$ for $k=0,1,\dots,q-2$. This is clear for $k=0$ (understanding $0^0$ as $1$). But $\alpha^q-\alpha=0$ for all $\alpha$ so $\alpha^{q-1}-1=0$ for all $\alpha\ne0$, and the result follows from the Newton identities.

Nice question. Let me first point out that the Riemann Hypothesis and $\mathsf{P}$-vs-$\mathsf{NP}$ are much simpler than $\Pi^1_2$: The former is $\Pi^0_1$, see this MO question, and the assertion that $\mathsf{P}=\mathsf{NP}$ is a $\Pi^0_2$ statement ("for every code for a machine of such and such kind there is a code for a machine of such other kind […]

For brevity's sake, say that a theory $T$ is nice if $T$ is a consistent theory that can interpret Peano Arithmetic and admits a recursively enumerable set of axioms. For any such $T$, the statement "$T$ is consistent" can be coded as an arithmetic statement (saying that no number codes a proof of a contradiction from the axioms of $T$). What […]

[…] de puissances, Fundamenta Mathematicae, 3, (1922), 52–58, or Fritz Herzog, George Piranian, Sets of convergence of Taylor series. I. Duke Math. J., 16, (1949), 529–534. Both papers prove more general results, by explicit […]