Jason Polak
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This is something I've never paid attention to until graduate school, but virtually every book uses the convention that formulae in display mode are part of the text. Every Springer text for instance ...

Inequalities! I don't think I've ever seen a course on inequalities, and there's certainly enough elementary material to cover in a one-semester course. Very few undergrads know much about ...

There are several different approaches you can use. As mentioned, health is important and you should go to the gym or exercise in some way, preferably with someone. Also reading about a variety of ...

I‘m all for talking about the applications of math and your research. I think such things can be very interesting. However, let me add one more point. I feel you should also express to the audience ...

Try coming up with counterexamples when you remove hypotheses. Play with the mathematics. The best way to know a theorem is to prove it. Try coming up with a different proof. Explain the theorem to ...

I think the answer is obvious. If you want to stay in mathematics, quit HTT and go back to the basics of what got you interested in math in the first place. It's easy for young people to be seduced ...

In the trivial case the abelianization comes from the short exact sequence $0\to J\to \mathbb{Z}G\to \mathbb{Z}\to 0$ where $J$ is the augmentation ideal. The homology $H_*(G,-)$ are just derived ...

Since this is not purely group theoretical and not a complete answer, this maybe should be more of a comment, but since you mentioned simplicial complexes perhaps you should check out the following ...

Kenzo and Chomp are for computing homology. Kenzo for instance can take an arbitrary abstract simplicial complex and compute the simplicial homology groups, and it has various spaces already built in. ...

The following answer will be an attempt for the 'general mathematical audience', and so contains a little imprecision. Here are two reasons (somewhat?) related to number theory why one might consider ...

Let $G$ be a split reductive group over a finite field $k$ (of sufficiently large characteristic), $X$ a smooth geometrically connected projective curve over $k$ and $\mathcal{M}$ the algebraic stack ...

If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which ...

"Essays In Group Theory" edited by S.M. Gersten, which in particular contains Gromov's paper "Hyperbolic Groups".

What strikes me most is that this letter is generic and not directed at you specifically, in the sense that there is no explained reason why they contacted you and not someone else. And "highest ...

According to Lusztig, it means locally trivial in the Zariski topology.

For joint collaboration I've used subversion. Having a central repository I think is a really good idea for a paper, although you can set up central repositories in bzr and git. I would stay away ...

I'll just mention some sources that are not already in the other question indicated by Franz's answer, in case you need to become more comfortable with the derived machinery itself. (The book by ...

Let $p$ is an odd prime and $C$ the $p$-Sylow of the class group of $\mathbb{Q}(\zeta_p)$. If $C^\sigma$ denotes the group fixed by complex conjugation then Vandiver's conjecture is that $C^\sigma = 0$...

Yes. It's essential for me to take notes or else I learn almost nothing unless it's something I've been thinking about for a while. I've discovered that my best learning technique is seeing symbols on ...

This doesn't directly answer your question, but some universities give guest access to visitors for limited periods of time (e.g. an hour a day), so if there are any resources you really want to check ...

Since you also mention algebraic geometry, you may want to take a look at "Model Theory and Algebraic Goemetry" edited by Elizabeth Bouscaren (Springer LNM 1696). Its primary purpose is to introduce ...

There's also Nicholson's Elementary Linear Algebra or the slightly more advanced Linear Algebra: With Applications. If your students react negatively to the intro of abstract vector spaces, I don't ...

There's also Lead2Amazon which can export in a variety of formats.

There does seem to be a difference, although I don't know how to distinguish them formally, but here are two cases which distinguish the two: *Let V be a vector space. A linear map f on V is ...

People use whatever is most useful. ZFC just happens to be a fairly simple formalization of the way people think of sets such that we can eliminate imprecision sufficiently to do good mathematics. ...

The notion of Frobenius algebra is still useful in the general case, but then group rings that are Frobenius algebras aren't necessarily quasi-Frobenius rings, as your example notes. However if $M$ is ...