Random Thing

On Math Overflow, I just saw an “answer” to a question, given by Scott Morrison, that I just had to share with anyone who hadn’t seen it.  The Message of the Day, on Oct 2, at Berkeley was the following:

Warning: Due to a known bug, the default Linux document viewer
        evince prints N*N copies of a PDF file when N copies requested.
        As a workaround, use Adobe Reader acroread for printing multiple
        copies of PDF documents, or use the fact that every natural number
        is a sum of at most four squares.

 

Math Overflow

Yeah, more of the non-posting stuff.  Have a couple of things that I’m working on, but haven’t written them up due to teaching and coursework.  In the meantime, I’ve gotten active over at this new site, Math Overflow.  Started up by some folks at Berkeley, funded by Ravi Vakil.  Great place to get quick answers to questions.  Though probably everyone who still bothers to subscribe to this feed reads the SBS anyway.

 

Blogroll Update Day

I’ve been a lot more active recently, now that my life has quieted down a bit into reading papers, running seminars, taking classes, and teaching a bit, instead of the craziness of a wedding.  So now, something I’ve been meaning to do, but which hasn’t been done yet: updating the blogroll.  So I ask that anyone who has a blog that discusses stuff like what we discuss, post here to be added.  Now, in the broadest sense, that means math, but most especially algebra, algebraic geometry, differential geometry, complex geometry, number theory, and representation theory.  But I’m most assuredly missing things, so if you think that we authors (or our readers) would be interested in your blog, post on this thread.

 

Local Fields

Dear Readers,

Let’s examine the role of topology in the study of fields and arithmetic. A topology on a field compatible with the field operations is given by an absolute value, which in turn defines a metric. Outside of number theory, people usually mean the standard real absolute  value when they talk about an absolute value. Note that on this absolute value carries the Archimedian property that is an unbounded set. What about the other case?

 

Shih’s Theorem

Dear Readers,

In spite of orals closing in a little more every day, I clearly haven’t been updating so much recently. I’d started a post about using Minkowski’s geometry of numbers to think about class numbers and unit groups and such things… but honestly that stuff is quite well-worn and at this point it wouldn’t be a good use of time to think carefully about how best to choose my words and explain this to the world when better expositors like William Stein or James Milne have already done so. Instead, I will talk about the expository part of my exam where I will talk about a particular case of the inverse Galois Problem.

Theorem(Shih,1974): If 2,3 or 7 is not a square in then is the Galois group of a polynomial with integral coefficients.

 

Dedekind Domains in Finite Galois Extensions

Last time we examined Dedekind domains in finite separable field extensions. One advantage to using a separable field extension that we did not use is that we can base extend to a finite Galois extension, where as we see the action of Galois forces the splitting of primes to be very uniform.

 

Dedekind Domains in finite separable extensions

Last time we took a look at Dedekind domains with fraction fields and found that if was any finite field extension of that the integral closure of in is Dedekind. The proof in this case is somewhat involved, but becomes slightly less so and shows that is also a finitely generated -module under the assumption that is also separable. In the following post we look at some of what makes life so much easier in the separable case.

 

Dedekind Domains, Krull-Akizuki and the Picard Group

As has been hinted in many previous posts, many facts about algebraic number theory tell us about geometric objects like elliptic curves. For instance, if you are working on a problem which primarily uses the affine geometry of a curve like the semistable reduction theorem for elliptic curves, the scheme you’re working on is opposite to what’s called a Dedekind Domain. We begin a series of posts on Dedekind Domains, beginning today with the very abstract and progressing to the concrete(which would of course be terrible for teaching this material but I mean these posts as more of a reference work).

 

Summer School: Day 1

As I mentioned, I’m participating in a summer school on the Geometry of Quantum Fields at Penn.  I’m in Katrin Wendland’s mentoring session this week, which means conformal fields and vertex algebras.

 

The next two weeks

I know I’ve been fairly bad about posting recently. Started teaching my first course. But that SHOULD end on Monday. Not the course, the silence. That’s when the Geometry of Quantum Fields summer school starts here at Penn.  For the first week, I’m going to be attempting to learn what a Conformal Field is with Katrin Wendland, and I’ll be attempting to blog about it.  The next week, I’m with Eric Sharpe, talking about Heterotic Compactifications and Quantum Cohomology.  The posts these next two weeks will be rather technical, but afterwards, I’ll probably attempt to distill them and provide some background and motivation beyond whatever else is covered.  Might also blog on some of the talks, but those are the two mentoring sessions I’m in, so they’ll be the most in-depth.