Geometry of polynomials

Here at PSU, Sergei Tabachnikov just finished giving a two-week mini-course on the “geometry of polynomials.” The collection of topics was diverse: various proofs of the fundamental theorem of algebra, resultant and discriminants, Chebyshev polynomials, harmonic functions in three-space, and a sketch of the proof of the four-vertex theorem. The lectures presupposed familiarity with no more than elementary analysis and linear algebra, though more advanced topics were referenced (without proofs).

Two weeks really means about seven days; class was cancelled because apparently people thought that most of us were interested in an art show.

For the benefit of the huddled masses yearning to be educated, here are the notes that I took from these lectures.  The file is rather large (40 MB) because of the insertion of jpg images that someone else in the REU drew. The djvu file is a lot smaller, but WordPress won’t let me post it, so email me if you want it.

The notes are mostly a faithful representation of what I took in class, but I have edited them lightly to moderate my tendency to embarrass myself.

Next week, Yakov Pesin is lecturing on fractal geometry and dynamics; I’ll post those notes when I’m done with them.

 

The road ahead

I graduated! Senior year was rather drawn-out, so it is good to be done.  My plans for the near future are also set.  During the summer, I will be attending the REU at Penn State (as I have already mentioned).  This actually starts in less than a week.  In the fall, I will be an undergraduate at Harvard.

The extent to which I keep posting will likely depend on how much time and energy I have at Penn State, along with the topic of my research project (about which I know next to nothing now).  However, I do know that I will resume blogging about more expository (and older) topics, since I have finished talking about my last project.  It would be criminal to leave off class field theory right before the Artin reciprocity law, so I—as a mildly self-respecting blogger—will post a few times more on the subject.

In addition, I’ve been trying to collect together the various posts I’ve done on algebraic number theory into some sort of sheaf of notes, but it has yet to attain even quasi-coherence.  I know I will regret this someday, but here is the current messy version.  I’ll try to flesh these out a bit over the next few weeks and clean it up.  So far, except for Chapter 0, it literally consists of my blog posts, one after another.

As an undergraduate, it is necessary for me to shore up the basics before pushing too far into fancier stuff like class field theory.  Nevertheless, I will try to keep the subject matter on this blog as advanced as I can, insofar as possible.  (Which is to say that I’m making no long-term promises, since  readers know full well that I tend to break them anyway.)

 

Return to Mount Bourbaki

I have a math blog? What is this?

The main excuse I had for ignoring Mount Bourbaki for the past month or so was the Intel science competition, which ended last week.  It was a lot of fun—I met many interesting people and enjoyed numerous pleasant conversations.

To my surprise, I ended up coming in third place.  I was quite stunned by this especially after hearing the finalists called before me–I have to say that I was genuinely amazed by every project that I saw.

Of course, I can’t resist a picture.   Here is one from the gala, of the top three:

I’m the guy on the left looking in the wrong direction.

I probably will do a technical post at some point about what my project was all about, but for now here is a non-technical video I made:

Other than this, I also know now what I’m doing this summer.  I’m going to do an REU at Penn State, where I’ll be working on a topic that I should probably find out about soon.   What I’m going to do next fall is still undetermined.

As of late, I’ve been reading up on some logic and model theory and a bit of ergodic theory (from Walters’ book, which I recommend).  I tried to study Spanier while I was at Intel though didn’t get very far.  And I sincerely will try to get some entries up soon.  I don’t know whether I will be able to keep my promise of Grothendieck topologies just yet.