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Intel day 4 March 14, 2010

Posted by Akhil Mathew in General.

I had the last day of judging today at Intel. The 40 finalists first went to the Capitol to take a bunch of pictures, then to the Einstein statue at the NAS for another one.  We then went in the project exhibition hall.  I met with seven judges, two of who were mathematicians.  In order that future generations of Intelists may face the day of judgment without crushing uncertainty in the morning, I shall briefly describe my experience.

The first two judges I had were mathematics judges.  The first one asked me what I would do if I were giving a talk about my project at a colloquium.  He asked me to explain one of my results, which I initially did incorrectly (having not looked through the older proof in quite some time) but fixed along the way.  He asked me how I had learned algebraic geometry (or, more precisely, that rather small subset I can claim to vaguely understand).   Interestingly, he referred to a specific result in my paper by number (3.10; I didn’t remember what that was for sure)—one of the differences between Intel and ISEF is that the judges read the papers.

The second mathematician asked me to give an overview of my project in detail, so I went into my usual spiel.  She asked me a few questions along the way about how the results were proved.  Finally, she asked where I was going to go to college.  I said that I didn’t know yet.   This was a somewhat longer interview. 

There were others who wanted a brief overview and then left.  A computer scientist who had asked me earlier about certain algorithms and an engineer that asked about the law of atmospheres chatted with me about extensions of those problems. 

The exhibits were then opened to the public.  I met a few RSI 2009 alumni from the D.C. area.  Most people were not mathematicians, which made explaining my project (on representation theory in complex rank) a somewhat difficult task, though there were some that knew, e.g. group theory.  I wasn’t envious of my neighbor Joshua Pfeffer with mobs of people craning to hear about the super Kahler-Ricci flow though owing to me extreme hoarseness despite my consuming two bottles of fluids.  Also, my parents stopped by to say hello and see the other projects.

I’m somewhat tired now, and there’s not that much more I really can say about it without going into technical details.

Intel day 3 March 13, 2010

Posted by Akhil Mathew in General.

(Today was the third day of the Intel STS competition.)

I had my third judging interview today at about 11:40 am.   The judging panel included a computer scientist who pointed to the seven wrapped chocolates on the desk and informed me that to each was assigned a number, and that I needed to discover which contained the median. I didn’t have the actual numbers, but I could compare any two.  In the end, I said something about O(n \log n) sorting algorithms (e.g. heapsort).  He then asked me about counting paths from (0,0) to (m,n) where one can move either right or up on each move; I stated a recursive formula, but got the wrong closed form expression (it’s a binomial coefficient, and I said it was a power of 2).  I was then asked by the other judge about what change I would make if I had to design the human body.  I suggested eliminating cognitive biases and improving rationality but that wasn’t legal; I then suggested various ideas such as removing vestigial organs and improving our ability to type, but settled on increasing the efficiency with which energy can be extracted from food.

I then had lunch and went to the National Academy of Sciences, where we set up our projects.   My poster had developed a slight tear from being sent through the mail, but I fixed it with construction paper.

Intel STS: Liveblogging day 2 March 12, 2010

Posted by Akhil Mathew in General.

I’m going to try liveblogging (insofar as possible) a science fair.  [9:24- It’s now pretty clear that what I’m doing is more like deadblogging.  Still, it’s better than nothing, I suppose.]

(9:45) So I’m at the 2010 Intel Science Talent Search in Washington, D.C.  Presumably many of the folk that come across this blog will have heard of it; it’s a science competition for high school seniors.  There are seven people from RSI here this year.  I’ve also met many interesting people among the other finalists for the first time, all of whom seem to be rather beastly.  Everybody arrived yesterday–I took the train–but nothing competitive actually happened.  Today, we will be judged by a panel of ten or eleven scientists and mathematicians who are going to ask us general questions about science in general, and not our projects.  My first interview is in about half an hour, so I’m basically procrastinating by writing this entry, if there was anything that I could do to prepare :-).

In any case, it was pretty cool to find that I’m in a room that has a TV in the bathroom.  The hotel is ridiculously fancy.

After this, I’m going to go back to random Wikipedia surfing about diverse scientific topics.   They told us this morning that the judges want to see the scientific process rather than technical knowledge–perhaps this is a license for me to babble?  I’ve always enjoyed idle pontification.  In any case, I promise more later after my judging interview. (more…)

Is early specialization good? March 4, 2010

Posted by Akhil Mathew in General.

Thomas Sauvaget asked a question on MO on whether specializing early is a good thing.  It got into an interesting discussion, which continues on his blog.  I have placed some of my own thoughts there, so I won’t ramble here.

In an unrelated note, if you haven’t already seen it, you ought to watch the MAA’s great \pi, e debate.  And that’s regardless what you think of the holiday “Pi Day”–I’m mostly in agreement with what John Armstrong has to say on this subject.  Also, cf. this MO thread for some good alternatives to memorizing digits.

Real and Complex Analysis, by C. Apelian and S. Surace, now published December 19, 2009

Posted by Akhil Mathew in analysis, General, math education.
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(This material is reposted from here.)

The book Real and Complex Analysis, by Christopher Apelian and Steve Surace, was recently released.

It’s mainly for an introductory upper-level undergraduate course in real and complex analysis, especially at small liberal-arts colleges.  In this post, I’ll describe this book and how I was involved in its production.

At the start of my freshman year, my analysis teacher, Professor Surace, asked me to check over the drafts of a book he and his colleague (and my former teacher) Prof. Apelian were working on. It was the textbook for the course. At the time, if I remember correctly, there were six chapters: on the real and complex spaces, basic topology, limits, continuity, convergence of functions, and derivatives. The complex analysis part of the book was in its infancy (e.g., there was only a rudimentary outline of one chapter, which had been written some time back and was typeset in Word—they wrote it well before before they had switched to LaTeX). (more…)

My new math blog: Climbing Mount Bourbaki November 16, 2009

Posted by Akhil Mathew in General.

I started a new mathematics blog at Climbing Mount Bourbaki.

Ultimately, as Steven observed in the comments yesterday, textbook-flavored posts–that is to say, those belonging to a long series on a given topic, aren’t quite what this blog is about.  In fact, I don’t think there has been much about “Mathematical research and problem-solving” as of late.  That said, I like doing such posts, and it helps me learn mathematics.  That’s why I started this new blath.

So, what do I plan to do here?  Of course, I’m still a contributor, though I probably will be less active than once-a-day.  I’m much less familiar with contest math-style problem-solving as some of the other contributors here.  At some point I will talk about my RSI project, but I’m still busy working on it.  Instead, I’ll probably aim to write more crisp, article-like posts that tell an interesting story without needing a whole series.  Those will appear here and on Climbing Mount Bourbaki.  The Bourbakist ones will be relegated to there.

New mathematical blogs November 8, 2009

Posted by Akhil Mathew in General.
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There are a bunch of new and not-so-new but interesting mathematical blogs on the internet.

First of all, there’s SymOmega.  This is by three mathematicians at the University of Western Australia.  There is a diverse collection of blog posts, ranging from etiquette at conferences to maximal subgroups of the symmetric group.  This will likely be especially interesting to those who like combinatorics, group theory, and geometry, though I don’t know enough to comment on the more technical posts.

Another is Andy Octavian’s blog.  The author is apparently an undergraduate interested in theoretical physics, though many of the posts are pure mathematics–especially differential geometry.  What he discuss is more sophisticated than what I’ve been doing here and what I’m likely to get to in the next few weeks, so I likely won’t overlap.  Anyway, it seems to deserve more linking to in the blathosphere (and the corresponding sphere for physics).

Yet another is Bounded Rationality.  It seems to be a mix of math and economics, by two students at UCLA.  On the math side, the blog includes discussions complex analysis and measure theory based on exercises from apparently Terence Tao’s notes.  In fact, it looks like they’re (like this blogger!) writing their posts to understand mathematics better.   I don’t know enough about economics to comment on the other half.

Two massively collaborative mathematical websites that readers may like October 17, 2009

Posted by Akhil Mathew in General, math education.
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I realize that I’m late to the party on this, but there is a new mathematical website precisely for answering questions: Math Overflow.   Modeled on the Stack Overflow site for programmers, Math Overflow seems to have done a nice job in attracting a large crowd of professional mathematicians and students.  Questions tend to be answered quickly, and there is an interesting “reputation” feature that measures one’s respect in the community.  This is probably a much better approach than tossing out blegs (for readers here that are bloggers) since many more people will read it, and since the questions will be available in a common source for other mathematicians.  Since there are other sites that have active communities for math help, Math Overflow restricts itself to questions that are “of interest to at least one mathematician.”   (more…)

Open source textbooks October 10, 2009

Posted by Akhil Mathew in General, math education.
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Well, it seems that the Bourbaki 2.0 idea I suggested some time back wasn’t entirely absurd: as a commenter pointed out, the Stacks Project is following a similar model.  Moreover, Nathan Dunfield of Low Dimensional Topology has proposed that the stacks model be applied to textbooks (I assume the stacks book is more of a reference).  Additionally, he asks why conventional textbook publishing, even for individual authors, is still necessary in the day of the internet when it is more efficient to distribute material online.   Some people have apparently listened to these ideas; Jacob Lurie, for instance, has put his treatise on higher topos theory on the arXiv, and Allen Hatcher has made available his well-known text on algebraic topology on his webpage

I’d very much like to see this trend continue; there are surely people out there who would like to learn mathematics beyond the introductory calculus and linear algebra level–when there are no longer massive surpluses of texts on one topic–but may not be affiliated with a university for various reasons, and may not want to fork over the substantial sums that conventionally printed textbooks cost these days.  At least for authors, I don’t think there’s much money to be made in algebraic topology writing, and math professors have nice salaries anyway, so why not?

A talk on the p-adic numbers September 16, 2009

Posted by Akhil Mathew in algebraic number theory, General, math education, number theory.
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The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed.  Things should clear up at least somewhat in a few more weeks.   In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory.  As is traditional, I gave a talk last May after the course on some aspects of the subject matter.  Several faculty members at the university and teachers in my school attended, along with some undergraduates there.  In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.