Prof Life


Whoa!

I had resolved myself to the idea that we might not prove our theorem in full generality. I accepted that we’d settle for writing a paper where we made a conjecture about the general case and wrote proofs for, I don’t know, up through n = 6 or something.

Note-to-Gibbons: You shouldn’t doubt your incredible REU group like that! We riffed on the general ideas in our proof and about 10 minutes ago, we proved the general case!

data-win

Now, I know it’s dangerous to declare that you have a proof right after coming up with it. However, it’s important to celebrate immediately after proving something just in case it isn’t an actual proof. Pro tip: this technique will maximize your personal happiness in the field of mathematics.

But the best part of all of it is that I don’t have to write any more code!


Advice for applying to REUs

I’ve had a few people ask me what I was looking for when I read through REU applications.  I thought I’d describe my process and my reasons, which you can take to be my rubric for putting together an excellent application.  As with all advice on this blog, these are my opinions.  You should gather a few other ideas for a complete picture, especially since this is my first time as an REU mentor.  Now, disclaimers aside, follow the jump to my advice.

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WMC REU: Stalking the Theorem

We’re pretty close to Theorem 1.

When I started designing my REU project, I worried that I was either too narrow or too ambitious.  I thought the students might finish in week 1…or not get anywhere at all.  After all, my experience mentoring student research has been with the amazing and talented Robert Huben.  On some days, I felt like I should be happy with less from Robert, and on other days, I thought I should push for more.  I’m sure it was baffling at times, working with Dr. Gibbons and Ms. Hyde.

Let me return to my point.  Based on research as an undergraduate and now my sample size of one research student, I wasn’t abundantly confident in my ability to design a challenging, satisfying, and fun project.  I’m overstating my anxiety a little bit; if I thought I couldn’t pull it off, I wouldn’t have agreed to mentor.  Nevertheless, I was anxious even into the first week.  (Luckily, in grad school, I learned to manage and even transform anxiety into a weird calm competence.)

Well, folks, even if my project had some flaws, it’s turned out to be a good project, and my group is working like a smoothly oiled machine.  Double induction? Ha! They eat double induction for breakfast.  Three cases with several subcases? No big deal.  I’ve taken a slightly more active role in the project now, trying to highlight some opportunities to make progress.  I think we’ll have Theorem 1 nailed down by the middle of next week.  There’s a second theorem I’d like them to prove, and I think that will be a bit more straightforward (knock on wood).  From these two theorems, we get some nice structure results.

I think part of our progress comes from a change of scenery.  We got locked out of our classroom, so we’ve been working in a common space with a couple of comfortable chairs.

Alas, my team didn’t win the first round of research symposium talks, but they have a chance to redeem themselves on Wednesday next week. Here’s the silver lining: I’d feel guilty if they won the research symposium talks every week.  They’re already exceeding expectations!

As for team-building, we had fun at the weekly picnic, we laughed our way through a session on “Do’s and Don’ts” for presentations, and we enjoyed a rousing game of “Rabbit Rabbit Moose Moose” at Friday Floats this week.  The fun will continue into the weekend: some of the students and some of the faculty are going to karaoke this weekend.  Watch out, Bowie.


The Linear Algebra of Voting

Mike Orrison, of Harvey Mudd, visited the REU on Wednesday and talked about the generalized Condorcet criterion and the underlying linear algebra used to prove a surprising result.

The Condorcet criterion is said to hold for a voting system if it guarantees that, if a candidate would win each head-to-head election, then that candidate would win the election.

In our usual voting system, we only ask voters to vote for their first choices. But you could imagine having voters list their preferences. For instance, if I had to rank my cats, I’d rank them M > T > S (Moo’s my favorite, then Tipper, and lastly, there’s Sophie).

Now, I could ask my friends to rank my cats, and maybe I’d come up with:

  • M > T > S – ten ballots
  • T > M > S – four ballots
  • T > M > S – five ballots
  • S > T > M – two ballots

We can take away a few things from these rankings:

  1. No one likes Sophie (poor Sophie!).
  2. Moo wins the election!
  3. I have a lot of friends (okay, okay: it’s a fictional example).
  4. Tipper beats Moo (11 to 10); Tipper beats Sophie (19 to 2); Moo also beats Sophie (19 to 2).

Item 4 means that Tipper is the Condorcet winner. However, if we use our usual “most first place votes” election, we’d see that Moo is the winner.*

If you have more than three candidates, the results get weirder. You can have someone who wins every head-to-head competition (call the 2-winner Bob), but someone else could win in every three-way battle (call the 3-winner Alice).

The surprising and cool result is that once Bob’s 2-winningness (my term) is overshadowed by Alice’s 3-winningness, Bob can’t go on to be a 4-winner. He can’t be any n-winner for n > 3. And the proof, my friends, rests in linear algebra.  You can read all about the work Mike did with his REU students in their paper: Generalized Condorcet winners.

*In reality, Moo would win by any metric because she’s fantastic.

Screenshot 2015-06-29 09.05.42


WMC REU – Week 4, or, “To Prove a Lemma”

The students in my research group are superlative! They’re funny, intelligent, hard-working; basically, they kick serious ass. It’s easy to get frustrated during the research process. The project I designed is no different, and we’re facing our first hurdle.

When you read someone else’s proof, often you’ll think, “Well, yeah. Duh. That’s easy enough.” But here’s the thing: every problem is hard (until it’s easy). At least half of the work is the process of framing the problem the right way so that you can use the right tools. We have a problem that we’re trying to solve, but there are a lot of competing pieces of information.

To try to help out with the framing process, I took the various moving parts and used them one at a time in a sequence of reductions (numbered 0 through 4, ish). After the reductions, we have a lot of powerful assumptions that we can use. At the end of our working day on Tuesday, I felt confident that we had set ourselves up for a quick slam-dunk to win the game.

…Ha. When I got home, I started playing around and found that I couldn’t get very far. As of this Wednesday morning, it turns out, no one got very far. Back to the whiteboard for us.