Posted by: holdenlee | March 16, 2014

Prime-producing polynomials


No nonconstant polynomial in one variable can produce only primes at integer values. But the polynomial x^2+x+41 does awfully well: it produces primes for 0\le x\le 39:

41 ,

43 ,

47 ,

53 ,

61 ,

71 ,

83 ,

97 ,

113 ,

131 ,

151 ,

173 ,

197 ,

223 ,

251 ,

281 ,

313 ,

347 ,

383 ,

421 ,

461 ,

503 ,

547 ,

593 ,

641 ,

691 ,

743 ,

797 ,

853 ,

911 ,

971 ,

1033 ,

1097 ,

1163 ,

1231 ,

1301 ,

1373 ,

1447 ,

1523.

Why is this the case? We’ll explore these remarkable prime-producing polynomials here (source, template). Note that you will need to know enough abstract algebra/algebraic number theory to understand factorization in quadratic rings.

(I found out about these polynomials from the post here.)

Posted by: holdenlee | March 15, 2014

Part III math talks


We’ve just concluded Lent term with a series of wonderful talks by Part III students. Here are some notes I took during the talks:

The notes are quite messy, so use at your own risk. If you edit the notes in any way, do let me know (hl422@cam.ac.uk) so I can update them.

Posted by: holdenlee | February 22, 2014

Predictably Irrational


Predictably Irrational by Dan Ariely is an intriguing book that shows us how we can make better decisions by understanding the patterns in our own irrational behavior. Here’s a summary of the main points in the book (by chapter).

Read More…

Posted by: holdenlee | February 18, 2014

Kazuo Ishiguro on Writing


This evening Richard Beard held a Masterclass in Creative Writing at Pembroke, and invited Kazuo Ishiguro to give a talk afterwards. Ishiguro shared his writing process.

Here’s some of the main points, paraphrased. (Again, any misrepresentations are entirely my fault.)

RB: Where do you come up with ideas?

KI: I’m not sure; the processes changes slightly every time. For me, the beginning of a project is very important. When should I start writing the actual words that will appear in the book? Some writers plunge right in; the words pour out; they rely on their improvisational powers. Others have to plan. I personally have to know a lot about the story before I write anything; it can be up to a year between when I decide to write a novel and when I start writing it. In that time I jot down notes, road-test various ideas, hold an “audition” for the narrator—having the right narrator is very important.

For me, ideas come out of themes and questions, not necessarily intellectual, and often emotional. I think about the premise of the story. For instance, what does it feel like if you reach a certain age and it dawns upon you that you’ve wasted much of your life? With experience, it becomes easier to recognize fruitful ideas.

I’ve found myself more drawn to certain topics—these topics have an unfathomed, rich feel that make me want to explore them more. I try to summarize them in one sentence, to pick out their essence.

I had to find where my territory is. When I was young I went through a “promiscuous” phase where I had a little go at writing many different things. After a while you feel a difference between the stories that you can happily write but that engage you only superficially, and those stories that are really deep and meaningful to you.

Read More…

Posted by: holdenlee | February 8, 2014

Time Bending


I once asked an acquaintance why she liked astronomy, and she said,

It’s reassuring to know we’re part of something bigger. It gives you so much perspective, that we’re just a small part of the universe, that x billion years from now the sun will explode and swallow up the earth.

Loquat’s Time Bending is a song about how small we are relative to the universe. Maybe we can’t explore everything, and can’t know everything because of some built-in limitation. But the song isn’t a gloomy song, it’s a beautiful one—because the vastness just makes the victories of what we can discover, and what we can explore, more beautiful.

Over break I made a music video for Time Bending, and I’ve finally finished editing it:

Lyrics (from http://artists.letssingit.com/loquat-lyrics-time-bending-kgm8wls#axzz2sm76kR00)

Can’t see it midday; it all comes at night.
If you’re lucky to see it – all the glittering light
You’ll know that there’s something else to it.
There are millions more, not just gas and core.
It’s the distance from suns, and there’s more than just one.

There’s so many of them, and they’re so far away.
We can’t bend time to get there, and can barely even see.
The pictures they take give us clues, yet each one is from the past.
I don’t know my purpose in this, but it’s obvious it’s minuscule at best.

Other oceans in space, busy with life:
They are parallel places, unreachable sights.
It’s strange, but there’s something else to it.
There are millions more, not just gas and core.

There’s so many of them, and they’re so far away.
We can’t bend time to get there, and can barely even see.
The pictures they take give us clues, yet each one is from the past.
I don’t know my purpose in this, but it’s obvious it’s minuscule at best.

You see the lights ahead, as they’re flying in?
Will they find us first, or will we find them?
If we could take the trip, it’d be a giant risk.
What if we never find our way back to our little atmosphere again?

There’s so many of them, and they’re so far away.
We can’t bend time to get there, and can barely even see.
The pictures they take give us clues, yet each one is from the past.
I don’t know my purpose in this, but it’s obvious it’s minuscule at best.

Posted by: holdenlee | February 3, 2014

The Scholar Speaks: Political Engagement and Information


Every Saturday, Erica Cao hosts The Scholar Speaks, a radio show where Gates Cambridge scholars share their research and stories. This past week she interviewed Farhan Samanani, who is currently doing a Ph.D. in anthropology at the University of Cambridge. (Recording here.) Farhan talked about understanding political engagement through personal stories, and how we might disseminate information to help people make better choices. Here are some highlights from the interview, very liberally summarized.

(Any misrepresentations are my fault entirely; please let me know.)

Read More…

Posted by: holdenlee | January 30, 2014

Mathematical Compression


1 Information compression in mathematics

1.1 For personal learning

I strongly believe that the best way to learn large amounts of mathematics—besides the problem-solving component—is to find a way to compress mathematical knowledge. A good math student does not memorize all the proofs and formulas; rather she remembers

  • what’s essential, and
  • the methods for computing the rest.

An example is trig formulas. You can get away with just knowing a few of them and then deriving the rest. Derive half-angle from double-angle, and double-angle from sum identities (or directly), and if you forget the sum identities you can even draw a picture with two triangles (like these pictures I drew when preparing for a ESP class). There are other ways to remember them: complex numbers, linear algebra. Multiple representations and many paths to the same result mean a lot less has to be memorized. When we talk about being smart at math, we refer to the “being able to compute” bit, not the “memorizing” bit. [I'm not using "compute" in the sense of rote, i.e., what we think computers do today.]

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Posted by: holdenlee | January 27, 2014

Solving an inequality with entropy


The following is Problem 5 from IMO 1992:

Let V be a finite subset of Euclidean space consisting of points (x,y,z) with integer coordinates. Let S_x,S_y,S_z be the projections of V onto the yz, xz, xy planes, respectively. Prove that

|V|^2\le |S_x||S_y||S_z|.

In this note, I’ll talk about how to solve this inequality using the idea of entropy. (source code)

Posted by: holdenlee | January 25, 2014

Math notes; Radio storytelling


Hi all!

1 This blog will live again

And rather than drop a few extremely well-thought out posts over a long period of time, I’ll write regularly about littler things as they come up.

2 Math notes (Cambridge Part III)

These have been moved to a dedicated page.

3 Radio storytelling

Recently I’ve gotten hooked on radio shows like RadioLab and 99% invisible.

These shows examine what we take for granted around us, and remakes them into stories. And what makes a story? One way to think of this is that stories involve both chance as intention–and these things seem to feed different parts of our brain:

  • chance reminds us that the world is unpredictable, that we should celebrate what we have because it might not have happened that way, to go with the flow and improv, and
  • intention reminds us of how people made things the way they are, and reaffirms a belief that we can make a difference. A quote from 99% Invisible captures this very well:

When you see the care that someone put into something, the genius of everyday decisions, I think it makes you pathologically optimistic.

I’ll summarize some stories I recently heard (likely with inaccuracies, as I’m doing it off memory, go listen to the shows!):

Read More…

Posted by: holdenlee | December 5, 2013

Szemeredi Regularity and Roth’s Theorem


I’m giving a talk today at the Part III seminars (at the University of Cambridge). Notes are available here: https://dl.dropboxusercontent.com/u/27883775/math%20notes/cnt.pdf

Abstract: The Szemeredi Regularity Lemma is the graph-theoretic analogue of the dichotomy between order and randomness. It states that any large enough graph can be described using a structured component of bounded complexity with small error, the error being the pseudorandom component of the graph. I’ll sketch a proof using the energy increment argument, and then apply it to prove Roth’s Theorem: that any set of positive density in the naturals has a 3-term arithmetic progression.

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