ESP Biography



HOLDEN LEE, MIT alum '13, Princeton math grad student




Major: Mathematics

College/Employer: MIT

Year of Graduation: 2013

Picture of Holden Lee

Brief Biographical Sketch:

Not Available.



Past Classes

  (Look at the class archive for more.)


Why your math textbook SUCKS in Splash! 2012 (Nov. 17 - 18, 2012)
Each year, many shiny new math textbooks with colored pictures and hundred-dollar price tags hit the market. Textbooks specially chosen by committees of educators (see all those smiling faces at the beginning?) after much deliberation. Textbooks revised to meet the new "standards" popping up like mushrooms. Imagine: your math textbook is the result of a century's worth of teachers' efforts to improve education. So why does your math textbook SUCK? Why do you want to repeatedly bash your math textbook against the table, or drop it off a tall building? Or repeatedly bash your head because your textbook isn't making sense? Or just fall asleep and drool on your textbook because it's SO BORING? In this class we will explore why so many textbooks are poorly written---an complex issue which involves factors ranging from educational practices in the classroom to the politics of the textbook companies. I will list the flaws of traditional textbooks, and show through example how to write better curriculum. Finally, I will give some strategies for learning despite that poorly written text. Bring your math textbooks and your opinions!


Hands-on game theory in Splash! 2012 (Nov. 17 - 18, 2012)
In this class you'll play some "games" with very simple rules but very counter-intuitive strategies and outcomes. Win a dollar... or not. You'll learn how these simple games are the beginnings of game theory, a branch of applied mathematics that arose in the 1900's and has had profound implications for diverse fields such as economics, psychology, and evolutionary biology.


Modular arithmetic in Splash! 2012 (Nov. 17 - 18, 2012)
How do you find the last digit of $$ 2012^{2012}$$? Modular arithmetic is just like math on the integers, except we only care about the remainder when we divide by 10. Or 5, or 7, or 2017. I'll cover Euler's Theorem, Fermat's Little Theorem, and talk about what this has to do with group theory and cryptography.


Trig Identities in Splash! 2012 (Nov. 17 - 18, 2012)
Trig identities don't come out of nowhere. In this class you will derive trig identities with nice geometry pictures. Hopefully, you will not forget them again.


Diophantine Approximation in Splash! 2012 (Nov. 17 - 18, 2012)
Rational numbers, numbers such as $$ -\frac{1}{2}$$ and $$ \frac{144}{89} $$, are nice. Sprinkled in between them are algebraic numbers---roots of polynomial equations, such as $$\frac{1+\sqrt 5}{2}$$ and $$ \sqrt[3]{2}$$---and transcendental numbers such as $$ e $$. In this class we'll explore the relationships between these types of numbers, and answer questions such as: * How closely can you approximate an algebraic number such as $$\frac{1+\sqrt 5}{2}$$ with fractions, if you try to use denominators as small as possible? * How "close" do two algebraic numbers have to be before you know they're equal? * Why is $$e$$ irrational? Why is $$e$$ transcendental? I'll cover Dirichlet's Theorem, and briefly talk about the Thue-Siegel-Roth Theorem---which tells us that numbers that are "too close" to rational numbers are actually transcendental! Finally, I'll describe some applications to deep questions in number theory such as Falting's Theorem---certain equations in the integers have only finitely many solutions.


Quadratic forms in Spark! 2012 (Mar. 10, 2012)
We'll take a tour of modern number theory by examining quadratic forms, from their roots in Gauss's time to our modern understanding. We'll touch on topics such as infinite descent, ideal factorization, p-adic numbers (Hasse-Minkowski), composition laws, modular forms, class field theory, and Chebotarev density. Emphasis will be on ideas rather than proofs. A quadratic form is a polynomial in several variables where each term has degree 2. We'll examine questions such as: Which numbers are the sum of 2 squares? How about 3 squares? 4 squares? How many representations are there? What proportion of primes can be written in the form $$x^2+ny^2$$, for a fixed $$n$$?


Play DDR like a pro! in Spark! 2012 (Mar. 10, 2012)
Want to know how to get good at DDR? Want to impress your friends? How the heck to read arrows flashing past at 400 beats per minute? Then this class is for you! In the first ten minutes we'll go over beginner and intermediate footwork tips, and introduce various step patterns such as gallops, cross-stepping, streams, and hands. In the next forty minutes you'll get the chance to practice, with individualized feedback. Students with health issues are not advised to take this class. Note: This class is not intended for you to show off your DDR skills. If you can regularly get B or better on 10-footers then we won't have too much to teach you.


Analytic Number Theory in Splash! 2011 (Nov. 19 - 20, 2011)
One main part of analytic number theory is the distribution of the prime numbers. We will sketch the proof of the Prime Number Theorem and Dirichlet's Theorem (with error bounds). On the way, we will introduce important techniques from complex analysis, Dirichlet series, and the Riemann zeta function.


Arithmetic and Algebra of Quadratic Forms in Splash! 2011 (Nov. 19 - 20, 2011)
A quadratic form is a polynomial in several variables where each term has degree 2. Quadratic forms over the integers are a rich source of problems for number theory. For example, we can ask, which numbers are the sum of 2 squares? How about 3 squares? 4 squares? How many representations are there? What about other forms such as $$ x^2+ny^2 $$? One of the first theorems in arithmetic geometry is Hasse-Minkowski: that we can get information about a form by looking at them over all the primes p - that is, by passing to what's known as the p-adic field $$ \mathbb{Q}_p $$. We will use this to prove Lagrange's Theorem as well as sketch the proof of the 15-theorem. On the algebraic side, if we restrict our attention to binary quadratic forms, we get an even nicer picture: it turns out that quadratic forms correspond to ideals in quadratic rings, giving a "composition law" for quadratic forms.


Factoring in Spark! 2011 (Mar. 12, 2011)
Why can integers be factored uniquely into primes? In proving this fundamental fact, we will discover how to generalize the idea of factoring to other number systems, such as complex numbers, as well as to polynomials.


Complex Analysis in Spark! 2011 (Mar. 12, 2011)
Calculus on the complex numbers has an entirely different attitude from calculus on the reals. The additional structure of the complex numbers greatly enriches the theory of differentiable functions. We will cover Cauchy's Theorem and power (Laurent) series, and prove the Fundamental Theorem of Algebra and the formula 1+1/2^2+1/3^2+...=pi^2/6. If time allows, I will discuss applications to elliptic curves.


Factoring in Spark! 2010 (Mar. 13, 2010)
Why can integers be factored uniquely into primes? In proving this fundamental fact, we will discover how to generalize the idea of factoring to other number systems, such as complex numbers, as well as to polynomials.


Linear Algebra in Splash! 2009 (Nov. 21 - 22, 2009)
Linear algebra? What could be harder than y=mx+b? A lot. To help demystify this challenging topic, we'll connect the abstract (vector spaces, linear transformations) with the concrete (matrices and linear equations). Have you wondered where the crazy formula for the determinant comes from? We'll derive it, from the ground up. Next, we'll explore eigenvalues and diagonalization. Finally, we'll see some surprising applications to combinatorics. The following topics may be included or substituted based on time and student interest: -Inner product spaces and orthogonality -Canonical forms -Multilinear forms -Sequences