Endpaper: Math has this funny property.
The Harvard College Mathematics Review, 2(1):101, 2008.
Mathematical minutiae: \(i\) has this funny property.
The Harvard College Mathematics Review, 2(1):75–77, 2008.
The Harvard College Mathematics Review.
FOCUS, 27(8):14, 2007.
Combinatorial properties of the Higman-Sims graph.
Final paper for Math 155: Designs and Groups, Spring 2010, Harvard Mathematics Department, taught by Professor Noam D. Elkies.
In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact it is strongly regular, where any two adjacent nodes have 0 common neighbors, and any two non-adjacent nodes have 6 common neighbors. We offer multiple constructions of the graph, based on the unique \(S(3,6,22)\) Steiner system (obtained by extending the projective plane \(\Pi_4\)), the Leech lattice, and maximal cocliques in the \((50, 7, 0, 1)\) strongly regular Hoffman-Singleton graph. Along the way, we compute the automorphism group of the Higman-Sims graph, proving that it is a double cover of a simple group, the so-called Higman-Sims group.
Tangles for knots and links.
Final paper for Knot Theory, Summer Tutorial 2009, Harvard Mathematics Department, taught by Tanya Kobylyatskaya and Ethan Street.
It is often useful to discuss only small "pieces" of a link or link diagram while disregarding everything else. For example, the Reidemeister moves describe manipulations surrounding at most 3 crossings, and the skein relations for the Jones and Conway polynomials discuss modifications on one crossing at a time. Tangles may be thought of a small pieces or local pictures of knots or links, and they provide a useful language to describe the above manipulations. But the power of tangles in knot and link theory extends far beyond simple diagrammatic convenience, and this article provides a short survey of some of these applications.
A summary of The Second Lefschetz Theorem on Hyperplane Sections, by Aldo Andreotti and Theodore Frankel.
Final paper for Math 99r: Morse Theory, Spring 2009, Harvard Mathematics Department, taught by Chen-Yu Chi.
In this paper we give a guided tour of the proof of the second Lefschetz hyperplane section theorem presented in [AF1969]. The present writeup is based primarily on my notes used while presenting the Andreotti-Frankel argument to the Harvard University Mathematics 99r Morse Theory seminar in Spring 2009. As such, this work uses much of the same notation as, and we follow the same general structure as, [AF1969].
Theta series as modular forms.
Final paper for Freshman Seminar 26k: Euclidean Lattices and Sphere Packings, Spring 2007, Harvard University, taught by Professor Benedict H. Gross.
In this paper we introduce the notion of the theta series \(\Theta_L\) of a lattice \(L\), a useful and powerful tool in Lattice theory, especially in the case when the underlying lattice is assumed to be even and unimodular.
Two solutions to a tiling problem.
My favorite problem: Bert and Ernie.
The Harvard College Mathematics Review, 1(2):78–83, 2007.
Unpublished, 2007. See also the talk based on this material.
There are generally two opposite approaches to Olympiad geometry. Some prefer to draw the diagram and simply stare (labelling points only clutters the diagram!), waiting for the interactions between problems' various elements to present themselves visually. Others toss the diagram onto the complex or coordinate plane and attempt to establish the necessary connections through algebraic calculation rather than geometric insight.
This article discusses an interesting way to visualize and approach a variety of geometry problems by combining these two common methods: synthetic and analytic. We'll focus on a theorem known as "The Fundamental Theorem of Directly Similar Figures" or "The Mean Geometry Theorem." Although the result is quite simple, it nevertheless encourages a powerful new point of view.
MOSP Training, 2007. See also a talk on the same material.
This lecture focuses on the Abel Summation formula, which is most often useful as a way to take advantage of unusual given conditions such as ordering or majorization, or simply a way to put a new look on an expression. But in addition to learning this formula, I want to emphasize good, motivated thinking for all of these problems. Don't simply hunt for the time and place to apply Abel Summation; hopefully, when and if the occasion arrises, you'll recognize it.
MOSP Training, 2007.
Multivariate generating functions and other tidbits.
Mathematical Reflections, March 2006.
This article is devoted to some of the methods and applications of generating functions as used in Olympiad problem solving. Their basic properties are well known to many problem solvers; this article is intended to explore some more advanced applications and ideas, the most prominent of which is the use of multivariate generating functions.