April, 2008 (to appear)

This is a 14-page overview of a part of my Ph. D. thesis-to-be. I presented a poster of this at FPSAC 2008; it will be published in the proceedings.

My patterns are pairs of 2x2 matrices, with all entries equal to 0 or 1. Grid shapes are arbitrary subsets of boxes on graph paper — Young diagrams are a very particular case. Choose a particular grid shape S, and some two patterns a and b (each is a pair of matrices). For some such choices, the number of ways of filling S with 0s and 1s so that it avoids a is equal to the number of fillings avoiding b. Then, we say a is equivalent to b in S — a close relative of Wilf-equivalence. This paper makes the "some" precise, systematically describing which equivalences hold in what classes of shapes.

April, 2007 (preprint)

This is joint work with Alex Postnikov, my Ph. D. thesis advisor.

Imagine that you are to build a system of space stations (graph vertices), which communicate via laser beams (edges). The edge directions were already chosen, but you must place the stations so that none of the beams miss their targets. In this paper, we let the edge directions be generic and independent, a choice that constrains vertex placement the most. We then give an elegant combinatorial characterization of the number of degrees of freedom in this system, and prove some other related results.

May, 2006

The first few pages of this paper give a quick overview of the results of my senior thesis. The remainder has some additional results about the decay of the spectrum near zero. Due to time constraints, I did not prove one key conjecture -- which later turned out to be known (see below). This was my final project for 18.318 in Spring 2006.

Update: In 2007, I repeated my regular search for "least positive tree eigenvalue", and discovered this paper, which completes my result on near-zero asymptotics:

C. D. Godsil, "Inverses of trees", Combinatorica 5 (1985), No. 1, 33-9

It proves that the least positive eigenvalue on even trees occurs on a path. As far as I know, the case of odd trees (conjectured to be a "forked path" — a 2n-path with an extra edge added to the second vertex), is still open.

May, 2004

This was my Princeton undergraduate senior thesis, written under Professor Yakov G. Sinai. It concerns studied the adjacency spectra of sparse random matrices. I established that the spectra are well-defined, gave some bounds on the tail asymptotics, provided a numerical overview of the various regimes, and made some steps towards quantifying the spectrum in the neighborhood of zero (see more on this here). Here are the relevant computer programs:

• Calculating moments (tgz): Includes a program to calculate the moments of the limiting adjacency spectrum of a sparse graph, and one for a gaussian-weight sparse graph.
• Tree enumeration (tgz): This is an implementation of my O(n) cpu, O(sqrt(n)) memory algorithm for enumerating free trees. It uses more memory than the O(1) of the Li-Ruskey algorithm, but provides every tree's automorphism group size at no cost. The implementation also computes tree adjacency spectra, and characteristic polynomials.

May, 2003

This was a small numerical investigation that I did during my junior year. It has some spectrum pictures that are not in the senior thesis, but is otherwise obsolete.

January, 2003

A pair of graph theory exercises that I solved in 2002 (they were suggested by Professor Paul Seymour for my junior paper). There are probably no new results; I'm not 100% certain that anybody else worked out at the torus case, but the rest is known.

May, 2003

Professor John H. Conway introduced me to this neat puzzle. Take 27 a by b by c blocks, and try to assemble them into an (a + b + c)³ cube. It's hard, but certainly doable. This paper shows how. If you think about it, this proves the arithmetic-geometric mean inequality for 3 variables! :)

I also tried to tackle the 4D case, but ran out of time. There was some progress made; it might be doable as well. The paper refers to this puzzle-solving code (tgz).

You can make your own copy of the puzzle! Take a 68-inch 2" x 4", and saw it into 27 2.5-inch bricks (half-an-inch to spare). Sand them a bit, and go impress your friends! Note: you're probably better off with a circular saw, not a hand saw, since the cuts should be rather straight.

May, 2004

This is a re-telling of a book (see reference in paper) chapter on solving Waring's problem with Linnik's approach, using Shnirel'man density. Written rather hastily (class final project), and not certainly not cribbed. Thus, it is probably less readable than the book. But, it is written from a different angle, so who knows, you might find it helpful.

May, 2005 (unfinished)

The paper is very sloppily written, the work is unfinished, and the proofs may be wrong. I'm putting it out here in the hope that it will be useful to somebody working on the same problem — I do not plan to come back to this soon.

I wrote this after Professor Igor Pak suggested I look at the paper "A new example of a flexible polyhedron" by Victor Alexandrov. The original paper uses some fairly tricky trigonometry to prove that the described 3D polyhedron is flexible, and has zero volume.

I constructs a generalization of Alexandrov's polyhedron in n dimensions, and sketches what might be a nicer proof (if correct :) of Alexandrov's result for dimension 3. It also argues that the construction in 4 or more dimensions would be rigid.

December 2006

This is a program that I wrote at Professor Eugene B. Dynkin's suggestion. It draws nice diagrams in HTML and EPS (for use in LaTeX). It lets you choose labels for the vertices of Dynkin diagrams. It has a catalog of all Dynkin diagrams (up to size 8), and of all extended and affine Dynkin diagrams (up to size 9). It can be used offline. It's written using a combination of JavaScript / XHTML / CSS, and runs in the major browsers (but works/looks best in Firefox).

You may also be interested in the Eugene B. Dynkin Collection of Mathematics Interviews.

April 2006

Well, I had this crazy idea, and this is the result of my 30-minute feasibility study. It turned out to work pretty well, in my opinion.

November 2003

A couple of scripts to replace XDM/GDM. Allows you to run many X sessions with ease, log directly into X from the console, and makes you impervious to window manager crashes.

Some assembly required — you should be at ease editing shell configuration files.

November 2003

This small patch for XV, the image viewer, which allows you to run scripts by pressing a key from F1 through F12. Included are scripts to losslessly rotate JPEGs, to edit JPEG comments, to run batch rotations, and to perform reversible crops. Another easy application of this patch is a script for face annotation (see README). If you don't know how, and ask nicely, I can write it for you. If you write one, let me know, and I'll include it.

Some assembly required, but detailed instructions are provided.

February 2008

During the semester's first installment of the Simple Person's Applied Math Seminar, I tried to convince attendees that LyX is a great way to create LaTeX documents. This page lists some caveats and "how to" ideas.

January 2005

The class was taught by Professor Victor Kac. This compilation of lecture notes was typed by the students. Exercises are stated and solved in every lecture write-up. PDFs are available for every lecture, with LaTeX source for most.

May 10, 2015

Trying to be a grown-up and have my own domain. Thanks for all the bits, Andrey!

June 1, 2011

The MIT math department doesn't host pages indefinitely, so I moved to the domain of my awesome friend Andrey Goder.

December 15, 2008

Added a resume, CV, some more papers, fixed some typos, improved writing and visual consistency.

February 8, 2008

Added a LyX notes link, hope to find time to add more material.

January 15, 2007

Things I hope to add in the future: some art/photos, more code and lecture notes, lots of math homeworks.

wasting your bandwidth since 1999

Before Facebook obviated the need for self-description, I made a couple of personal homepages. The current one, I hope, is more of a "here's some interesting stuff" homepage. Here are the links to the previous iterations:

Princeton era (2003-2005)

Ithaca High era (1999-present) [warning :) ugly]