I have moved to UC Berkeley as of Fall 2013; my new homepage is here. This page will no longer be updated.
(If you can read that this is actually Greta's own poster, you don't need glasses)
This is the academic webpage of Yan Zhang (I publish under Yan X Zhang). I received my Ph. D. in mathematics at MIT under Richard Stanley in June 2013. Starting Fall 2013, I will be a Morrey Visiting Assistant Professor at UC Berkeley. I do research in a variety of problems related to combinatorics, mostly algebraic.
If you are looking for more personal stuff, see my personal page.
A more updated version of "Adinkras for Mathematicians." Contains some exposition and new results related to adinkras, a delightful combinatorial structure used to encode information about supersymmetry representations created by the "DFGHIL" collaboration. It has some neat results relating the combinatorics of adinkras to homological algebra, posets, coding theory, graph coloring, and so forth. I particularly like the elementary but cute idea of Stiefel-Whitney classes of codes.
This problem came out as a tangent to some enumeration techniques I was/am developing with posets. As a possible stepping stone in the difficult problem of enumerating (3+1)-avoiding posets, we successfully enumerate graded (3+1)-avoiding posets with generating function techniques.
Half of this paper is a mathematician (in particular combinatorialist) -targetting exposition of adinkras, a delightful combinatorial structure used to encode information about supersymmetry representations created by the "DFGHIL" collaboration of mathematicians and physicists. I then extract the purely combinatorial problems and make some structural and enumerative theorems using elementary poset-theoretical and homological techniques.
The parallel chip-firing game is an interesting dynamical system with some history that is far from well-understood. In this paper, we introduce the neat concept of "motors" to better study localized behavior of the game. Furthermore, we give the first complete characterization of the periodic behavior that can occur in this game.
We note the cute theme that certain properties (such as being an edge polytope, or normality) behave very well for edge polytopes when they are separated by hyperplanes. We make this precise and outline some related algorithms and structural properties.
This paper solves the problem of seeing if six graduate students can write a joint paper. It also uses a hodgepodge of ideas to look at enumerating matrices over finite fields with certain conditions. We think of them as q-analogues of permutations and explain strange coincidences. An earlier draft is available on the arXiv.
We generalize the Frobenius problem of determining the maximal integral value such that a given set of integral denominations cannot produce to higher dimensions. Many fundamentals we'd naturally want for such a theory are proven here.
A discussion of four problems where I give solutions of various forms: enumerating graded posets, graded (2+2)-avoiding posets, graded (3+1)-avoiding posets, and graded (2+2)- and (3+1)-avoiding posets. The (3+1)-avoiding poset work is joint work with Joel Lewis.
During the MIT IAP period of 2011-2012, I worked about 14 hours a day (with Sasha Rush) to create a heads-up NL Hold'em program for the first MIT Pokerbots competition. We had some weird bugs but still won the dubious "Best Strategy Report" award. This was an invited lecture given for the second-year competitors about some lessons I learned, ranging from strategy to programming practices, but also some war stories of blood, sweat, and tears.
A not-so random walk through some basic topics in probability and machine learning. I did this by juxtoposing some seemingly disparate topics that I bundle in my mental model: binary hypothesis testing, Naive Bayes (and its lesser-known sister, Noisy-Or), and basic causality concepts (e.g. d-separation). Theme: being naive is good.
I debunk some common misconceptions about the role of mathematics in poker, ranging from overemphasis ("all you have to do is play by the odds") to over de-emphasis ("poker is not about numbers, it is about guts"). In mathematician-style, I introduce successive toy games to show how certain "psychological" aspects such as bluffing arise naturally from game-theory and how different "action ranges" form given the players' options. The toy games are happily rich enough to create some heuristics directly applicable in actual play.
Abstract: I will attempt to illuminate the shady "umbral calculus" classically delegated to algebraic manipulations that annoyed mathematicians much like physics still does (in both senses of being dreadfully non-rigorous and delightfully useful), starting with Blissard's Symbolic Method which thrived in the 1860's development of classical invariant theory. The talk will then scan across a couple of other seemingly unrelated disciplines, including proving that $e = 2$ in finite calculus and giving a way to solve a general cubic/quartic. Short ninja lesson included upon request.
This is based on one of my projects at my REU in Duluth. I generalize slightly a result of Greene and contemplate another angle of thinking about permutation containment and avoidance by splitting permutations into classes based on RSK.
For non-academic things, please see my personal page.