Research (Yan X Zhang)

Papers and Preprints:

In roughly reverse-chronological order:

  • Combining GHOST and Casper. (with Vitalik Buterin, Diego Hernandez, Thor Kamphefner, Khiem Pham, George Qiao, Danny Ryan, Juhyeok Sin, and Ying Wang) 2020. We combine LMD GHOST and Casper FFG for an abstract blockchain design that serves as the backbone of the Ethereum 2.0 beacon chain.
  • Applying Fourier Analysis to Judgment Aggregation. 2018. We try to get a similar theorem as the celebrated Arrow’s Theorem from social choice theory, except in the theory of judgment aggregation. Inspired by Gil Kalai, we use Fourier methods. Our theorem has a fun appearance of classical logical functions AND, OR, and XOR.
  • Enumerative Gadget Phenomena for (4,1)-Adinkras (with Isaac Friend and Jordan Kostiuk). 2018. We explain algebraically some phenomena that comes up with Jim Gates’ ‘gadget’ from the adinkras literature. We also philosophize what the right definitions could be for higher dimensions.
  • A Unified Enumeration of 1-dimension Garden Algebras and Valise Adinkras. Annales de L’Institut Henri Poincare. 2018. We clean up some enumeration of garden algebras/valise adinkras from the adinkras literature, with the help of techniques developed from my thesis.
  • Decompositions of Amplituhedra (with Steven Karp and Lauren Williams). Annales de L’Institut Henri Poincare. 2017. We study the decomposition of “the amplituhedron,” another mysterious object where physicists have strong intuitions that are elusive to prove, with connections to the Grassmannian and total positivity. We prove these strong intuitions for $n=2$ with some surprisingly technical combinatorics.
  • Convex Neural Codes in Dimension 1 (with Zvi Rosen). 2017.  Inspired by recent work on neural codes, this paper studies “1-dimensional discrete convexity” which really just ends up being the consecutive-ones-property. Divides the problem space into different regimes and gives algorithmic/enumerative results.
  • Discrete Envy-free Division of Necklaces and Maps (with Roberto Barrera, Kathryn Nyman, Amanda Ruiz, Francis Su). 2015. Uses some elementary techniques to discuss discrete cake-cutting problems, where we no longer have guarantee of envy-freeness that usual cake-cutting problems do. However, we relax the condition to the natural “epsilon-Envy-free” condition and prove some simple but useful theorems that are applicable to 2 dimensions (map-cutting) as well.
  • Structural Theory of 2-d Adinkras (with Kevin Iga). Advances in High Energy Physics, vol. 2016, Article ID 3980613, 12 pages, 2016. A continuation of the earlier Adinkras work. We classify possible Adinkras under the additional constraints that come with 2 dimensions (1+1 space and time). As before, there are nice connections to doubly-even codes and the theory of graph switching.
  • Four Variations on Graded Posets. 2015. Explores four classes of graded posets, adding a nice observation of counting certain classes of combinatorial objects where switching between the exponential and ordinary generating functions is easy. Uses a lot of matrix tranfer methods. This was the work I took a hiatus on when I started working with Joel Lewis for graded (3+1)-avoiding posets; many similar techniques come up.
  • The Combinatorics of Adinkras. 2013. MIT Thesis. 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.
  • Enumeration of Graded (3+1)-Avoiding Posets (with Joel Lewis). Journal of Combinatorial Theory, Series A. 2013. Vol. 120, Issue 6.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.The extended abstract presented to FPSAC 2012 in Nagoya, Japan can be found here. It won the award for best student paper.
  • Adinkras for Mathematicians. Transactions of the American Mathematical Society. 366 (2014), 3325-3355.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.
  • Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game (with Tian-yi Jiang and Ziv Scully). 2012. Preprint.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.
  • Separating hyperplanes of edge polytopes (with Takayuki Hibi and Nan Li). Journal of Combinatorial Theory, Series A, 120 (2013), 218-231.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.
  • Matrices with restricted entries and q-analogues of permutations (with Joel Lewis, Ricky Liu, Alejandro Morales, Greta Panova, and Steven Sam). J. Combinatorics, 2 (2011), pp. 355-396 (issue 3).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.
  • The multidimensional Frobenius problem (with Jeffrey Amos, Iuliana Pascu, Vadim Ponomarenko, and Enrique Trevino). Involve, Vol. 4 (2011), No. 2, 187-197.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.

Talks and posters:

In roughly reverse-chronological order. Research talks refer to papers and will have no additional information. Other (such as expository) talks will contain an abstract or similar information.

  • Automated Cryptography, San Jose State University Colloquium. 2018.
  • Fourier Analysis in Judgment Aggregation, Berkeley Combinatorics Seminar, 2018.
  • Combinatorics for Adinkranauts, Adinkras Math/Physics Hangout, Brown University. 2017.
  • Unfair Poker With Just 3 Cards, San Jose State University Research Seminar. 2017.
  • Voting “Theorems” and Misapplied Math, San Francisco State University, 2017.
  • A Combinatorial Approach to Supersymmetry, UIUC, 2017. This is an updated version of the Adinkras series, with more emphasis on combinatorics and some computational results.
  • Voting Theory “Theorems” and Misapplied Math, SJSU, 2016. This talk discusses some classical results and show how they can be applied and misapplied, focusing on the celebrated Arrow’s Theorem and the slightly less-well-known May’s Theorem, both “theorems” about how innocuous constraints on a voting system can force it to have certain restrictive, and often very strange, properties.  We also discuss work in progress (joint with Mahendra Prasad, UC Berkeley) about a generalization of May’s Theorem and implications for the social sciences.
  • Adinkras, Progress and Problems, Santa Clara University, 2016
  • Academia: The Game / How to Tinker with Problems, Meta-Math Meetup (MMM) MIT, 2016. This was a short workshop for (mostly) MIT math majors. These talks were about navigating the undergraduate phase of academia, and the difference between “real life” research and classwork.
  • Adinkras, Progress and Problems, SJSU, 2016.
  • Problem-Solving: Strategies and Community, UC Berkeley MUSA Seminar, 2016. I talk about problem-solving in the context of competitions such as the Math Olympad or Putnam, my personal experience as a competitor and community member, and strategies / tactics for the Putnam itself.
  • Adinkras, Progress and Problems, Shanghai Jiaotong University, 2015.
  • Adinkras, Progress and Problems, UCLA, 2015.
  • Adinkras, Progress and Problems, Minnesota, 2014. Similar to the “Adinkras for Mathematicians” talk, but includes more things as the program has advanced to 2 dimensions. Basically things from “Adinkras for Mathematicians” plus “Structural Theory of 2-d Adinkras.”
  • Combinatorics, Topology, and Fair Division, UC Berkeley MUSA Seminar, 2014. I talk about elementary topology with a combinatorial flavor applied to “real-life” problems such as splitting cakes or dividing rent, talking about Sperner’s Lemma, Tucker’s Lemma, and Brouwer’s Fixed-Point Theorem.
  • Patterns in the Chip-firing Game, Stanley@70, 2014.
  • Adinkras for Mathematicians, San Jose State University, 2013.
  • Adinkras for Mathematicians, San Francisco State University, 2013.
  • Patterns in the Chip-firing Game, BAD Math Day, 2013.
  • Variations on Graded Posets, Berkeley Combinatorics Seminar, 2013.
  • Adinkras for Mathematicians, UC Davis Combinatorics Seminar, 2013.
  • Variations on Graded Posets, MIT Combinatorics Seminar, 2012.
  • Variations on Graded Posets, LaCim Combinatorics Seminar, 2012. 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.
  • Adinkras for Mathematicians, Dartmouth Combinatorics Seminar, 2012.
  • Adinkras for Mathematicians, Berkeley Combinatorics Seminar, 2012.
  • Pokerbots Lessons, MIT Pokerbots Competition, 2012. 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.
  • Adinkras for Mathematicians, MIT Combinatorics Seminar , 2011.
  • On Being Naive, MIT Applied Math Graduate Student Seminar , 2011.A not-so random walk through some basic topics in probability and machine learning. I did this by juxtaposing 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.
  • Adinkras for Mathematicians, Lehigh University Graduate Student Seminar , 2011.
  • Matrices with restricted entries and q-analogues of permutations, FPSAC, 2011.
  • Myths of Poker Mathematics, MIT Applied Math Graduate Student Seminar , 2011. 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.
  • Shadow Calculus, MIT Pure Math Graduate Student Seminar , 2010.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.
  • On Permutation Avoidance, MAA/AMS Joint Meetings, 2006.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.
  • The multidimensional Frobenius Problem, Young Mathematicians Conference, 2005.