Ruoguang (Roger) Tian

Current Office: 3129 MSB
E-mail: rgtian[at]math.ucdavis.edu (replace [at] with @)

I am a graduate student in the Mathematics Department at UC Davis. My advisor is  Anne Schilling.

My research focus is on card shuffling (from a combinatorial perspective), Young tableaux and their statistics, and generally combinatorics on words.

My other interests include algebraic geometry and its applications to number theory.

Teaching

2011 Summer Session II

Math 16B

2011 Fall

Math 21D

2012 Summer Session I

Math 16C

2013 Summer Session I

Math 16B

2015 Summer Session II

Math 145

Papers

The following is a list of mathematical papers I've written.

  • Generalizations of an Expansion Formula for Top to Random Shuffles. To appear in Annals of Combinatorics, arXiv version (2014).
    Here I give a bijective proof for an expansion formula derived (non-bijectively) by Adriano Garsia in his paper On the Powers of Top to Random Shuffling (link here), and then use this bijection to generalize the expansion formula in various directions.

  • Symmetric Decomposition in the Column Case. Preprint (2015).
    A major open problem is the identification of the symmetric and antisymmetric components (called "plethysms") of the power of a Schur function. In the case of the square of a Schur function, this problem has already been solved (link here), where the symmetric and antisymmetric components were described using a statistic called "spin" on domino tableaux, but the proof is advanced and non-combinatorial. Here I give a simple bijective proof in the case of the column Schur function, describing the symmetric and antisymmetric components with a different statistic called "energy" on pairs of semistandard Young tableaux.

  • Even and Odd-Lengthed Subwords of a Given Word. Preprint (2014).
    A basic result of enumerative combinatorics is that there is a bijection between the even-sized subsets and odd-sized subsets of a given set. Here I extend this result to multisets and arbitrary words; this work is related to my work described above.

  • On the Diophantine Equation 2^a3^b + 2^c3^d = 2^e3^f + 2^g3^h . Preprint, arXiv version (2009).
    Here I solve, through elementary means, the Diophantine equation (m+1)^k(m^p-1) = m^q-1 for the case that m is even.

  • Identities of the Function f(x,y) = x^2 + y^3 . Preprint, arXiv version (2009).
    Here I give some partial results on an open problem of Harvey Friedman.

    • Talks Given

  • Expansion Formulae for Top to Random Shuffling, October 2015, AMS Central Fall Sectional Meeting (link)
  • Expansion Formulae for Top to Random Shuffling, March 2015, UC Davis/Berkeley Combinatorics Gathering II
  • Expansion Formulae for Top to Random Shuffling, October 2014, Student-Run Applied & Math Seminar, UC Davis
  • Generalizing an Expansion Formula in the Algebra of Shuffles, October 2013, Student-Run Applied & Math Seminar, UC Davis
  • Expansion Formulae in the Algebra of Shuffles, December 2012, Student-Run Applied & Math Seminar, UC Davis
  • Representation of Quivers, June 2012, Student-Run Algebraic Geometry Seminar, UC Davis
  • On the Diophantine equation (m+1)^k(m^p-1) = m^q-1, October 2010, Student-Run Applied & Math Seminar, UC Davis

    • Conferences/Workshops Attended

  • 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), June - July 2014, DePaul University, Chicago, Illinois (link)
  • ICERM workshop Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series, January - May 2013, Brown University, Providence, RI (link)
  • Automorphic Forms, Representations, and Combinatorics (A Conference in Honor of Daniel Bump), August 2012, Stanford University, Palo Alto, CA (link)