Publications (Annotated list with abstracts available here):
Dungeons and Dragons: Combinatorics for the dP3 Quiver with Tri Lai
Higher Cluster Categories and QFT Dualities with Sebastian Franco
Cluster Algebraic Interpretation of Infinite Friezes with Emily Gunawan and Hannah Vogel
Paths to Understanding Birational Rowmotion on Products of Two Chains with Tom Roby, to appear in Algebraic Combinatorics
Counting Arithmetical Structures on Paths and Cycles with Benjamin Braun, Hugo Corrales, Scott Corry, Luis David Garcia Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin and Carlos E. Valencia, Discrete Mathematics. (2018), 341, 2949--2963.
Beyond Aztec Castles: Toric Cascades in the dP3 Quiver with Tri Lai, Comm. in Math. Phys. (2017), Volume 356, Issue 3, 823--881.
On Maximal Green Sequences For Type A Quivers with Alexander Garver, Journal of Algebraic Combinatorics
Vol. 45, Number 2 (2017), 553--599
The Greedy Basis Equals the Theta Basis: A Rank Two Haiku with
Man Wai Cheung, Mark Gross, Greg Muller, Dylan Rupel, Salvatore Stella, and Harold Williams, JCTA (2017), Volume 145, 150--171.
T-Path Formula and Atomic Bases for Cluster Algebras of Type D with Emily Gunawan, SIGMA 11 (2015), 060, 46 pages
Aztec Castles and the dP3 Quiver with
Megan Leoni, Seth Neel, and Paxton Turner, J. Phys. A: Math. Theor. 47 (2014), 474011
The Cyclotomic polynomial topologically with Vic Reiner, J. Reine Angew. Math. 687 (2014), 113-132.
Gale-Robinson sequences and brane tilings with In-Jee Jeong and Sicong Zhang, DMTCS Proceedings vol. AS, 25th International Conference on
Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 707--718.
Involutions on standard Young tableaux and divisors on metric graphs with
Rohit Agrawal, Vladimir Sotirov, and Fan Wei
Electronic Journal of Combinatorics. Vol. 20(3) (2013) #P33, 1-23.
Bases for cluster algebras from surfaces with Ralf Schiffler and Lauren Williams
Compos. Math. 149 (2013), Issue 02, 217-263.
Matrix Formulae and skein relations for cluster algebras from surfaces with Lauren Williams
Int. Math. Res. Notices, 2013, No. 13, 2891-2944.
Linear Systems on Tropical Curves with Christian Haase and Josephine Yu
Math. Zeitschrift, Volume 270, 3-4 (2012), 1111-1140.
Also on the Arxiv.
A Compendium on the Cluster Algebra and Quiver Package in SAGE with Christian Stump,
Seminaire Lotharingien de Combinatoire 65 (2011), Article B65d
Positivity for cluster algebras from surfaces with Ralf Schiffler and Lauren Williams
Advances in Mathematics, Vol. 227, Issue 6 (2011), 2241--2308.
Also on the Arxiv.
A Graph Theoretic Expansion Formula for Cluster Algebras of Classical Type
Annals of Combinatorics, Volume 15, Number 1 (2011) 147--184
Also on the Arxiv, and presented at FPSAC 2008.
Cluster expansion formulas and perfect matchings with Ralf Schiffler
Journal of Algebraic Combinatorics, Vol. 32, Number 2 (2010), 187--209
Also on the Arxiv.
The critical groups of a family of graphs and elliptic curves over finite fields. Journal of Algebraic Combinatorics: Vol. 30, Issue 2 (2009), 255--276
Preprint also presented at FPSAC '07 at Tianjin, China.
Invariants, Kronecker Products
and Combinatorics of some Remarkable Diophantine systems (Extended Version)
with Adriano Garsia, Nolan Wallach, and Guoce Xin.
Adv. in Appl. Math. 42 (2009), no. 3, 392--421
A new characterization for the m-quasiinvariants
of S_n and explicit basis for two row hook shapes
with Jason Bandlow. J. Combin. Theory Ser. A, 115 (2008), no 8, 1333-1357 Also presented at FPSAC 2008.
Combinatorial Aspects of Elliptic Curves
Seminaire Lotharingien de Combinatoire 56 (2007), Article B56f
Also presented at FPSAC '06 at San Diego.
Combinatorial Interpretations for Rank-Two Cluster Algebras of Affine Type
with Jim Propp
Electronic Journal of Combinatorics. Vol. 14 (R15), 2007.
Quasiinvariants of S_3 with Jason Bandlow.
J. Combin. Theory Ser. A 109 (2005), no 2, 281-298
Slides from selected Talks:
Combinatorics of Toric Cascades: Aztec Castles and Beyond : Worshop on Postive Grassmannians, CRM, July 28, 2015
Combinatorics of the Del-Pezzo 3 Quiver:
Aztec Dragons, Castles, and Beyond : Integrable Combinatorics Special Session at AMS Spring Sectional Meeting at Michigan State University, Lansing, March 15, 2015
Combinatorics of the Del-Pezzo 3 Quiver: Aztec Dragons, Castles, and Beyond: Conference on Cluster Algebras in Combinatorics and Topology, KIAS, December 14, 2014
Aztec Castles and Other Instances of Brane Tilings in Combinatorics: Workshop on Gauge Theories: Quivers, Tilings and Calabi Yaus, ICMS, May 12, 2014 (Versions also given at
ICERM, University of Minnesota and Oberwolfach)
Brane Tilings and Cluster Algebras: Algebra, Geometry, and Combinatorics Day 9, November 9, 2013 (Versions also given at
2013 Joint Meetings of the AMS and MAA at San Diego, UC Berkeley, MSRI, and the 2012 Canadian Summer Math Meeting)
Bases for Cluster Algebras from Surfaces: Bay Area Discrete Math Day, November 17, 2012 (Also given at
UC Davis, University of Toronto, University of Minnesota, BIRS, ICERM)
A Topological Interpretation of the Cyclotomic Polynomial, FPSAC, June 16, 2011 (Also given at MIT, University of Minnesota, and UQAM)
Linear Systems on Tropical Curves, Combinatexas, April 16, 2011 (Also given at FPSAC 2010, SIAM DM10, the 2010 Canadian Summer Math Meeting, UCSD, and UC Berkeley)
Elliptic Curves and Chip-Firing Games on Graphs: University of Minnesota, Dec. 3, 2009
(Also given at UC Berkeley, University of Kentucky, UCSD, Dartmouth, UC Davis, MIT, University of Kansas, Tulane University, Georgia Tech, and San Francisco State University, Stanford University, Arizona State University, MSRI, and USC)
Cluster Algebras of Triangulated Surfaces and Snake Graphs: Brown University, Feb. 9, 2010
(Also given at University of Michigan, at the
Maurice Auslander Distinguished Lectures and International Conference, at UQAM, MIT, University of Washington, Texas A&M, University of Minnesota, Ohio State University, and University of Pittsburgh.
Abridged versions given at FPSAC 2009, the AMS Fall 2009 Eastern Sectional at Penn State, and MSRI.)
Graph Theoretical Cluster Expansion Formulas: International Conference on Cluster Algebras and Related Topics - Mexico City - December 19, 2008
A Graph Theoretic Interpretation for Cluster Algebras of Classical Type: Northeastern University - Representation Theory Seminar - March 14, 2008
(Also given at UMass Amherst, University of Pennsylvania, and MIT)
The Zeta Function of a Cyclic Language with Connections to Elliptic Curves and Chip-Firing Games: Recent Progress in Combinatorics on Words - CRM - March 14, 2007 (Similar talks given at FPSAC 2006, UQAM, and UCSD)
The m-Quasiinvariants of S_n: Joint Meetings of the AMS and MAA at Phoenix - January 9, 2004
Theses and other Documents:
My Ph.D. Thesis- A Combinatorial Comparison of Elliptic Curves and Critical Groups of Graphs
My undergraduate thesis - Cluster Algebras, Somos Sequences, and Exchange Graphs.
Combinatorial Interpretations for the Markoff Numbers (Unpublished preprint) with Andy Itsara, Jim Propp, and Rui Viana, as part of the REACH program
Schoof's Algorithm for Counting Points on E(F_q)
(Math 168 Project under direction of William Stein)