### Publications

Mixed Dimer Configuration Model in Type D ClusterAlgebras II: Beyond the Acyclic Case with Libby Farrell and Kayla Wright.

Matrix Formulae for Decorated Super Teichmüller Spaces
with N. Ovenouse and S. Zhang.

Labeled Chip-firing on Binary Trees with S. Nguyen.

Two Formulas for F-Polynomials with F. Lin and T. Nakanishi.

Mixed Dimer Model in Type D ClusterAlgebras with Kayla Wright.

Cluster scattering diagrams and theta functions for reciprocal generalized cluster algebras with M.-W. Cheung and E. Kelley. Annals of Combinatorics (to appear)

Double Dimer Covers on Snake Graphs from Super Cluster Expansions
with N. Ovenouse and S. Zhang. Journal of Algebra 608 (2022): 325-381.

An Expansion Formula for Decorated Super-Teichmuller Spaces with Nick Ovenhouse and Sylvester Zhang, SIGMA, Volume 17 (2021), 080, 34 pages..

Quiver Mutations, Seiberg Duality and Machine Learning with Jiakang Bao, Sebastian Franco, Yang-Hui He, and Yan Xiao, Phys. Rev. D. 102, 086013 (2020), 1--28.

Dungeons and Dragons: Combinatorics for the dP3 Quiver with Tri Lai, Annals of Combinatorics, Volume 24 (2020), no. 2, 257--309.

Cluster Algebraic Interpretation of Infinite Friezes with Emily Gunawan and Hannah Vogel, European Journal of Combinatorics, Volume 81 (2019), 22--57.

Paths to Understanding Birational Rowmotion on Products of Two Chains with Tom Roby, Algebraic Combinatorics, Volume 2 (2019) no. 2, 275--304.

Higher Cluster Categories and QFT Dualities with Sebastian Franco, Phys. Rev. D. 98, 046021 (2018).

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:

Double Dimer Covers on Snake Graphs from Super Cluster Expansions: Isaac Newton Cluster Algebra Seminar, November 2, 2021

Combinatorial Expansion Formulas for Decorated Super Teichmuller Spaces: Paris Algebra Seminar, April 26, 2021

Double-dimer configurations and quivers of dP3 (del Pezzo) type: Online Cluster Algebra Seminar, November 17, 2020

Higher Cluster Categories and QFT Dualities: University of Minnesota Combinatorics Seminar, October 4, 2019

Dimer Interpretations of toric cluster variables associated to del Pezzo quivers: From Dungeons to Dragons: RIMS, June 21, 2019

Applications of New F-polynomials Formulas in terms of C-Vectors, Cluster Algebras and Related Topics: Special Session at AMS Spring Sectional Meeting at University of Connecticut, Hartford, April 13, 2019

A Combinatorial Formula for Birational Rowmotion on Rectangular Posets: Combinatorics and Graph Theory Seminar, Michigan State, October 24, 2017

Beyond Aztec Dragons and Castles: Toric Cluster Variables for the dP3 Quiver: MIT Combinatorics Seminar, April 8, 2016

Cluster Algebras and Brane Tilings: University of Connecticut Colloquium, April 7, 2016

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.

FPSAC 2007 Conference Proceeding- Elliptic Curve Groups and Chip-Firing Games

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)