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About Me

I received my Ph.D. in 2013 from the University of Minnesota, under the supervision of Dennis Stanton; the title of my dissertation was "Cataland". I then had the opportunity to spend 2+1 wonderful years at LaCIM as a postdoc—my original offer was for two years, but I was able to stay for one additional year because Hugh Thomas came to LaCIM with a Canada Research Chair and generously extended my postdoc. (Thanks to this extra year, Hugh and I were able to invert the sweep map!)

After one year at the University of California, Santa Barbara as Jon McCammond's postdoc, I started as an assistant professor at the University of Texas at Dallas in 2017. I became a tenured associate professor in 2024.

I am roughly Γ() years old. I am neither the most distinguished Nathan, nor the most distinguished Williams in algebraic combinatorics.


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Selected Papers

My research is in algebraic combinatorics, centered around reflection, braid, and Lie groups. I have been very fortunate to have had many brilliant collaborators, with whom I have written some very nice papers. Here are a few of my favorites:


Promotion and Rowmotion (with J. Striker) unifies various results in what J. Propp calls "Dynamical Algebraic Combinatorics". Later related work includes Rowmotion in Slow Motion and Independence Posets (with H. Thomas), and Semidistrim Lattices (with C. Defant).
Doppelgängers (with Z. Hamaker, R. Patrias, and O. Pechenik) uses K-theoretic jeu-de-taquin to give an explicit bijection between plane partitions in a rectangle and plane partitions in a trapezoid. This solves an old problem considered by both R. Proctor and J. Stembridge--as R. Proctor wrote "the question of a combinatorial correspondence...seems to be a complete mystery."
Sweeping up Zeta (with H. Thomas) inverts D. Armstrong, N. Loehr, and G. Warrington's sweep map, with applications to (q,t)-combinatorics. Inverting sweep is unexpectedly equivalent to a different problem that Hugh and I solved in 2012. Later related work includes Fixed Points of Parking Functions (with J. McCammond and H. Thomas).
Cataland: Why the Fuss? (with C. Stump and H. Thomas) was a 132-page monograph that extends N. Reading's Coxeter-sortable elements and the cluster complex to the positive braid monoid---extending D. Armstrong's Fussification of noncrossing partitions (after P. Edelman) to the other Coxeter-Catalan families.

Rational Noncrossing Catalan Combinatorics (with P. Galashin, T. Lam, and M.-T. Trinh) is a 42-page preprint that uses braid varieties to give the first uniform construction of rational noncrossing Catalan objects and the first uniform enumeration of noncrossing objects. These were both longstanding open problems, considered in both Why the Fuss? and Rational associahedra and noncrossing partitions (with D. Armstrong and B. Rhoades).
Coxeter Pop-Tsack Torsing (with C. Defant) studies the Bessis dual version of C. Defant's Coxeter Pop-Stack Sorting map--that is, we modify the definitions by replacing all instances of S (the simple reflections) with T (the set of all reflections). Later related work with C. Defant includes Crystal pop-stack sorting and type A crystal lattices and The Ungar games (with N. Kravitz).
Charmed Roots and the Kroweras Complement (with B. DequĂȘne, G. Frieden, A. Iraci, F. Schreier-Aigner, and Hugh Thomas) continues the saga of trying to find a bijection between nonnesting partitions and noncrossing partitions. The title is an unfortunate portmanteau of Germain Kreweras's last name and rowmotion. As I said at FPSAC: "I was hired for this project as the marketing consultant. They told me to write the introduction to the paper. At some point, about halfway through writing this introduction, I decided I should check out what the paper was all about. And that's what I'm here to tell you about today."

Papers and Preprints

These profiles likely have a fairly up-to-date list of my work:
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Selected Talks

Recording Event/Slides Year
Catalan Recollections
Lecture 1 Lecture 2
SLC93
2025
Charmed Roots and the Kroweras Complement
FPSAC 2024
2024
Coxeter Combinatorics and Braid Varieties
ICMS
2023
Rational Noncrossing Catalan Combinatorics
OPAC
2022
Semidistrim Lattices
BIRS
2021
Strange Expectations
UCLA Combinatorics Seminar
2020
Independence Posets
Triangle Lectures in Combinatorics
2020

Selected Posters

Poster Event Year
Pop, Crackle, Snap (and Pow): Some Facets of Shards
FPSAC 2023
2021
Strange Checkspectations
FPSAC 2021
2021
Independence Posets
FPSAC
2020

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Summary

Since Fall 2017, I have taught 33 courses over 15 semesters at UTD— In 2020, I won UTD's School of Natural Science and Mathematics tenure-track teaching award. I completed the ACUE course in "Effective College Instruction" in Spring 2021.

I have advised four independent study courses, two senior capstone projects, five honors senior theses (two of which resulted in publications), and one Ph.D. thesis.

 

Current Courses (Spring 2025)

Course Name Notes
MATH 3315: Discrete Math and Combinatorics

Past Courses

Course Name Notes Semesters (click for reviews)
MATH 2417: Calculus I F17, F18a, F18b, F20
MATH 2419: Calculus II S22, S23
MATH 3315: Discrete Math and Combinatorics S19, F19, S20, F20, S21, F21, S22, F22, S23, F23, S24, F24
MATH 6311: Algebra I
(graduate course)
S18, S19, S20, S21, F21, F22, F23
MATH 6312: Combinatorics and Graph Theory
(graduate course)
F24
Honors Reading
Topics in Algebraic Combinatorics
F19, F20, F21, F22, F23, F24
MATH 2419: Calculus II
(honors section)
S24

Selected Student Supervision

Name Degree Project Year
Weston Miller Honors Bachelor's Thesis Rational Catalan Numbers for Complex Reflection Groups
Project code available here
2024
Priyojit Palit Ph.D. Dual Braid Presentations and Cluster Algebras
Won ``Best Graduate Blitz'' for his video presentation at the DFW Area 2021 Virtual Poster Competition in Mathematics
2022
Yusuf Ahsanullah Honors Bachelor's Thesis Quantum Groups and Cluster Fans 2022
Deep Desai, Ramesh Kanakala, Ryan Lofton, Savitha Venkatesh, Ariba Tahsin Bachelor in Data Science - Capstone Project Software implementation of independence posets
Capstone Project Presentation Award: Honorable Mention - Creativity
2022
Michelle Sferrazza Honors Capstone Project EL-shellability of noncrossing partition lattices 2022
Joshua Marsh Honors Bachelor's Thesis Nesting Nonpartitions
Published in Journal of Integer Sequences
2020
Benjamin Cotton Honors Bachelor's Thesis A core model for G2
Published in Involve
2020

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Algebraic Combinatorics

Organization Title
Annals of Combinatorics Associate Editor

UTD Department of Mathematical Sciences

Chair of the website and computer committee

Member of the outreach committee:
  • Yearly Weeks of Welcome recreational math activity table (with Maxim Arnold and Vladimir Dragovic)
  • $\pi$-day Buffon's noodle presentation (with Maxim Arnold)

  • Member of the K-12 mathematics competitions committee
  • MATH COUNTS

  • Faculty organizer for the Putnam Club and William Lowell Putnam Mathematical Exam

    Community

    Link Subject News Stations
    2022 Mega Millions lottery jackpot CBS 11 (also NBC 5)
    2018 NCAA men's basketball bracket WFAA and on over 15 television stations nationwide

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    Grants under Review

    Title (click for proposal) Funding agency Year

    Grants Funded

    Title (click for proposal) Funding agency Reviews Year
    New Techniques in Algebraic Combinatorics Simons (Not provided) 2018
    Combinatorics and Braid Varieties NSF Reviews Panel Summary Officer Summary 2023+

    Grants Declined

    Title (click for proposal) Funding agency Reviews Year
    Independence Posets and Dynamical Algebraic Combinatorics NSF Reviews Panel Summary Officer Summary F21
    CAREER: Dynamical Algebraic Combinatorics NSF Reviews Panel Summary Officer Summary S21
    Independence Posets NSF Reviews Panel Summary Officer Summary F20
    Combinatorial Tools in Representation Theory NSF Reviews Panel Summary Officer Summary F19
    Geometry of Braid Groups in Combinatorics NSF Reviews Panel Summary Officer Summary F18
    New bijective techniques in algebraic combinatorics NSF Reviews Panel Summary Officer Summary F17