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).
Nathan Williams
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(0) years old. I am neither the most distinguished Nathan, nor the most distinguished Williams in algebraic combinatorics. (Nor, as it turns out, am I the most well-known Nathan Williams at UTD.)
I have been the department's mathematics graduate advisor for 0 days. Visit my graduate advising page for more information.
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(0) years old. I am neither the most distinguished Nathan, nor the most distinguished Williams in algebraic combinatorics. (Nor, as it turns out, am I the most well-known Nathan Williams at UTD.)
I have been the department's mathematics graduate advisor for 0 days. Visit my graduate advising page for more information.
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:
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.
Some time after publication (which took a short decade or so), I was in Fargo at a sectional. Browsing through the AMS book fair room, I asked the representative if they didn't also possibly have things published as Memoirs:
Rep: "No, why? Do you have one?"
Me: "Well, yes—actually, it was a bit tricky to get published because of the marginalia..."
Rep: "Oh. We all heard about that one."
Some time after publication (which took a short decade or so), I was in Fargo at a sectional. Browsing through the AMS book fair room, I asked the representative if they didn't also possibly have things published as Memoirs:
Rep: "No, why? Do you have one?"
Me: "Well, yes—actually, it was a bit tricky to get published because of the marginalia..."
Rep: "Oh. We all heard about that one."
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. Dequêne, 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:
Selected Talks
Video Recordings
| Recording | Event/Slides | Year |
|---|---|---|
| Charmed Roots and the Kroweras Complement Video recording  |
FPSAC 2024  |
2024 |
| Coxeter Combinatorics and Braid Varieties Video recording  |
ICMS  |
2023 |
| Rational Noncrossing Catalan Combinatorics Video recording  |
OPAC  |
2022 |
| Semidistrim Lattices Video recording  |
BIRS  |
2021 |
| Strange Expectations Video recording  |
UCLA Combinatorics Seminar  |
2020 |
| Independence Posets Video recording  |
Triangle Lectures in Combinatorics  |
2020 |
| Combinatorics of the “coincidental” reflection groups (on behalf of Vic Reiner) Video recording |
FPSAC 2019 — Ljubljana VideoLectures.NET · CC BY 4.0 |
2019 |
| Mathematical clarity versus an entrenched, out-of-touch bureaucracy Video recording Kevin Jon Williams*, Lewis Katz School of Medicine, Temple University (USA); Sahlgrenska Academy, Gothenburg University (Sweden) |
Special Session on Dynamical Systems, Algebraic Combinatorics, and Nepotism AMS Sectional (Hawaii) |
2019 |
Slides and Presentations
| Presentation | Event / Materials | Year |
|---|---|---|
Catalan Combinatorics and Decategorification of Knot Invariants |
Diagrammatics for Combinatorics, Algebra and Topology  Interactive slides |
2026 |
Rowmotion Is...Inevitable |
AMS Sectional Interactive slides |
2026 |
| Cell Decompositions of Hecke Traces and Link Polynomials |
FPSAC 2025  |
2025 |
| Catalan Recollections Lecture 1 Lecture 2  |
SLC93  |
2025 |
Fixed Points of Parking Functions / Inverting Parking Zeta  |
UTD UTD slides November 22, 2019 LaCIM slides October 5, 2017 |
2017, 2019 |
Sweeping up Zeta  |
UTD Job talk October 9, 2017 |
2017 |
Cataland: Why the Fuss?  |
FPSAC 2016 July 6, 2016 |
2016 |
Selected Posters
| Poster | Event | Year |
|---|---|---|
Triangulating the Permutahedron  |
FPSAC 2026  |
2026 |
Pop, Crackle, Snap (and Pow): Some Facets of Shards  |
FPSAC 2023  |
2023 |
Strange Checkspectations  |
FPSAC 2021  |
2021 |
Independence Posets  |
FPSAC  |
2020 |
Research Code
As with most academics, documentation, version control, and usability have always been things...I've definitely considered learning more about one day. This casual disregard for the future is a gift that my past self often passes on to me, and now bestows upon you. As such, complaints can only be accepted upon proper delivery to my past self by time machine.
Trim Rowmotion and Independence Posets
Build maximal and tight orthogonal-pair posets, compute rowmotion, and explore Galois-graph and Cambrian-lattice examples.
Mixed Shifted Insertion
Implements mixed shifted insertion and its inverse, with examples and random generation for skew tableaux of staircase shape minus a rectangle.
Sweeping up Zeta
Implements the modular sweep map, its inverse, and the equitable-partition routines used for rational Catalan objects.
Dual-braid rewriting
A documented GAP3/CHEVIE toolkit for rewriting pure braids in dual generators, including an E8 driver and machine-readable exceptional data.
HOMFLY signed-subword counts
A SageMath companion that computes the q = 1 HOMFLY a-coefficients of a knot or link as signed subword counts.
Mathematica visualizations
- LOAD.nb — An undocumented Mathematica file I began in graduate school. Use at your own risk.
18 additional Mathematica visualizations
Summary
Since Fall 2017, I have taught 38 course sections over 18 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), one completed Ph.D. thesis, and three current Ph.D. students.
I have advised four independent study courses, two senior capstone projects, five honors senior theses (two of which resulted in publications), one completed Ph.D. thesis, and three current Ph.D. students.
Student supervision
Algebraic Combinatorics
| Organization | Title |
|---|---|
Annals of Combinatorics
|
Associate Editor |
Outreach
UTD Department of Mathematical Sciences
|
Chair of the website and computer committee
Member of the outreach committee:
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 |
Grants under Review
| Title (click for proposal) | Funding agency | Reviews | Year |
|---|---|---|---|
| No current submissions listed | - | - | - |
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 |