Portrait of 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.

Family photo

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? cover image. 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 Catalan equation diagram.
Animated braid path preview.
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:

Selected Talks

Recording Event/Slides Year
Rowmotion Is...Inevitable
Preview image for the Rowmotion Is...Inevitable presentation.
AMS Sectional
Interactive slides
2026
Cell Decompositions of Hecke Traces and Link Polynomials
Animated braid path preview.
FPSAC 2025
FPSAC 2025 logo.
2025
Catalan Recollections
Lecture 1 Lecture 2
Nonnesting and noncrossing Catalan diagram.
SLC93
SLC93 group photo.
2025
Charmed Roots and the Kroweras Complement
Recording preview for Charmed Roots and the Kroweras Complement.
FPSAC 2024
Slide preview for Charmed Roots and the Kroweras Complement.
2024
Coxeter Combinatorics and Braid Varieties
Recording preview for Coxeter Combinatorics and Braid Varieties.
ICMS
Slide preview for Coxeter Combinatorics and Braid Varieties.
2023
Rational Noncrossing Catalan Combinatorics
Recording preview for Rational Noncrossing Catalan Combinatorics.
OPAC
Slide preview for Rational Noncrossing Catalan Combinatorics.
2022
Semidistrim Lattices
Recording preview for Semidistrim Lattices.
BIRS
Slide preview for Semidistrim Lattices.
2021
Strange Expectations
Recording preview for Strange Expectations.
UCLA Combinatorics Seminar
Slide preview for Strange Expectations.
2020
Independence Posets
Recording preview for Independence Posets.
Triangle Lectures in Combinatorics
Slide preview for Independence Posets.
2020
Mathematical clarity versus an entrenched, out-of-touch bureaucracy
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
Sweeping up Zeta
Preview image for the Sweeping up Zeta presentation.
UTD Job talk
October 9, 2017
2017
Inverting Parking Zeta
Animated preview for the Inverting Parking Zeta presentation.
Algebra and Combinatorics at Lacim, A conference for the 50th anniversary of the CRM
October 5, 2017
2017
Cataland: Why the Fuss?
Preview image for the Cataland: Why the Fuss? presentation.
FPSAC 2016
July 6, 2016
2016

Selected Posters

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

Summary

ACUE Effective College Instruction certificate preview.
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.

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

Algebraic Combinatorics

Organization Title
Annals of CombinatoricsAnnals of Combinatorics journal cover. Associate Editor

Outreach

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)Weeks of Welcome recreational math activity table.
  • $\pi$-day Buffon's noodle presentation (with Maxim Arnold)
Member of the K-12 mathematics competitions committee
Faculty organizer for the Putnam Club and William Lowell Putnam Mathematical Exam

Community

Link Subject News Stations
CBS segment preview for the Mega Millions lottery jackpot. 2022 Mega Millions lottery jackpot CBS 11 (also NBC 5)
WFAA segment preview for an NCAA basketball bracket story. 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