# Joshua P. Swanson

## Contact

I'm an NTT RTPC Assistant Professor of Mathematics at the University of Southern California (USC) in Los Angeles working with Greta Panova. From Fall 2018 to Spring 2021, I was a Stefan E. Warshawski Visiting Assistant Professor at the University of California, San Diego (UCSD) under the mentorship of Brendon Rhoades. I graduated from the University of Washington in June 2018 with a PhD in Mathematics. My doctoral advisor was Sara Billey. My PhD thesis was on "Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics".

My research is in algebraic combinatorics and analytic combinatorics. Algebraic combinatorics is a field of mathematics which is broadly interested in applying a wide variety of combinatorial methods (e.g. generating functions, Möbius inversion, recursive constructions, explicit bijections, polytopes) to analyze a vast array of algebraic structures (e.g. cohomology rings, irreducible decompositions, independent sets, Grothendieck groups, graphs). Analytic combinatorics seeks to give effective asymptotic estimates of combinatorial quantities, often by exploiting generating function identities. Such estimates make frequent use of tools from real and complex analysis (e.g. contour integrals, the saddle point method) and probability theory (e.g. the method of moments). A famous example combining both areas is the Hardy--Ramanujan estimate for the number of ways to write n as an unordered sum of positive integers. Much of my research is related to Young tableau, coinvariant algebras, and the surrounding combinatorics, commutative algebra, and representation theory, especially major index statistics. More technical topics of interest are listed below.

Research interests: Algebraic combinatorics, analytic combinatorics, combinatorial representation theory, symmetric functions, complex reflection groups, Coxeter groups, major index statistics, tableaux combinatorics, dynamical algebraic combinatorics, invariant theory, cyclic sieving, $\{$classical, super, diagonal$\}$ coinvariant algebras, $q$-analogues, generating function factorizations, free Lie algebras, cumulants, limit laws, local limit theorems, plabic graphs, Stirling numbers

## Publications and preprints

1. $q$-Stirling numbers in type $B$ (2022).
With Bruce Sagan.
Submitted. 46 pages. Available at arXiv:2205.14078.
2. Curious cyclic sieving on increasing tableaux (2022).
With Christian Gaetz, Oliver Pechenik, and Jessica Striker.
Published in Enumer. Combin. Appl. ECA 2:3 (2022) Article S2R17. 9 pages. Available at arXiv:2112.09228. pdf
3. Tanisaki witness relations for harmonic differential forms (2021).
Submitted. 29 pages. Available at arXiv:2109.05080.
4. Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds (2021).
With Nolan R. Wallach.
Submitted. 48 pages. Available at arXiv:2109.03407.
5. The metric space of limit laws of $q$-hook formulas (2022).
With Sara C. Billey.
To appear in Combinatorial Theory. 59 pages. Available at arXiv:2010.12701.
6. On the distribution of the major index on standard Young tableaux (2020).
With Sara C. Billey and Matjaž Konvalinka.
Talk at FPSAC 2020. Extended abstract published in Sém. Lothar. Combin, 84B (2020), Art. 44, 12 pp. Available at arXiv:2005.10341. pdf
7. Harmonic differential forms for pseudo-reflection groups I. Semi-invariants (2021).
With Nolan R. Wallach.
Published in J. Combin. Theory Ser. A 182 (2021), Paper No. 105474. 30 pages. Available at arXiv:2001.06076. pdf
8. Thrall’s problem: cyclic sieving, necklaces, and branching rules (2019).
With Connor Ahlbach.
Talk at FPSAC 2019. Extended abstract published in Sém. Lothar. Combin, 82B (2019), Art. 37, 12 pp. pdf
9. Existence and hardness of conveyor belts (2020).
With Molly Baird, Sara C. Billey, Erik D. Demaine, Martin L. Demaine, David Eppstein, Sándor Fekete, Graham Gordon, Sean Griffin, Joseph S. B. Mitchell.
Published in Electron. J. Combin. 27 (2020), no 4. Paper 4.25. 21 pages. Available at arXiv:1908.07668. pdf
10. Alternating super-polynomials and super-coinvariants of finite reflection groups (2019).
Superceded by joint work with Nolan Wallach, Harmonic differential forms for pseudo-reflection groups I. Semi-invariants. 18 pages. Available at arXiv:1908.00196.
11. Asymptotic normality of the major index on standard tableaux (2020).
With Sara C. Billey and Matjaž Konvalinka.
Published in Adv. in Appl. Math. 113 (2020), 101972. 28 pages. Available at arXiv:1905.00975.
12. On a Theorem of Baxter and Zeilberger via a Result of Roselle (2022).
Published in Ann. Comb. (2022). 9 pages. Available at arXiv:1902.06724. pdf
13. Tableau posets and the fake degrees of coinvariant algebras (2020).
With Sara C. Billey and Matjaž Konvalinka.
Published in Advances in Mathematics 371 (2020). 45 pages. Available at arXiv:1809.07386.
14. Cyclic sieving, necklaces, and branching rules related to Thrall's problem (2018).
With Connor Ahlbach.
Published in Electron. J. Combin. 25 (2018), no. 4, Paper 4.42. 38 pages. Available at arXiv:1808.06043. pdf
15. Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics (2018).
Ph.D. Thesis at the University of Washington under Sara Billey. ISBN 978-0438-52242-8. 206 pages. pdf
16. Refined cyclic sieving on words for the major index statistic (2018).
With Connor Ahlbach.
Published in European J. Combin. 73 (2018), 37-60. 24 pages. Available at arXiv:1706.08631.
17. On the Existence of Tableaux with Given Modular Major Index (2018).
Published in Algebraic Combinatorics Volume 1 (2018), no. 1 pp. 3-21. 20 pages. Available at arXiv:1701.04963. pdf
18. Refined cyclic sieving (2017).
With Connor Ahlbach.
Poster presentation at FPSAC 2017. Extended abstract published in Sém. Lothar. Combin, 78B (2017), Art. 48, 12 pp. pdf
19. Standard tableaux and modular major index (2017).
Poster presentation at FPSAC 2017. Extended abstract published in Sém. Lothar. Combin, 78B (2017), Art. 50, 9 pp. pdf

## In progress

The following projects are in various stages of completion and should eventually appear publicly.
• Web-like graphs for four row standard tableaux (2022+).
With Christian Gaetz, Oliver Pechenik, Stephan Pfannerer-Mittas, and Jessica Striker.
• $q$-Stirling numbers for complex reflection groups (2022+).
With Bruce Sagan.
• Schur--Weyl duals of free Lie superalgebras (2020+).
• The dihedral sieving phenomenon (2020+).
• Cyclotomic generating functions (2020+).
With Sara Billey.
• Generalized bivariate $\gamma$-positivity (2019+).
With Sheila Sundaram.
• A joint local limit law for inv and maj on permutations (2019+).
• Canonical inclusions and rim hook tableaux (2018+).
• Euler--Mahonian refined cyclic sieving (2017+).
With Connor Ahlbach and Brendon Rhoades.

## Invited talks

• (11/7/2022) Higher coinvariant algebras, q-Stirling numbers, and Coxeter-like complexes. Cornell Discrete Geometry and Combinatorics Seminar. web
• (11/2/2022) TBD. USC Combinatorics Seminar. web
• (10/26/2022) Higher coinvariant algebras, q-Stirling numbers, and Coxeter-like complexes. MIT-Harvard-MSR Combinatorics Seminar. web
• (10/13/2022) "Spinny pictures": connecting Catalan combinatorics, alternating sign matrices, and plabic graphs. NDSU Mathematics Department Colloquium. web
• (6/7/2022) q-Stirling numbers in type B. AlCoVE poster presentation. pdf web
• (3/10/2022) Type B q-Stirling numbers. University of Waterloo Algebraic and Enumerative Combinatorics Seminar. pdf web
• (11/16/2021) DUSTPAN distributions as limit laws for Mahonian statistics on forests. UCSD Combinatorics Seminar. pdf web
• (10/7/2021) Combinatorics of harmonic polynomial differential forms. UCLA Combinatorics Seminar. pdf web
• (9/27/2021) Differential Coinvariant Algebras. USC Algebra Seminar. pdf web
• (3/23/2021) Cyclotomic generating functions. ICERM workshop on Geometry and Combinatorics of Root Systems. pdf web
• (2/27/2021) An Introduction to Pure Mathematics Research. UCSD Graduate Road Map 2021. pdf
• (2/20/2021) DUSTPAN distributions as limit laws for Mahonian statistics on forests. CombinaTexas 2021. pdf
• (1/27/2021) DUSTPAN distributions as limit laws for Mahonian statistics on forests. Michigan State University Combinatorics and Graph Theory Seminar. pdf
• (12/15/2020) DUSTPAN distributions as limit laws for Mahonian statistics on forests. University of Ljubljana Discrete Math Seminar. pdf
• (10/25/2020)* (Canceled due to COVID-19). BIRS Workshop on Dynamical Algebraic Combinatorics, Banff, Alberta, Canada. web
• (7/13/2020) On the distribution of the major index on standard Young tableaux. FPSAC 2020 Online (Contributed). pdf web
• (6/1/2020)* Harmonic differential forms for pseudo-reflection groups. SIAM Conference on Discrete Mathematics (DM20) in Portland, Oregon. web
• (5/2/2020)* Harmonic differential forms for pseudo-reflection groups. AMS Spring Western Sectional Meeting at Cal State Fresno: Special Session on Combinatorics Arising from Representations. web
• (11/3/2019) Alternating super-polynomials and super-coinvariants of finite reflection groups. SIAM TX/LA Section at SMU, Dallas. pdf web
• (7/2/2019) Thrall's problem: cyclic sieving, necklaces, and branching rules. FPSAC 2019, University of Ljubljana, Slovenia (Contributed). pdf web
• (5/4/2019) Asymptotics of Mahonian statistics. Southern California Discrete Math Symposium (Contributed). pdf web
• (3/24/2019) Cyclotomic generating function asymptotics. CombinaTexas 2019 at Texas A-M, College Station (Contributed). pdf web
• (2/13/2019) Cyclotomic generating function asymptotics. USC Combinatorics Seminar. pdf web
• (10/27/2018) Tableaux posets and the fake degrees of coinvariant algebras. AMS Fall Western Sectional Meeting at SFSU: Special Session on Combinatorial and Categorial Aspects of Representation Theory. pdf web
• (5/23/2018) Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics (PhD thesis defense). University of Washington Combinatorics Seminar. pdf web
• (3/26/2018) Inv and Maj Asymptotics. University of Washington Probability Seminar. pdf web
• (1/19/2018) Major Index Asymptotics. University of Michigan Combinatorics Seminar. pdf web
• (1/13/2018) Refined Cyclic Sieving on Words and Tableaux. JMM in San Diego: AMS Special Session on Dynamical Algebraic Combinatorics. pdf web
• (11/4/2017) Major Index Asymptotics. AMS Fall Western Sectional Meeting at UC Riverside: Special Session on Combinatorial Representation Theory. pdf web
• (2/27/2017) On the existence of tableaux with given modular major index. UCSD Combinatorics Seminar. pdf
• (11/4/2016) Refined cyclic sieving on words. University of Minnesota Combinatorics Seminar. pdf web
• (3/11/2015) Schubert multiplication rules and Bruhat chains (Candidacy exam). University of Washington Combinatorics Seminar. pdf web

*Postponed or canceled due to COVID-19

## Informal talks

• (10/16/2020) Asymptotics of Mahonian statistics. UCSD Student Colloquium. pdf
• (11/12/2019) A gentle introduction to coinvariant algebras. UCSD Postdoc Seminar. pdf
• (6/1/2018) Lehrer--Solomon cohomology decomposition. University of Washington Hyperplane Arrangements graduate course. pdf
• (1/17/2018) Introduction to $\lambda$-rings and Plethysms. University of Washington CAT seminar. pdf
• (10/12/2017) A zoo of coinvariant algebras. University of Washington 123 Seminar. pdf
• (10/10/2017) Introduction to Hopf monoids. University of Washington CAT seminar. pdf
• (5/30/2017) The Hilbert Scheme of Points in the Plane is Connected. University of Washington Applied Algebraic Geometry course. pdf
• (5/1/2017) Irreducible decompositions. University of Washington Applied Algebraic Geometry course. pdf
• (4/13/2017) Asymptotic normality. University of Washington CAT seminar. pdf
• (4/5/2017) Monomial orderings. University of Washington Applied Algebraic Geometry course. pdf
• (2/22/2017) Intro to complex reflection groups. University of Washington Reflection Groups graduate course. pdf
• (1/18/2017) Math 308 Wiki Project Discussion. University of Washington informal talk. pdf
• (11/3/2016) Cyclic sieving and Springer's regular elements. University of Minnesota Student-Run Combinatorics Seminar. pdf web
• (10/6/2016) Symmetric group characters as symmetric functions (two talks). University of Washington CAT Seminar. pdf
• (4/7/2016) $n!$ Conjecture Seminar (six talks). University of Washington $n!$ Seminar. pdf
• (2/11/2016) On eigenvalue of representations of reflection groups and wreath products (two talks). University of Washington CAT Seminar. pdf
• (1/12/2016) Three examples of algebraic geometry in algebraic combinatorics. University of Washington 123 Seminar. pdf
• (12/9/2015) The fundamental theorem of Galois theory. University of Washington gradudate Algebra course. pdf
• (11/5/2015) Introduction to cyclic sieving. University of Washington CAT seminar. pdf
• (10/1/2015) Diagonal coinvariants and Tesler matrices. University of Washington Humphreys reading seminar. pdf
• (7/13/2015) The PBW Theorem. University of Washington Humphreys reading seminar. pdf
• (6/7/2014) Duality between Quasi-Symmetric Functions and the Solomon Descent Algebra. University of Washington graduate combinatorics course. pdf
• (5/30/2014) Serre spectral sequence. University of Washington graduate group cohomology course. pdf

## Mathematics notes

I've taken various notes for courses, seminar series, qualifying exams, etc. Some are well-edited, and some are not.

• (2017) The Hilbert scheme of points in the plane is connected, 3 pages. Project for the Applied Algebraic Geometry course at the University of Washington in Spring 2017. pdf mp4
• (2016) Combinatorics and Geometry of Polytopes, 64 pages. Notes for a graduate course on polytopes taught by Isabella Novik at the University of Washington in Spring 2016. Includes material on basics (e.g. equivalence of convex hull and intersection definitions), f and h numbers, duality, face lattices, simple and simplicial polytopes, the upper bound theorem, shellability, simplicial complexes, the Kruska--Katona theorem, the $g$-theorem, the lower bound theorem, rigidity and frameworks, matroid polytopes. The first half has been edited, though the second half has not and likely has typos. pdf zip
• (2016) Introduction to Algebraic Geometry, 101 pages. Notes for a graduate course on algebraic geometry taught by S\'andor Kov\'acs at the University of Washington in Winter and Spring 2016. Includes material on classical algebraic geometry, sheaves, rational maps, blow ups, singularities, normality, Serre's conditions, finite morphisms, constructable sets, Zariski's main theorem, ringed spaces, coherent sheaves, schemes, fibers, dimension, divisors, invertible sheaves, differentials. The first portion has been edited, though later lectures have not been and likely have typos. pdf zip
• (2016) Background for the $n!$ theorem, 24 pages. Lecture notes for an informal graduate seminar on Haiman's 2004 proof of the $n!$ conjecture. Discusses classical symmetric function theory, Kostka--Foulkes polynomials, Springer fibers, Hall--Littlewood symmetric functions, Macdonald symmetric functions, Garsia--Haiman modules, $k$-Schur functions, diagonal coinvariants, and sketches some of the broad outlines of Haiman's proof. pdf
• (2015) Algebraic Number Theory, 62 pages. Notes for a graduate course on algebraic number theory taught by Bianca Viray at the University of Washington in Fall 2015. Includes material on rings of integers, Dedekind domains, fractional ideals, ramification, decomposition and inertia groups, cyclotomic fields, Gauss' reciprocity law, class groups, Minkowski's theorem, lattices, Dirichlet's unit group theorem, p-adics, the archimidean property, Ostrowski's theorems, valuations, Hensel's lemma, local fields, Newton polygons, ramification group, topological and profinite groups, induction, group cohomology, inflation and restriction. It is largely unedited. pdf zip
• (2015) Humphreys section 23.3, 7 pages. An expanded account of the part of Humphreys' "Introduction to Lie Algebras and Representation Theory" proving Harish--Chandra's theorem. pdf
• (2014) Algebraic Combinatorics, 93 pages. Notes for a graduate course on algebraic combinatorics taught by Sara Billey at the University of Washington in Spring 2014. Includes material on symmetric group representations, the Pieri rule, symmetric functions, coalgebras, Sweedler notation, bialgebras, Hopf algebras, antipodes, homological properties of Hopf algebras, duality, and student lecturs on various papers. pdf zip
• (2014) Group Cohomology, 61 pages. Notes for a graduate course on group cohomology taught by Julia Pevtsova at the University of Washington in Spring 2014. Includes material on derived functors, associated graded objects, spectral sequences, double complexes, the universal coefficient theorem, Cartan--Eilenberg resolutions, hyper-derived functors, Grothendieck spectral sequence, Lyndon--Hochschild--Serre spectral sequence, group cohomology rings, restriction, induction, corestriction, Frobenius reciprocity, G-invariants, Leray--Serre spectral sequence, the Gysin sequence, cohomology of homogeneous spaces, Eilenberg--MacLane spaces, finite generation, \v{C}ech and sheaf cohomology. pdf zip
• (2014) Advanced Commutative Algebra, 53 pages. Notes for a graduate course on advanced commutative algebra taught by S. Paul Smith at the University of Washington in Fall 2014. Includes material on tensor-hom adjunction, flatness, support, primes, duality conditions, injectives, local cohomology, Matlis duality, injective resolutions, depth, projective dimension, Auslander--Buchsbaum formula, Rees' theorem, regular sequences, Krull dimension, principal ideal theorems, Krull's intersection theorem, Hilbert series, Gelfand--Kirillov dimension, associated graded rings, regular local rings, minimal resolutions, global dimension, Cohen--Macaulay rings. pdf zip
• (2014) Algebraic Groups, 55 pages. Notes for a graduate course on algebraic groups taught by Julia Pevtsova at the University of Washington in Fall 2014. Includes material on affine group schemes, Hopf algebras, Weil restriction, \'etale algebras, Cartier duality, comodules, representations, characters, induction, connected components, K\"ahler differentials, Lie algebras, algebraic groups. pdf zip
• (2014) Lie Groups and Representation Theory, 71 pages. Notes for a graduate course on compact Lie groups and their representation theory taught by Robin Graham at the University of Washington in Fall 2014. Includes material on Lie algebras, semisimplicity, Cartan's criterion, real forms, complexification, reductive Lie algebras, signature, quaternions, classifications, classical Lie groups, Cartan decomposition, Haar measure, representations, intertwining operators, the exponential map, complete reducibility, spherical harmonics, the Peter--Weyl theorem. Not well edited--beware of typos. pdf zip
• (2013) Complex prelim notes, 16 pages. A summary of the key results used in the University of Washington Complex Analysis graduate preliminary exam as of 2013. Includes material on basics (e.g. Schwarz' lemma, open mapping theorem), integral formulas (Cauchy's, Schwarz', Jensen's), analytic extensions, root finding, uniform approximations, normal families, harmonic and subharmonic functions, inequalities, series, products, analytic continuation, residues, and conformal maps. pdf
• (2013) Spring 2013 algebra notes, 16 pages. A summary of a graduate course at the University of Washington taught by S. Paul Smith. Topics included Noether normalization, a classical introduction to varieties, Ext, chain complexes, adjoint functors, homological dimensions, tensor products, exactness, Tor, Dedekind domains. pdf

## Courses Taught

• (2022 FA) Math 407: Probability Theory, USC. On Blackboard.
• (2022 SP) Math 126g: Calculus II, USC. On Blackboard.
• (2021 FA) Math 225: Linear Algebra and Differential Equations, USC. On Blackboard.
• (2021 WI) Math 11: Introductory Probability and Statistics, UCSD. Course site.
• (2021 WI) Math 20C: Calculus III, UCSD. Course site.
• (2020 FA) Math 20C: Calculus III, UCSD. Course site.
• (2020 FA) Math 109: Mathematical Reasoning, UCSD. Course site.
• (2020 SP) Math 184: Enumerative Combinatorics, UCSD. Course site.
• (2020 WI) Math 15A: Introduction to Discrete Mathematics, UCSD. Course site.
• (2020 WI) Math 109: Mathematical Reasoning, UCSD. Course site.
• (2019 FA) Math 20C: Calculus III, UCSD. Course site.
• (2019 SP) Math 184A: Combinatorics, UCSD. Course site.
• (2019 SP) Math 11: Introductory Probability and Statistics, UCSD. Course site.
• (2019 WI) Math 20C: Calculus III, UCSD. Course site.
• (2018 FA) Math 109: Mathematical Reasoning, UCSD. On TritonEd.
• (2018 SP) Math 308: Matrix Algebra with Applications, University of Washington. Course site.
• (2017 SU) Math 308: Matrix Algebra with Applications, University of Washington. Course site.
• (2017 SP) Math 307: Introduction to Differential Equations, University of Washington. Course site.
• (2016 FA) Math 308: Matrix Algebra with Applications, University of Washington. Course site.
• (2015 SU) Math 307: Introduction to Differential Equations, University of Washington. Course site.
• (2015 SP) Math 308: Matrix Algebra with Applications, University of Washington. Course site.
• (2013 SU) Math 126: Calculus with Analytic Geometry III, University of Washington. Course site.

## Courses TA'd

• (2018 WI) Math 126: Calculus with Analytic Geometry III, University of Washington.
• (2017 FA) Math 126: Calculus with Analytic Geometry III, University of Washington.
• (2016 SP) Math 506: Graduate Algebra, University of Washington.
• (2016 WI) Math 505: Graduate Algebra, University of Washington.
• (2015 FA) Math 504: Graduate Algebra, University of Washington.
• (2015 WI) Math 125: Calculus with Analytic Geometry II, University of Washington.
• (2014 AU) Math 124: Calculus with Analytic Geometry I, University of Washington. Course site.
• (2014 SP) Math 126: Calculus with Analytic Geometry III, University of Washington. TA site.
• (2014 WI) Math 125: Calculus with Analytic Geometry II, University of Washington.
• (2013 AU) Math 126: Calculus with Analytic Geometry III, University of Washington. TA site.
• (2013 SP) Math 126: Calculus with Analytic Geometry III, University of Washington.
• (2013 WI) Math 126: Calculus with Analytic Geometry III, University of Washington.
• (2012 AU) Math 125: Calculus with Analytic Geometry II, University of Washington.