# Josh Swanson

Mathematics Home Page

## Contact

arXiv, MathSciNet, Google Scholar, ORCID

## About me

I'm a postdoc 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 advisor was Sara Billey. My thesis was on "Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics".

My research is in Algebraic Combinatorics. This field of mathematics 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). Much of my own work is related to Young tableau and the surrounding combinatorics and representation theory, especially major index statistics. I am also interested in exploiting combinatorial generating functions for enumerative and asymptotic purposes.

Research interests: Coxeter groups, representation theory of reflection groups, symmetric functions, tableaux combinatorics, invariant theory of reflection groups, coinvariant algebras, diagonal coinvariants, cyclic sieving, major index statistics, descent statistics, generating function factorizations, plethysms, Thrall's problem, induced characters, cumulants, limit laws

## Publications

1. Tableaux posets and the fake degrees of coinvariant algebras (2018+).
With Sara C. Billey and Matjaž Konvalinka. Submitted to Advances in Mathematics. Available at arXiv:1809.07386.
2. Cyclic sieving, necklaces, and branching rules related to Thrall's problem (2018+).
With Connor Ahlbach. To appear in Electronic Journal of Combinatorics. Available at arXiv:1808.06043.
3. Refined cyclic sieving on words for the major index statistic (2018).
With Connor Ahlbach. European J. Combin. 73 (2018), 37-60. Available at arXiv:1706.08631.
4. On the Existence of Tableaux with Given Modular Major Index (2018).
Algebraic Combinatorics Volume 1 (2018), no. 1 pp. 3-21. Available at arXiv:1701.04963.
5. Refined cyclic sieving (2017).
With Connor Ahlbach. Poster presentation at FPSAC 2017, extended abstract in Sém. Lothar. Combin, 78B (2017), Art. 48, 12 pp. pdf
6. Standard tableaux and modular major index (2017).
Poster presentation at FPSAC 2017, extended abstract in Sém. Lothar. Combin, 78B (2017), Art. 50, 9 pp. pdf

## In progress

• On a theorem of Baxter and Zeilberger via a result of Roselle (2018+).
• Canonical inclusions and rim hook tableaux (2018+).
• Asymptotic normality of the major index on tableaux and limit laws for cyclotomic generating functions (2018+).
With Sara C. Billey and Matjaž Konvalinka.
• New results on conveyor belts (2018+).
With Molly Baird, Sara C. Billey, Erik D. Demaine, Martin L. Demaine, David Eppstein, Sándor Fekete, Graham Gordon, Sean Griffin, Joseph S. B. Mitchell.
• Euler--Mahonian refined cyclic sieving (2017+).
With Connor Ahlbach and Brendon Rhoades.

## Invited talks

• (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

## Informal talks

• (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

## Notes

• (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

• (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.
• (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.