Office: Fine Hall 407
Email: joshuaxw@princeton.edu
I'm a Veblen Research Instructor at Princeton and the IAS. I'm also an NSF Postdoctoral Fellow, and I'm affliated with a Simons Collaboration.
Previously, I was an NSF and Simons postdoc at MIT after completing my PhD in mathematics at Harvard, advised by Peter Kronheimer. I was an undergraduate at Princeton before that, and I'm very happy to be back!
My main areas of research are in low-dimensional topology: Floer theory and categorification. My training was in applications of gauge theory to low-dimensional topology, while my more recent interests touch representation theory, algebraic combinatorics, and other parts of geometry and topology.
Free loop spaces and link homology
in preparation
A minimality property for knots without Khovanov 2-torsion
with Onkar Singh Gujral
Algebraic and Geometric Topology, under revision | arXiv
The Gysin sequence and the sl(N) homology of T(2,m)
Proceedings of Symposia in Pure Mathematics 109: 233-251 (2024) | arXiv
Colored sl(N) homology and SU(N) representations I: the trefoil and the Hopf link
preprint | arXiv
Split link detection for sl(P) link homology in characteristic P
On sl(N) link homology with mod N coefficients
Link Floer homology also detects split links
Bulletin of the London Mathematical Society 53 (4): 1037-1044 (2021) | arXiv
The cosmetic crossing conjecture for split links
A combinatorial proof of invariance of double-point enhanced grid homology
with Timothy Ratigan and Luya Wang
preprint | arXiv
Fall 2024 - MAT 216: Multivariable Analysis and Linear Algebra I
Spring 2024 - 18.099: Independent Study in Mathematics (Low-dimensional topology)
Fall 2023 - 18.099: Independent Study in Mathematics (The geometry of complex analysis)
Fall 2021 - Math 21a: Multivariable calculus
Spring 2021 - Tutorial: Low-dimensional manifolds
Summer 2020 - Tutorial: Differential forms in algebraic topology
Summer 2019 - Tutorial: Knot invariants and category theory with Morgan Opie
I co-organized the MIT DRP during 2023-2024, and I mentored two reading projects.
I started and co-organized the Harvard DRP during 2018-2023, and I mentored five reading projects.
I'm mentoring a high school research project through MIT PRIMES during 2024.
Minor thesis (Ph.D. requirement) - Hodge theory for matroids
Some photos of knots and links made from pretzels, baked at a birthday get-together.