Photo of me Joshua Wang

Princeton University

Office: Fine Hall 407
Email: joshuaxw@princeton.edu


Institute for Advanced Studies

Office: MOS 112
Email: jxwang@ias.edu


Curriculum Vitae

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.



Research

  1. Free loop spaces and link homology

    in preparation

  2. A minimality property for knots without Khovanov 2-torsion

    with Onkar Singh Gujral

    Algebraic and Geometric Topology, accepted | arXiv

  3. The Gysin sequence and the sl(N) homology of T(2,m)

    Proceedings of Symposia in Pure Mathematics 109: 233-251 (2024) | arXiv

  4. Colored sl(N) homology and SU(N) representations I: the trefoil and the Hopf link

    preprint | arXiv

  5. Split link detection for sl(P) link homology in characteristic P

    Journal of Topology 16 (2): 806-821 (2023) | arXiv

  6. On sl(N) link homology with mod N coefficients

    Quantum Topology 15 (1): 87-121 (2024) | arXiv

  7. Link Floer homology also detects split links

    Bulletin of the London Mathematical Society 53 (4): 1037-1044 (2021) | arXiv

  8. The cosmetic crossing conjecture for split links

    Geometry & Topology 26 (7): 2941-3053 (2022) | arXiv

  9. A combinatorial proof of invariance of double-point enhanced grid homology

    with Timothy Ratigan and Luya Wang

    preprint | arXiv



Teaching and Mentoring

Courses at Princeton

Fall 2024 - MAT 216: Multivariable Analysis and Linear Algebra I


Courses at MIT

Spring 2024 - 18.099: Independent Study in Mathematics (Low-dimensional topology)

Fall 2023 - 18.099: Independent Study in Mathematics (The geometry of complex analysis)


Courses at Harvard

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


Directed Reading Programs (DRP)

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.


Research mentoring

I'm mentoring a high school research project through MIT PRIMES during 2024.



Exposition

Minor thesis (Ph.D. requirement) - Hodge theory for matroids



Personal

Some photos of knots and links made from pretzels, baked at a birthday get-together.