Henry Liu

Research

Very broadly, my research interests lie at the intersection of algebraic geometry, representation theory, and mathematical physics. I like to study various moduli of sheaves, especially on surfaces or 3-folds (or 4-folds, on adventurous days), and their enumerative invariants and (quantum) symmetries.

Lately I've been thinking about vertex algebras.

Links to: my arXiv, my ORCID.

Papers and preprints

1. Wall-crossing for invariants of equivariant CY3 categories
   with Nick Kuhn and Felix Thimm
   In progress (2024)
2. Invariance of elliptic genus under wall-crossing
   Submitted (2024) [arXiv:2405.12587]
3. The 3-fold K-theoretic DT/PT vertex correspondence holds
   with Nick Kuhn and Felix Thimm
   Submitted (2024) [arXiv:2311.15697] [slides]
4. Semistable refined Vafa-Witten invariants
   Submitted (2023) [arXiv:2309.03673]
5. The 4-fold Pandharipande-Thomas vertex
   J. Geom. Phys. 198 (2024), 105104 [arXiv:2306.12923] [Journal]
6. Multiplicative vertex algebras and quantum loop algebras
   Submitted (2022) [arXiv:2210.04773]
7. Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories
   Preprint (2023) [arXiv:2207.13546]
   This paper is currently undergoing a mild revision/elaboration.
8. A representation-theoretic approach to qq-characters
   SIGMA 18 (2022), 090 [arXiv:2203.07072] [Journal]
9. Asymptotic representations of shifted quantum affine algebras from critical K-theory
   PhD. Thesis (2021), Columbia University [Academic Commons]
10. Quasimaps and stable pairs
    Forum Math. Sigma 9 (2021), e32 [arXiv:2006.14695] [Journal]
11. Self-duality in quantum K-theory
    Preprint (2019) [arXiv:1906.10824]
    I intend to revisit this project at some point in the future.
12. No Laplacian Perfect State Transfer in Trees
    with Gabriel Coutinho
    SIAM J. Discrete Math., 29(4) (2015), 2179-2188 [arXiv:1408.2935] [Journal]
13. Cardinality estimation using neural networks
    with Mingbin Xu, Ziting Yu, Vincent Corvinelli and Calisto Zuzarte
    CASCON 2015, 53-59 [Journal]
    A project from a past life.

Software

I have written and am maintaining some code (GitHub repository) for box-counting computations in SageMath. Here are some of the more useful objects that are implemented:

• 3d partitions (with legs)
• fully equivariant 3-leg DT and PT vertices in cohomology and K-theory
  (checked to be equal, after normalization, up to legs of size 4 and 6 extra boxes)
• the elliptic Hall algebra \(U_{q,t}(\hat{\hat{\mathfrak{gl}}}_1)\) and its Fock representation on symmetric polynomials. Here is an exposition of the algorithm for performing such computations, using nothing more than a generators-and-relations description.
• (new) The K-theoretic stable basis (at all slopes) for the Hilbert scheme.

If you are also counting boxes and would like a new feature implemented, feel free to let me know.