Papers
- ACC for F-signature: a likely counterexample (with C. Adams, A.
Simpson). Exp. Math., 2025.
[pdf]
[arXiv]
Let \( k = \overline{\mathbb{F}_2} \) and let \( 0 \neq \alpha \in k \). We present a conjecture supported by computer experimentation involving the Brenner–Monsky quartic \( g_\alpha = \alpha x^2 y^2 + z^4 + x y z^2 + (x^3 + y^3) z \in k[[x, y, z]] \). If true, this conjecture provides a formula for the Hilbert–Kunz multiplicity and \( F \)-signature of the family of four-dimensional hypersurfaces defined by \( u v + g_\alpha \in k[[x, y, z, u, v]] \) which depends on \( [\mathbb{F}_2(\alpha) : \mathbb{F}_2] \), giving an infinite increasing chain of strict inequalities of \( F \)-signatures. Additionally, we obtain for any \( t \in \mathbb{N} \) a formula for the Hilbert–Kunz multiplicity and \( F \)-signature of the \( t \)-parameter family of \(3 t + 1 \)-dimensional hypersurfaces defined by \( u v + \sum_{i=1}^t g_{\alpha_i}(x_i, y_i, z_i) \).
- On localization of tight closure in line-S4 quartics (with L. Borevitz,
N. Nader, A. Shapiro, A. Simpson, J. Zomback). J. Pure Appl. Algebra, 2024.
[pdf]
[arXiv]
Building on work of Brenner and Monsky from 2010 and on a Hilbert–Kunz calculation of Monsky from 1998, we exhibit a novel example of a hypersurface over \( \overline{\mathbb{F}_2} \) in which tight closure does not commute with localization. Our methods involve a tiling argument using Sierpiński triangles, as well as an inspection of a certain dynamical system in characteristic two.
- Realizing pairs of multicurves as cylinders on translation surfaces
(with J. Aygun, J. Barkdoll, A. Calderon, J. Lorman), Algebr. Geom. Topol., 2025.
[pdf]
[arXiv]
Any pair of intersecting cylinders on a translation surface is “coherent,” in that the geometric and algebraic intersection numbers of their core curves are equal (up to sign). In this paper, we investigate when a pair of multicurves can be simultaneously realized as the core curves of cylinders on some translation surface. Our main tools are surface topology and the “flat grafting” deformation introduced by Ser-Wei Fu.
- Virtual multicrossings and petal number for virtual knots and links
(with C. Adams, C. Even-Zohar, J. Greenberg, R. Kaufman, D. Lee, D. Li, D. Ping, X.
Wang). J. Knot Theory Ramifications, 2023.
[pdf]
[arXiv]
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.