Carl Lian

Papers

I work in algebraic geometry, with an emphasis on moduli spaces of curves and related objects, e.g. branched covers of curves (Hurwitz spaces), [limit] linear series on curves (Brill-Noether theory), stable maps (Gromov-Witten theory), and abelian varieties. I'm especially interested in the intersection theory (enumerative geometry) of such moduli spaces and various combinatorial problems that emerge in related calculations.

Research articles (listed in reverse order of publication, not necessarily in order of writing):

• (with Naufil Sakran) Enumerating log rational curves on some toric varieties (arXiv), submitted

• (with Alessio Cela) Complete quasimaps to Bl_{P^s}(P^r) (arXiv), submitted

• (with Alessio Cela) A Brill-Noether Theorem for (toric) surfaces (arXiv), submitted

• (with Alessio Cela) Fixed-domain curve counts for blow-ups of projective space (arXiv), submitted

• (with Alessio Cela) Curves on Hirzebruch surfaces and semi-stability (arXiv), to appear in Michigan Math. J.

• (with François Greer) d-elliptic loci and the Torelli map (arXiv), to appear in Math. Res. Lett.

• (with François Greer, appendix by Naomi Sweeting) Modularity of d-elliptic loci with level structure (arXiv, journal), J. Lond. Math. Soc. 112 (2025), e70212.

• (with Rahul Pandharipande) Enumerativity of virtual Tevelev degrees (arXiv, journal), Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 (2025), 71-89.

• Asymptotic geometric Tevelev degrees of hypersurfaces (arXiv, journal), Michigan Math. J. 75 (2025), 119-139.

• Degenerations of complete collineations and geometric Tevelev degrees of P^r (arXiv, journal, video), J. Reine Angew. Math. 817 (2024), 153–212.

• (with Roya Beheshti, Brian Lehmann, Eric Riedl, Jason Starr, and Sho Tanimoto) On the asymptotic enumerativity property for Fano manifolds (arXiv, journal, video), Forum Math. Sigma 12 (2024), e112.

• The HHMP decomposition of the permutohedron and degenerations of torus orbits on flag varieties (arXiv, journal), Int. Math. Res. Not. IMRN (2024), no. 20, 13380-13399.

• Torus orbit closures and 1-strip-less tableaux (arXiv, journal), Algebr. Comb. 7 (2024), 1103-1121.

• (with Gavril Farkas) Linear series on general curves with prescribed incidence conditions (arXiv, journal), J. Inst. Math. Jussieu 22 (2023), 2857-2877.

• (with Alessio Cela) Generalized Tevelev degrees of P^1 (arXiv, journal), J. Pure Appl. Algebra 227 (2023), 107324.

• (with Riccardo Moschetti) Z/rZ-equivariant covers of P^1 with moving ramification (arXiv, journal), Israel J. Math. 253 (2023), 487-500.

• Enumerating pencils with moving ramification on curves (arXiv, journal), J. Algebraic Geom. 32 (2023), 143-182.

• Non-tautological Hurwitz cycles (arXiv, journal), Math. Z. 301 (2022), 173-198.

• d-elliptic loci in genus 2 and 3  (arXiv, journal), Int. Math. Res. Not. IMRN (2021), no. 20, 15959-16007. The published version of this paper contains some errors in some formulas stemming from a mistake in Proposition 5.1, see this erratum. The errors have been fixed in the version linked here and in the arXiv version.

• The H-tautological ring (arXiv, journal), Selecta Math. (N.S.) 27 (2021), Article Nr. 96.

• (with Joshua Alman and Brandon Tran) Circular Planar Electrical Networks: Posets and Positivity (arXiv1, arXiv2, journal). J. Comb. Theory Ser. A 132 (2015), 58-101.

Other writing:

• Counting linear series on curves: old and new results, Oberwolfach report for Classical Algebraic Geometry, June 2021

• (with Raymond Cheng and Takumi Murayama, in collaboration with Yordanka Kovacheva and Monica Marinescu, mentored by Davesh Maulik) Projectivity of the Moduli of Curves, exposition of Kollár's Projectivity of complete moduli, initiated at the Stacks Project Workshop in Ann Arbor, August 2017

Teaching

Previous courses:

• Combinatorics Spring 2025 (Tufts)

• Algebraic Geometry Spring 2024 (Tufts)

• Abstract Algebra 1 Fall 2024,  Fall 2023 (Tufts)

• Linear Algebra  Spring 2024 (Tufts), Summer 2020 (Columbia)

• Calculus 1 Fall 2023 (Tufts), Summer 2019, Fall 2018 (Columbia)

Other programs I have taught in: 

• Virtual Bootcamp (SRI at Colorado State, Summer 2025)

• VERSEIM-REU (Tufts, Summer 2024)

• Summer School on Algebraic Curves (SLMath/MSRI, Summer 2024)

• CMUSR (Columbia, Summer 2016) 

• MathROOTS (MIT, Summer 2015)

• PRIMES-USA (MIT, 2015)

• PRIMES Circle (MIT, 2013)