Carl Lian


I am a fifth-year graduate student in mathematics at Columbia University, advised by Aise Johan de Jong. I was previously supported by an NSF Graduate Research Fellowship. Before that, I graduated from MIT in June 2015 with an S.B. in mathematics and minor in music.


I'm broadly interested in algebraic geometry. Some specific things I like thinking about are:

  • Moduli spaces of curves and branched covers (Hurwitz spaces), and their intersection theory
  • Brill-Noether theory, limit linear series, and related enumerative problems
  • Combinatorial aspects of the above, e.g. Schubert calculus
  • Syzygies of curves
  • Interpolation of curves in higher-dimensional varieties
  • Linear systems of singular divisors, e.g. the SHGH conjecture and related problems

    Here are my papers:

  • Enumerating pencils with moving ramification on curves, arXiv 1907.09087.
  • The class of the d-elliptic Locus in genus 2, arXiv 1907.08991. Here is a video of my talk at ICTP on this work. Update (December 2019): I am in the process of completely rewriting this paper to include new results in higher genus and for pointed curves; please e-mail me if you'd like a detailed status update.
  • (with Joshua Alman and Brandon Tran) Circular Planar Electrical Networks: Posets and Positivity. J. Combinatorial Theory, Ser. A 132, 58-101 (2015).
  • Representations of Cherednik Algebras Associated to Symmetric and Dihedral Groups in Positive Characteristic, arXiv 1207.0182.

    Other Writing

  • (with Raymond Cheng, Yordanka Kovacheva, Monica Marinescu, and Takumi Murayama) Projectivity of moduli of curves, written at the inaugural Stacks Project Workshop in August 2017 with the guidance of Davesh Maulik.

    Old Links (past teaching, seminars organized, etc.)