__Carl Lian__

I am a final year graduate student in mathematics at Columbia University, advised by Aise Johan de Jong. I was previously an undergraduate at MIT.

A draft (version of 18 February 2020) of my thesis, entitled "On the enumerative geometry of branched covers of curves," is here. The content is essentially that of [3] and [4] below.

Starting Fall 2020, I will be an NSF Postdoctoral Fellow at Humboldt-Universität zu Berlin working with Gavril Farkas.

**Research**

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:

[4] Enumerating pencils with moving ramification on curves, arXiv 1907.09087.
[3] The class of the d-elliptic Locus in genus 2, arXiv 1907.08991. **Note (February 2020): This version is out-of-date, and a completely rewritten account with improved results can be found in Chapter 3 of my thesis, posted above. I plan to update the preprint after my thesis is finalized.**
[2] (with Joshua Alman and Brandon Tran) Circular Planar Electrical Networks: Posets and Positivity. J. Combinatorial Theory, Ser. A 132, 58-101 (2015).
[1] 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.)