# Anand Deopurkar

Mathematical Sciences Institute

The Australian National University

Canberra, Australia

`anand.deopurkar`

at `anu`

dot `edu`

dot `au`

(pgp)

4.56 Hanna Neumann Building

+61 2 6125 4628

## Table of Contents

I am a mathematician working in algebraic geometry and related areas. I study the interplay between systems of algebraic equations and the geometric shapes they describe.

This what I have been up to lately.

- With Jayan Mukherjee, I am trying to understand syzygies of some non-reduced algebraic curves.
- With Valery Alexeev and Changho Han, I am working on describing compact moduli spaces of some classes of K3 surfaces.
- With Asilata Bapat and Tony Licata, I am studying Bridgeland stability conditions on some triangulated categories.
- With Rachel Newton, Vaidehee Thatte, and Rosa Winters, I am exploring arithmetic ramifications of the geometry of certain Hurwitz spaces.
- I am teaching a fun course called Games, graphs, and machines, in which we talk about graphs, Markov chains, automata, regular expressions, and games like Nim and chomp.
- I am also teaching a reading course based on Bott and Tu’s famous book Differential forms in algebraic topology.
- I am supervising Charlie Sherrat, who has been exploring the moduli space of cubic surfaces arising from 3x3x4 rubik’s cubes.
- I am supervising Luke Doherty, who is looking into the geometry of linkages.
- With John Cobb, Sione Ma’u, and Hal Schenk, I am organising a special session at the joint NZMS/AustMS/AMS meeting in December 2024.

## Papers/pre-prints

- Equivariant classes of orbits in GL(2)-representations. (pdf, arxiv, source/comments)

*Pre-print*. - Orbits of linear series on the projective line. (pdf, arxiv, journal)

*International Mathematics Research Notices*, with Anand Patel. - Spherical objects and stability conditions on CY2 quiver categories. (arxiv)

*Mathematische Zeitschrift*, with Asilata Bapat, Anthony Licata. - Vector bundles and finite covers. (pdf, arXiv)

*Forum of Mathematics, Sigma*, with Anand Patel.

Talks: Adelaide (slides), Jeju Island (notes). - A universal formula for counting cubic surfaces. (arxiv)

*Pre-print*, with Anand Patel, Dennis Tseng.

Talk: Cambridge (notes). - Stable log surfaces, admissible covers, and canonical curves of genus 4. (pdf, arxiv)

*Transactions of the Americal Mathematical Society*, with Changho Han.

Talk: Sydney (notes). - A Thurston compactification of the space of stability conditions. (arxiv)

*Pre-print*, with Asilata Bapat, Anthony Licata.

Talks: Bonn (slides), Mumbai (slides), Amidale (slides), Sydney (notes). - Ramification divisors of general projections. (pdf, arxiv)

*Documenta Mathematica*, with Eduard Duryev, Anand Patel.

Talks: Mexico (slides), San Diego (slides). - Anticanonical tropical cubic del Pezzos contain exactly 27 lines. (arxiv)

*Pre-print*, with María Angélica Cueto. - Covers of stacky curves and limits of plane quintics. (pdf, arxiv)

*Transactions of the Americal Mathematical Society*.

Talks: Boston (notes), Salt Lake City (poster). - Syzygy divisors on Hurwitz spaces. (pdf, arxiv)

*Contemporary Mathematics*, with Anand Patel. - The canonical syzygy conjecture for ribbons. (pdf, arxiv)

*Mathematische Zeitschrift*.

Talk: Melbourne (notes). - Toward GIT stability of syzygies of canonical curves. (arxiv, journal)

*Algebraic Geometry*, with Maksym Fedorchuk, David Swinarski.

Talks: Daejeon (slides), Pohang (notes). - The Picard rank conjecture for the Hurwitz spaces of degree up to five. (pdf, arxiv, journal)

*Algebra & Number Theory*, with Anand Patel. - Groebner techniques for ribbons. (pdf, journal)

*Albanian Journal of Mathematics*, with Maksym Fedorchuk, David Swinarski. - Compactifications of Hurwitz spaces. (pdf, arxiv, journal)

*International Mathematics Research Notices*.

Talks: Palo Alto (notes), Cambridge (notes). - Class of the Hodge eigenbundle using orbifold Riemann-Roch. (pdf)

*Pre-print, appendix to*.*Cyclic covering morphisms on \(\overline M_{0,n}\)*by Maksym Fedorchuk - Sharp slope bounds for sweeping families of trigonal curves. (pdf, arxiv, journal)

*Mathematical Research Letters*, with Anand Patel.

Talk: Boston (poster). - Modular compactifications of the space of marked trigonal curves. (pdf, arxiv)

*Advances in Mathematics*.

Talk: Cambridge (poster). - Alternate compactifications of Hurwitz spaces. (pdf)

*Thesis, Harvard, 2012*.

Talks: Palo Alto (notes), Cambridge (notes).

## Teaching

Notes and other materials from some of my courses:

- Games, graphs, and machines, Australian National University, 2024.
- The language of mathematics, Australian National University, 2022.
- Algebraic Geometry (Algebra 3), Australian National University, 2021.
- Foundations of Algebraic Geometry (Schemes), Australian National University, 2020.
- Algebraic Geometry (Algebra 3), Australian National University, 2019.
- Algebraic curves, University of Georgia, 2017.
- Moduli of curves, Columbia University, 2014.

## Talks

Notes or slides for many of my talks are linked in the section on papers/pre-prints. Here are some additional talks:

- The geometry and combinatorics of Harder–Narasimhan filtrations. (slides)

Braids in representation theory and algebraic combinatorics, Institute for Computational and Experimental Research in Mathematics, Providence, Rhode Island. - The geometry of Fermat-like equations (expository). (notes)

Trimester program on triangle groups, Belyi uniformization, and modularity, Bhaskaracharya Pratishthana, Pune, India. - Algebraic curves and Belyi’s theorem (expository). (notes)

Trimester program on triangle groups, Belyi uniformization, and modularity, Bhaskaracharya Pratishthana, Pune, India. - Error correcting codes (expository). (slides)

ANU Mathematics Extension Program, Canberra, Australia. - The work of Claire Voisin (expository). (slides)

Women in mathematics day, Australian National University, Canberra, Australia. - Geometry of Hurwitz spaces. (notes)

Character varieties and topological quantum field theory, University of Auckland, Auckland, New Zealand. - The work of Caucher Birkar (expository). (notes)

Colloquium, Australian National University, Canberra, Australia. - How to count using (co)homology (expository). (notes)

Tata Institute of Fundamental Research, Mumbai, India. - Quadrature and algebraic geometry. (slides)

Workshop on algebraic geometry approximation, and optimization, MATRIX, Creswick, Victoria, Australia.

## Odds and ends

- A sketch of the Thurston compactification of the stability manifold for a generic non-algebraic K3 surface This is an instance of the construction proposed in the paper with Bapat and Licata.
- GIT for syzygies of genus 7 curves

This is an ongoing project to determine GIT stability for syzygies of genus 7 canonical curves. This is the first genus where canonical syzygies define an interesting GIT quotient, so it would be nice to understand the complete picture. One day… - Categorical braid group actions

With Asilata Bapat, I wrote`Sage`

code to make explicit computations in the homotopy category of projective modules over a ring. In particular, this code can compute braid actions on complexes of projective modules over the zig-zag algebra of a quiver. - MSI course graph

This is an interactive dependency graph of ANU math department’s course catalogue. - A mystery hunt style puzzle
- Some visualisations
- Notes from some past seminars
- DbCoh: ANU, 2021
- Mixed hodge modules: UGA, 2017
- Stable rationality: Columbia, 2016

- I am an advocate of free software and a user of emacs. I occassionally contribute to the development of org mode.