Anand Deopurkar

• 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.
• Anand Patel and I threw every tool we knew at a cool enumerative problem (counting 3-uple Veronese surfaces) whose answer comes out to \(4246\). We would love a numerical verification, so if you know something about bertini or homotopy continuation, please write to us.
• 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.
• This semester, I am teaching Algebra 2 (Fields and Galois theory).

If you are thinking of doing a PhD with me, please read this page for prospective PhD students.

• The Thurston compactification of the stability manifold of a generic analytic K3 surface. (repository)
  Pre-print.
• Compactifications of moduli spaces of K3 surfaces with a higher-order nonsymplectic automorphism. (arxiv)
  Pre-print, with Valery Alexeev, Changho Han.
• Syzygies of canonical ribbons on higher genus curves. (arxiv)
  Pre-print, with Jayan Mukherjee.
• Counting 3-uple Veronese surfaces. (arxiv)
  Pre-print, with Anand Patel.
• Equivariant classes of orbits in GL(2)-representations. (arxiv, repository)
  Pre-print.
• Orbits of linear series on the projective line. (arxiv, journal)
  International Mathematics Research Notices, with Anand Patel.
• A universal formula for counting cubic surfaces. (arxiv)
  Algebraic Geometry (to appear), with Anand Patel, Dennis Tseng.
• Spherical objects and stability conditions on 2-Calabi–Yau quiver categories. (arxiv, journal)
  Mathematische Zeitschrift, with Asilata Bapat, Anthony Licata.
• Vector bundles and finite covers. (arXiv, journal)
  Forum of Mathematics, Sigma, with Anand Patel.
• Stable log surfaces, admissible covers, and canonical curves of genus 4. (arxiv, journal)
  Transactions of the Americal Mathematical Society, with Changho Han.
• A Thurston compactification of the space of stability conditions. (arxiv)
  Pre-print, with Asilata Bapat, Anthony Licata.
• Ramification divisors of general projections. (arxiv, journal)
  Documenta Mathematica, with Eduard Duryev, Anand Patel.
• 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. (arxiv, journal)
  Transactions of the Americal Mathematical Society.
• Syzygy divisors on Hurwitz spaces. (arxiv, journal)
  Contemporary Mathematics, with Anand Patel.
• The canonical syzygy conjecture for ribbons. (arxiv)
  Mathematische Zeitschrift.
• Toward GIT stability of syzygies of canonical curves. (arxiv, journal)
  Algebraic Geometry, with Maksym Fedorchuk, David Swinarski.
• The Picard rank conjecture for the Hurwitz spaces of degree up to five. (arxiv, journal)
  Algebra & Number Theory, with Anand Patel.
• Groebner techniques for ribbons. (journal)
  Albanian Journal of Mathematics, with Maksym Fedorchuk, David Swinarski.
• Compactifications of Hurwitz spaces. (arxiv, journal)
  International Mathematics Research Notices.
• Class of the Hodge eigenbundle using orbifold Riemann-Roch.
  Pre-print, appendix to Cyclic covering morphisms on \(\overline M_{0,n}\) by Maksym Fedorchuk.
• Sharp slope bounds for sweeping families of trigonal curves. (arxiv, journal)
  Mathematical Research Letters, with Anand Patel.
• Modular compactifications of the space of marked trigonal curves. (arxiv, journal)
  Advances in Mathematics.
• Alternate compactifications of Hurwitz spaces.
  Thesis, Harvard, 2012.

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.

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.
  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).
  Trimester program on triangle groups, Belyi uniformization, and modularity, Bhaskaracharya Pratishthana, Pune, India.
• Algebraic curves and Belyi’s theorem (expository).
  Trimester program on triangle groups, Belyi uniformization, and modularity, Bhaskaracharya Pratishthana, Pune, India.
• Error correcting codes (expository).
  ANU Mathematics Extension Program, Canberra, Australia.
• The work of Claire Voisin (expository).
  Women in mathematics day, Australian National University, Canberra, Australia.
• Geometry of Hurwitz spaces.
  Character varieties and topological quantum field theory, University of Auckland, Auckland, New Zealand.
• The work of Caucher Birkar (expository).
  Colloquium, Australian National University, Canberra, Australia.
• How to count using (co)homology (expository).
  Tata Institute of Fundamental Research, Mumbai, India.
• Quadrature and algebraic geometry.
  Workshop on algebraic geometry approximation, and optimization, MATRIX, Creswick, Victoria, Australia.

• 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
  • The blow up of a planar triple point (sage code)
  • A conic fibration on a cubic surface (sage code)
• Notes from some past seminars
  • DbCoh: ANU, 2021
  • Mixed hodge modules: UGA, 2017
  • Stable rationality: Columbia, 2016

• Does course engagement correlate with final scores? I did a bit of data science based on a course in 2024; see the results.
• I am an advocate of free software and a user of emacs. I occassionally contribute to the development of org mode.