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.
- 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. - Last semester, I taught a course of about 150 students. I wanted to check if continuous engagement in the course correlates with higher scores on the final, so I did a bit of data science; see the results.
- Last semester, I taught a reading course based on Bott and Tu’s famous book Differential forms in algebraic topology; see the take-home final exam.
- 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.
- I am travelling to India (mostly Pune) from mid December to mid January. If you will be around, please write!
Papers/pre-prints
- Counting 3-uple Veronese surfaces. (arxiv)
Pre-print. - 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 wroteSage
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.