Professor Daniel Chan

Research Papers and articles

Publications

1. Twisted Multi-Homogeneous Coordinate Rings in Journal of Algebra vol. 223 (2000).
2. Noncommutative Rational Double Points in Journal of Algebra vol. 232 (2000).
3. Morita Dualities and Dualizing Complexes [PDF] with Q.S. Wu and J. Zhang. In Israel Journal of Maths vol. 132 (2002).
4. Del Pezzo Orders with Rajesh Kulkarni in Advances in Math. vol. 173 (2003).
5. Noncommutative Coordinate Rings and Stacks [PDF] with Colin Ingalls, in Proc. of the LMS vol. 88 (2004).
6. Splitting Bundles over Hereditary Orders Comm in Algebra vol. 333(7) (2005) p.2193-9.
7. The minimal model program for orders over surfaces [PDF] with Colin Ingalls. Inventiones Math. vol. 161 (2005)
8. Numerically Calabi-Yau orders on surfaces with Rajesh Kulkarni, Journal of the LMS vol. 72 (2005).
9. Noncommutative cyclic covers and maximal orders on surfaces [PDF] in Mike Artin's 70th birthday issue of Advances in Math. vol. 198(2) (2005)
10. Canonical Singularities of Orders over Surfaces with Colin Ingalls and Paul Hacking, Proc. of the LMS vol. 98 p. 83-115 (Jan, 2009)
11. McKay correspondence for canonical orders. Trans. AMS vol. 362 (2010), p. 1765-95
12. Hilbert schemes for quantum planes are projective. Algebras & Repr. Theory vol. 13 (2010), p. 119-26
13. Moduli of bundles on exotic del Pezzo orders with Rajesh Kulkarni, American Journal of Math. vol. 133, no. 1, (Feb. 2011) p.273-93
14. Conic bundles and Clifford algebras with Colin Ingalls. Contemp. Math. vol. 562 (2012) p.53-75
15. Twisted rings and moduli stacks of "fat" point modules in non-commutative projective geometry Advances in Mathematics vol. vol. 229 (2012) p.2184-209
16. Noncommutative Mori contractions and P1-bundles with Adam Nyman, Advances in Mathematics vol. 245 (2013) p.327-81
17. Rational curves and ruled orders on surfaces with Kenneth Chan. Journal of Algebra vol. 435 (2015) 52-87
18. Species and non-commutative P1's over non-algebraic bimodules.with Adam Nyman. Journal of Algebra vol. 460 (2016) 143-80
19. 2-hereditary algebras and almost Fano weighted surfaces. Journal of Algebra vol. 478 (2017) 92-13
20. Moduli stacks of Serre stable representations in tilting theory with Boris Lerner. Advances in Math. vol. 312 (2017) 588-635
21. A representation theoretic study of noncommutative symmetric algebras with Adam Nyman. Proc. Edinburgh Mathematical Society vol. 62 (2019) 875-887
22. Low dimensional orders of finite representation type with Colin Ingalls. vol. 297 (2021) Math. Z. 1161-1190
23. Morphisms to noncommutative projective lines with Adam Nyman. vol. 149 (2021) Proc. AMS 2789-2803
24. The minimal model program for b-log canonical divisors and applications with Kenneth Chan, Louis de Thanhoffer de Volcsey, Colin Ingalls, Kelly Jabbusch,Sandor Kovacs, Rajesh Kulkarni, Boris Lerner, Basil Nanayakkara, Shinnosuke Okawa and Michel Van den Bergh. vol. 303 Math. Z. (2023)
25. Tensor stable moduli stacks and refined representations of quivers with Tarig Abdelgadir. Journal LMS (online 2023)
26. Noncommutative linear systems and noncommutative elliptic curves with Adam Nyman. Trans. AMS volume 377, no. 3 (2024) 1957-1987
27. Terminal orders on arithmetic surfaces with Colin Ingalls, Algebra and Number Theory, vol. 18, (2024) 2027--2046

Here are some preprints.

In 2010, I gave a series of lectures on the theory of orders. The main purpose was to provide details on ramification theory so that one can read papers on recent work on concerning orders on surfaces. These LECTURES ON ORDERS have been typed up by Boris Lerner and should be appropriate for graduate students.

Honours, Masters and PhD students

If you are interested in doing an honours project in algebra, geometry or number theory, feel free to pop in to my office at any time or browse my Adventures in Pure Mathematics: Youtube videos above. Some suggestions for thesis topics. You can also check out my past students below and their theses.

Past Honours/Masters Students

1. Antony Orton (2003) wrote an excellent thesis on Algebraic Geometry and the Generalisation of Bezout's Theorem.
2. Kenneth Chan (2004) wrote a thesis on Riemann surfaces and the Jacobian variety On a proof of Torelli's theorem.
3. Dave Cock (2004) wrote a thesis on The Weyl algebras.
4. Piotr Horodynski wrote a thesis on Grobner Bases.
5. Maiyuran Arumugam (2005) wrote a thesis on A theorem of homological algebra: the Hilbert-Burch theorem.
6. James Maclaurin (2006) wrote a thesis on The resolution of toric singularities.
7. Boris Lerner (2007) wrote his thesis on The Brauer-Manin obstruction to the Hasse principle.
8. Koushik Panda (2007) wrote his thesis on Twisted rings of differential operators on the projective line and the Beilinson-Bernstein theorem.
9. Nathan Menzies (2007) wrote a thesis entitled An introduction to A-infinity algebras.
10. Steve Ozvatic (2009) wrote a thesis on Factorisation theory in a non-commutative algebra.
11. Daniel Smyth (2010) wrote a thesis on Finitely generated powerful pro-p groups.
12. Anthony Christie (2011) wrote a thesis on Classification of simple plane curve singularities and their Auslander-Reiten quiver.
13. Matthew Brassil (2012) wrote his thesis on Geometric invariant theory.
14. Hanning Zhang (2013) wrote a wonderful thesis on homotopical algebra.
15. Steve Siu (2013) wrote a thesis on K-theory and the Adams operation.
16. Dorothy Cheung (2015) wrote a thesis on Classification of quadratic forms with Clifford algebras.
17. Timothy Chan (2016) wrote a thesis on Dimer models and their characteristic polygons.
18. Adrian Miranda (2017) wrote a masters thesis on Bicategories and higher categories.
19. Zac Murphy (2017) wrote a thesis on Quotient categories and Grothendieck's splitting theorem
20. Matthew Evat (2017) wrote a thesis on Generating functions associated to polynomial invariants.
21. Rumi Salazar (2020) wrote a thesis entitled A link between quotients in symplectic and algebraic geometry
22. Jackson Ryder (2020) wrote a thesis entitled How can we understand an abelian category
23. Abdellah Islam (2021) wrote a thesis entitled Homology of Toric Varieties.
24. Elie Sikh (2021) wrote a thesis entitled Intersection Theory from a Differential Viewpoint.
25. Tasman Fell (2022) wrote a thesis entitled The McKay Correspondence on Kleinian Singularities
26. Fahim Rahman (2022) wrote a thesis entitled An investigation into invariant theory: the Shephard-Todd theorem
27. Tal Zwikael (2022) Geometric Invariant Theory and Representations of Quivers
28. Simon Bohun (2023) wrote a thesis on Beilinson-Bernstein localization on the base affine space and the Jantzen filtration for SL2 (joint supervision with Anna Romanov).
29. Dominic Matan (2023) wrote his thesis on Del Pezzo surfaces.
30. Matthew Bignell (2024) wrote his thesis on Quiver varieties and the Hilbert scheme of points.

Current Honours/Masters Students

• James Davidson is looking at tilting theory in algebraic geometry

Postgraduate Students

1. Kenneth Chan (2010) wrote a thesis entitled Resolving singularities of orders on surfaces. He had postdocs at the University of Washington and the Mathematical Sciences Research Institute, Berkeley.
2. Hugo Bowne-Anderson (2011) wrote a thesis entitled Explicit construction of orders on surfaces. He had postdocs at Max Planck Institute, Dresden and Yale University.
3. Boris Lerner (2012) wrote a thesis entitled Line bundles and curves on a del Pezzo order. He did a postdoc at Nagoya University and with me at UNSW and is now at Amazon.
4. Sean Lynch (2022) wrote a thesis entitled The slivered into parts method for semilocal Zeta functions.
5. Jackson Ryder is studying noncommutative algebraic geometry
6. Dom Matan

Postdocs Past and Current

• Boris Lerner (see above)
• Tarig Abdelgadir is now at Loughborough University
• Sean Lynch is now at Westlake University, Hangzhou
• Joseph Baine

Teaching materials for current courses

See Extra Material for teaching materials related to past years.

MATH3711/5706 Higher Algebra

A former student, Charles Qin typed up some lecture notes for this course. The course has only changed a little since then.

MATH3701/5700 Higher Topology and Differential Geometry/Modern Geometry (2022)

Here are the lecture notes which you should read. The lecture slides on the other hand can be found on Moodle. Those can be printed off before class.

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture 29 Lecture 30 Lecture 31 Lecture 32 Lecture 33 Lecture on classification of curves

MATH1141 Higher Mathematics 1A (Algebra) (2022)

Here are the lecture slides Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5

Extension III Mathematics

This is an informal self-paced course for talented high school students who have finished the secondary school curriculum early and want to be extended before entering university. It is particularly aimed at students who found Extension 2 mathematics easy and want exposure to new mathematics in a much more explorative format. If you are interested, go to Extension III Mathematics

Extra material

• Conference and seminar talks (videos and beamer slides).
• Click here for old lecture notes, problem sets etc concerning the following courses: Higher Algebra: Group and Ring Theory, Group theory, Homology and homological algebra, Algebraic Topology, Galois theory, MATH1141: algebra, MATH1241: algebra, Algebraic geometry, Commutative algebra, SCIF1121: Elementary mathematical modelling, MATH2601: Higher linear algebra, MATH5735: Modules and representation theory
• Here's a short parabola article I wrote on quaternions and their role in rotating objects in computer animation. It is aimed at high school students but should have some interesting tidbits for talented MATH1241 students.