Head of Pure Mathematics

School of Mathematics and Statistics

UNSW, Sydney, Australia

E-mail: danielc * followed by shift two* unsw period edu period au

Office: Red Centre (East Wing) 4104

Consultation Times: TBA

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If you know the basic language of pure mathematics (i.e. what's a
group, ring, topological space), then you may be interested my
Youtube channel videos which give snapshots of more advanced pure mathematics.
These *adventures in pure mathematics* are mostly aimed at honours students,
and more generally, anyone who has completed MATH3711 and MATH3611. The goal
is to present important ideas and results in mathematics without the burden of
going through heavy duty proofs. For the latter, you just need to
do the hard work.

DanielChanMaths Youtube channel.

- Twisted Multi-Homogeneous Coordinate Rings in Journal of Algebra vol. 223 (2000).
- Noncommutative Rational Double Points in Journal of Algebra vol. 232 (2000).
- Morita Dualities and Dualizing Complexes [PDF] with Q.S. Wu and J. Zhang. In Israel Journal of Maths vol. 132 (2002).
- Del Pezzo Orders with Rajesh Kulkarni in Advances in Math. vol. 173 (2003).
- Noncommutative Coordinate Rings and Stacks [PDF] with Colin Ingalls, in Proc. of the LMS vol. 88 (2004).
- Splitting Bundles over Hereditary Orders Comm in Algebra vol. 333(7) (2005) p.2193-9.
- The minimal model program for orders over surfaces [PDF] with Colin Ingalls. Inventiones Math. vol. 161 (2005)
- Numerically Calabi-Yau orders on surfaces with Rajesh Kulkarni, Journal of the LMS vol. 72 (2005).
- Noncommutative cyclic covers and maximal orders on surfaces [PDF] in Mike Artin's 70th birthday issue of Advances in Math. vol. 198(2) (2005)
- Canonical Singularities of Orders over Surfaces with Colin Ingalls and Paul Hacking, Proc. of the LMS vol. 98 p. 83-115 (Jan, 2009)
- McKay correspondence for canonical orders. Trans. AMS vol. 362 (2010), p. 1765-95
- Hilbert schemes for quantum planes are projective. Algebras & Repr. Theory vol. 13 (2010), p. 119-26
- Moduli of bundles on exotic del Pezzo orders with Rajesh Kulkarni, American Journal of Math. vol. 133, no. 1, (Feb. 2011) p.273-93
- Conic bundles and Clifford algebras with Colin Ingalls. Contemp. Math. vol. 562 (2012) p.53-75
- Twisted rings and moduli stacks of "fat" point modules in non-commutative projective geometry Advances in Mathematics vol. vol. 229 (2012) p.2184-209
- Noncommutative Mori contractions and P1-bundles with Adam Nyman, Advances in Mathematics vol. 245 (2013) p.327-81
- Rational curves and ruled orders on surfaces with Kenneth Chan. Journal of Algebra vol. 435 (2015) 52-87
- Species and non-commutative P1's over non-algebraic bimodules.with Adam Nyman. Journal of Algebra vol. 460 (2016) 143-80
- 2-hereditary algebras and almost Fano weighted surfaces. Journal of Algebra vol. 478 (2017) 92-13
- Moduli stacks of Serre stable representations in tilting theory with Boris Lerner. Advances in Math. vol. 312 (2017) 588-635
- A representation theoretic study of noncommutative symmetric algebras with Adam Nyman. Proc. Edinburgh Mathematical Society vol. 62 (2019) 875-887
- Low dimensional orders of finite representation type with Colin Ingalls. vol. 297 (2021) Math. Z. 1161-1190
- Morphisms to noncommutative projective lines with Adam Nyman. vol. 149 (2021) Proc. AMS 2789-2803
- 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.

Here are some preprints.

- Tensor stable moduli stacks and refined representations of quivers with Tarig Abdelgadir. submitted

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.

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.

- Antony Orton (2003) wrote an excellent thesis on Algebraic Geometry and the Generalisation of Bezout's Theorem.
- Kenneth Chan (2004) wrote a thesis on Riemann surfaces and the Jacobian variety On a proof of Torelli's theorem.
- Dave Cock (2004) wrote a thesis on The Weyl algebras.
- Piotr Horodynski wrote a thesis on Grobner Bases.
- Maiyuran Arumugam (2005) wrote a thesis on A theorem of homological algebra: the Hilbert-Burch theorem.
- James Maclaurin (2006) wrote a thesis on The resolution of toric singularities.
- Boris Lerner (2007) wrote his thesis on The Brauer-Manin obstruction to the Hasse principle.
- Koushik Panda (2007) wrote his thesis on Twisted rings of differential operators on the projective line and the Beilinson-Bernstein theorem.
- Nathan Menzies (2007) wrote a thesis entitled An introduction to A-infinity algebras.
- Steve Ozvatic (2009) wrote a thesis on Factorisation theory in a non-commutative algebra.
- Daniel Smyth (2010) wrote a thesis on Finitely generated powerful pro-p groups.
- Anthony Christie (2011) wrote a thesis on Classification of simple plane curve singularities and their Auslander-Reiten quiver.
- Matthew Brassil (2012) wrote his thesis on Geometric invariant theory.
- Hanning Zhang (2013) wrote a wonderful thesis on homotopical algebra.
- Steve Siu (2013) wrote a thesis on K-theory and the Adams operation.
- Dorothy Cheung (2015) wrote a thesis on Classification of quadratic forms with Clifford algebras.
- Timothy Chan (2016) wrote a thesis on Dimer models and their characteristic polygons.
- Adrian Miranda (2017) wrote a masters thesis on Bicategories and higher categories.
- Zac Murphy (2017) wrote a thesis on Quotient categories and Grothendieck's splitting theorem
- Matthew Evat (2017) wrote a thesis on Generating functions associated to polynomial invariants.
- Rumi Salazar (2020) wrote a thesis entitled A link between quotients in symplectic and algebraic geometry
- Jackson Ryder (2020) wrote a thesis entitled How can we understand an abelian category
- Abdellah Islam (2021) wrote a thesis entitled Homology of Toric Varieties.
- Elie Sikh (2021) wrote a thesis entitled Intersection Theory from a Differential Viewpoint.
- Tasman Fell (2022) wrote a thesis entitled The McKay Correspondence on Kleinian Singularities

- Fahim Rahman is looking at the Shephard-Todd theorem in Invariant theory
- Tal Zwikael is looking at Geometric Invariant Theory and Quivers
- Simon Bohun as looking at D-modules (joint supervision with Anna Romanov)

- 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.
- 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.
- 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.
- Sean Lynch is studying zeta functions in noncommutative algebra.
- Jackson Ryder is studying noncommutative algebraic geometry

- Boris Lerner (see above)
- Tarig Abdelgadir is now at Loughborough University

See Extra Material for teaching materials related to past years.

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

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

- 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.