PhD candidate

Department of Mathematics and Statistics

University of Melbourne, Australia

jchan3 AT student.unimelb.edu.au

Curriculum vitae: General Academic

I'm a Mathematics PhD candidate at the University of Melbourne in Australia, expecting to submit by the end of 2015. My doctoral thesis advisor is Paul Norbury. My thesis is titled On Hyperbolic Monopoles. I earned my Bachelors of Advanced Science with Honours at Monash University. My honours thesis advisor was Robert Bartnik.

Here you will find my academic (mathematics and physics) and non-academic interests. In particular, I study

- Yang-Mills-Higgs integrable systems (instantons, monopoles, Higgs bundles);
- Gauge field theory, gauge-theoretic invariants and related techniques;
- Quantum field theory and string theory;
- geometric Langlands conjecture.

Some papers of interest:

- Szabo's
**hot new**preprint, N=2 gauge theories, instanton moduli spaces and geometric representation theory. - Elliot and Gukov, Exceptional knot homology. This manages to link three topics in my life right now: BPS instantons, knot invariants and double affine Hecke algebras (thanks, Arun).
- Hitchin, Langlands duality and the G_2 spectral curve. G2 is on my mind...

I will be delivering a lecture series titled Riemann Rocks, Atiyah Sings on every Tuesday and Thursday of November. You can find out more on my teaching page.

I have started reading Halverson and Lewandowski's RSK Insertion for Set Partitions and Diagram Algebras as mentioned by Arun Ram, as well as the fresh preprint Shnir and Zhilin, G2 monopoles.

I am currently reading

- Bullimore-Dimofte-Gaiotto: The Coulomb branch of 3d N=4 theories
- Do-Dyer-Mathews: Topological recursion and a quantum curve for monotone Hurwitz numbers
- Ward: Symmetric instantons and discrete Hitchin equations
- Dunin-Barkowski et al.: Quantum spectral curve for the Gromov-Witten theory of the complex projective line

In July/August, I attended my first conference outside of Australia, "New Developments in TQFT" at the QGM in Aarhus, Denmark. I was inspired to look into some topics which I heard about there. I am grateful for travel funding from the University of Melbourne and QGM.

I am currently tutoring Metric and Hilbert Spaces. Exam consultation times are 10am Tu,Th SwotVac and 10am Wed the next week. Location: Russell Love Theatre. You can find supplementary material on my teaching page.

I am currently reading Eynard: A short overview of the "Topological recursionâ€ť.

I just put my first paper up on arxiv.

I will be in Europe for the next six weeks. I will be in Germany, the Netherlands, Italy and Denmark. As you can imagine, I'm very excited!

My interests lie on the boundary between pure mathematics and theoretical particle physics. Here is a brief description of the topics in mathematics and physics which I think about:

**Yang-Mills-Higgs integrable systems:** An instanton is an anti-self-dual curvature 2-form on a four-dimensional manifold; such a 2-form minimises the classical Yang-Mills action. Physically, an instanton represents a particle's instantaneous transition between two states, such as in tunnelling. Monopoles and Higgs bundles are reductions of instantons to three and two dimensions respectively. All three are unified in their treatment with twistor, Fourier-Mukai/ADHM-Nahm type transforms and spectral curve techniques.

**Gauge field theory, gauge-theoretic invariants and related techniques:** In supersymmetric gauge theories, the instanton solutions (known as BPS states) contribute to the partition function. Pandharipande and Thomas in their paper 13/2 ways of counting curves, explain how counting BPS states is a method of counting curves in a variety satisfying some conditions. Other gauge-theoretic invariants such as Gromov-Witten and Donaldson-Thomas invariants also count curves. Interesting techniques such as Floer cohomology and topological recursion are involved in the study of such invariants.

**The Geometric Langlands Conjecture:** In electromagnetism, Maxwell's equations are invariant under the exchange of electric and magnetic fields. We call this electric-magnetic duality and S-duality is a generalisation of electric-magnetic duality.

A paper of Kapustin and Witten explains that under the conditions of N=4 supersymmetry, S-duality implies the Geometric Langlands Conjecture. The Geometric Langlands Conjecture says that there is a correpondence between D-modules on the moduli space of vector bundles on an algebraic curve (for example, a torus) and flat connections on the same curve. In this story, there is a reductive group on one side and its Langlands dual on the other. For example, SL(2,ℂ) and PSL(2,ℂ) are Langlands duals.

You can see me dancing with the Melbourne University Dancesport Club in the 2015 Committee Dance here.

I speak Malay, Cantonese and Mandarin, in decreasing order of fluency. I am also learning German and Italian.

In my spare time, I

- dance the Argentine tango and other latin dances. I currently dance with the Melbourne University Dancesport Club;
- play the piano and the violin;
- swim;
- sing with the Melbourne Gay and Lesbian Youth Chorus; and
- draw mathematical objects - some of which you can see on this site.

I compete in the annual MUMS and SUMS puzzlehunts with Melissa Neish and Anthony Carapetis under the team name The Quick Brown Fox.