ARC DECRA Fellow
School of Mathematical Sciences
Faculty of Engineering, Computer and Mathematical Sciences
I am a mathematical physicist, with expertise in K-theory, operator algebras, algebraic topology, noncommutative geometry and index theory, and applications to topological phases of matter and string theory.
I work on applications of K-theory, algebraic and differential topology, operator algebras, index theory, and noncommutative geometry to the phenomena of topological phases of matter in condensed matter physics and dualities in string theory.
Here's an introductory presentation that I gave for a general audience.
Current interests: discovering new mathematical dualities from physical phenomena, and in reverse, understanding certain physical systems through dualities. I am curently finding applications of coarse geometry and index theory in physics.
One type of duality is the bulk-boundary correspondence. I recently proved rigorously that the Chern class invariant for topological insulators leads dually to quantised unidirectional metallic boundary currents which can follow the edge of a material around arbitrary corners and imperfections, as theoretically suggested and experimentally verified by physicists. This duality is a physical manifestation of the index theory of certain semigroup C*-algebras. An even more powerful approach uses coarse geometry methods (in progress with M. Ludewig)
In return , I was able to show with M. Ludewig that the K-theory of a group C*-algebra has the interpretation of obstructions to existence of "good atomic" models in solid state physics. In this work, and in an ongoing collaboration with K. Gomi and Y. Kubota, I gave the first direct physical interpretation of the Baum-Connes assembly map which computes the above K-theory obstruction from physically determined K-homology data.
K. Gomi and I recently discovered the notion of crystallographic T-duality. This is closely related to a super-version of the Baum-Connes conjecture, and implements a duality of twisted equivariant K-theories with interesting computational consequences when complemented with "traditional" spectral sequence methods. The intuition for this duality came from studying position and momentum space versions of topological invariants in solid state physics, and the general concept of T-dualities as topological Fourier transforms.
An exciting new result (published in Physical Review Letters, Japanese press release), which I obtained in collaboration with theoretical and experimental physicists in spintronics, proves that the classical phenomenon of magnetostatic spin waves (MSSWs), used in the Nobel-winning discovery of Giant Magnetoresistence, has a differential-topological origin.
Earlier work on mathematics of topological insulators, semimetals, and T-dualities: My earlier contributions include a rigorous mathematical analysis of the general classification problem for topological insulating phases, the classification of topological semimetal phases, and the formulation of bulk-boundary correspondences. Roughly speaking, topological phases are equivalence classes of physical systems subject to certain spectral and symmetry conditions, labelled by intereting topological invariants. These invariants are typically "invisible" until boundary conditions are introduced, whence the they manifest as analytic boundary-localised zero modes via some index theorem. Some notes for a lecture series given in Feb-Mar '17 in Leiden are available here, and notes for a lecture series given in Feb-March in Seoul and Taiwan are available here.
As general tools, I also study the mathematics of T-duality, which has historically found deep applications in the analysis of D-branes in string theory. In particular, I seek to apply concepts from string dualities to the condensed matter setting. For instance, I introduced the notion of T-duality of topological insulators in a paper with V. Mathai. Together with K. Hannabuss, we demonstrated the conceptual and computational utility of T-duality in simplifying and providing geometric intuition for bulk-boundary correspondence for topological insulators.
Euclidean symmetry of non-relativistic dynamics (of an electron) can be broken into crystallographic group symmetry, with far-reaching consequences that have recently gained attention in the form of topological crystalline phases. Utilising a concrete physical model and applying the heuristics of a "crystallographic bulk-edge correspondence", I discovered, with K. Gomi, a new mod 2 "super-index theorem".
I have also studied the global topology of semimetals through Poincare duality, or "Dirac stringy" methods. Topological semimetals have the potential to realise exotic topologically stable fermions which are characterised by subtle topological invariants. In particular, there are intriguing links between semimetal topology, and Seiberg-Witten invariants, Kervaire semicharacteristics, and torsions of manifolds. In the presence of time-reversal symmetry, semimetals realise a new exotic type of monopole which acts as a charge for the famous mod 2 invariant of Kane-Mele.
Another general direction is the study of topological phases in different geometries. The idea that many-body effects can change the effective geometry "felt" by a single electron had previously been used to model the fractional quantum Hall effect. Utilising a variant of T-duality for Riemann surfaces, I formulated a bulk-boundary correspondence for fractional indices for the first time.
Some years ago, I dabbled in algebraic quantum field theory, and was a researcher in quantum information theory at the Centre for Quantum Technologies, National University of Singapore.
Date Position Institution name 2017 ARC DECRA Fellow University of Adelaide 2015 - 2017 ARC Research Associate University of Adelaide 2010 - 2010 Research assistant Centre for Quantum Technologies
Date Institution name Country Title — University of Oxford United Kingdom DPhil — University of Cambridge United Kingdom MASt (Distinction) — National University of Singapore Singapore Bachelor of Science (First Class Honours)
Australian Research Council Discovery Projects (Chief Investigator, joint with V. Mathai and P. Hochs). $507,438, 2020-2022.
Project: Coarse Geometry: a novel approach to the Callias index & topological matter.
Australian Research Council Discovery Early Career Researcher Award (DECRA). $357,000, 2017-2019
Project: T-duality and K-theory: Unity of condensed matter and string theory.
Office of the Deputy Vice-Chancellor (Research) Establishment Grant, University of Adelaide. $25,000, 2017
University of Adelaide Vice-Chancellor's Research Fellowship, 1-year salary and $10,000 grant, 2018
Project: Mathematics at the Intersection of string theory and topological materials.
Lecturer at University of Adelaide for:
Semester 1 2020 and 2019: Functional Analysis
Semester 2 2018: Introduction to topological K-theory
Tutor at University of Oxford for:
2014: Linear Programming
Teaching Assistant at University of Oxford for:
Calculus, Probability, Dynamics, Fourier series and PDEs, Statistics, Vector Calculus, Optimisation, Geometry, Electromagnetism and Quantum Mechanics
Past Higher Degree by Research Supervision (University of Adelaide)
Date Role Research Topic Program Degree Type Student Load Student Name 2018 - 2019 Co-Supervisor Analytic Pontryagin Duality Doctor of Philosophy Doctorate Full Time Mr Johnny Khai Yang Lim
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