Dr Guo Chuan Thiang
ARC DECRA Fellow
School of Mathematical Sciences
Faculty of Engineering, Computer and Mathematical Sciences
I am a mathematical physicist, with expertise in Ktheory, operator algebras, algebraic topology, noncommutative geometry and index theory, and applications to topological phases of matter and string theory.
I work on applications of Ktheory, 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 bulkboundary 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 Ktheory 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 BaumConnes assembly map which computes the above Ktheory obstruction from physically determined Khomology data.
K. Gomi and I recently discovered the notion of crystallographic Tduality. This is closely related to a superversion of the BaumConnes conjecture, and implements a duality of twisted equivariant Ktheories 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 Tdualities 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 Nobelwinning discovery of Giant Magnetoresistence, has a differentialtopological origin.
Earlier work on mathematics of topological insulators, semimetals, and Tdualities: 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 bulkboundary 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 boundarylocalised zero modes via some index theorem. Some notes for a lecture series given in FebMar '17 in Leiden are available here, and notes for a lecture series given in FebMarch in Seoul and Taiwan are available here.
As general tools, I also study the mathematics of Tduality, which has historically found deep applications in the analysis of Dbranes 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 Tduality of topological insulators in a paper with V. Mathai. Together with K. Hannabuss, we demonstrated the conceptual and computational utility of Tduality in simplifying and providing geometric intuition for bulkboundary correspondence for topological insulators.
Euclidean symmetry of nonrelativistic dynamics (of an electron) can be broken into crystallographic group symmetry, with farreaching 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 bulkedge correspondence", I discovered, with K. Gomi, a new mod 2 "superindex 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 SeibergWitten invariants, Kervaire semicharacteristics, and torsions of manifolds. In the presence of timereversal symmetry, semimetals realise a new exotic type of monopole which acts as a charge for the famous mod 2 invariant of KaneMele.
Another general direction is the study of topological phases in different geometries. The idea that manybody 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 Tduality for Riemann surfaces, I formulated a bulkboundary 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.

Appointments
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 
Education
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) 
Research Interests

Journals

Conference Papers
Australian Research Council Discovery Projects (Chief Investigator, joint with V. Mathai and P. Hochs). $507,438, 20202022.
Project: Coarse Geometry: a novel approach to the Callias index & topological matter.
Australian Research Council Discovery Early Career Researcher Award (DECRA). $357,000, 20172019
Project: Tduality and Ktheory: Unity of condensed matter and string theory.
Office of the Deputy ViceChancellor (Research) Establishment Grant, University of Adelaide. $25,000, 2017
University of Adelaide ViceChancellor's Research Fellowship, 1year 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 Ktheory
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 CoSupervisor Analytic Pontryagin Duality Doctor of Philosophy Doctorate Full Time Mr Johnny Khai Yang Lim
Connect With Me
External Profiles