Dr Guo Chuan Thiang
Position  ARC DECRA Fellow 

Org Unit  Mathematical Sciences 
guochuan.thiang@adelaide.edu.au  
Telephone  +61 8 8313 4762 
Location 
Floor/Room
6 60
,
Ingkarni Wardli
,
North Terrace


Biography/ Background
I am an ARC DECRA Fellow (20172020) specialising in mathematical physics.
***New*** Chief Investigator, with V. Mathai and P. Hochs, ARC Discovery Projects grant 20202022 ($507k)
I was awarded a University of Adelaide Research Fellowship for 2018.
In 20152016, I was an ARC Postdoctoral Research Associate at the University of Adelaide.
I completed a DPhil in mathematics at the University of Oxford in December 2014 (conferred March 2016). Prior to this, I studied physics and mathematics at the National University of Singapore and the University of Cambridge.
In 2010, I was a research assistant at the Centre for Quantum Technologies, Singapore.

Research Interests
I work on applications of Ktheory, operator algebras, index theory, algebraic and differential topology, 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.
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).
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.
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. This topologylocalisation duality is a basic tenet of "topological quantum chemistry". 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.
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.
Events
In September 2016, I organised a conference on mathematical topics at the interface of string theory, condensed matter physics, Ktheory, operator algebras, and geometry. [Website]
In 2017, I coorganised workshops on string geometries and dualities [Website], gauge theory and higher geometry [Website], and the AustraliaChina conference in noncommutative geometry and related areas. [Website]
In 2018 and 2019, I am the convenor of the Differential Geometry Seminar in the University of Adelaide [Website]

Publications
Preprints
 Edgefollowing topological states. [1908.09559]
 Good Wannier functions in Hilbert modules associated to topological insulators (with M. Ludewig). [1904.13051]
Refereed papers
 Topological phases on the hyperbolic plane: fractional bulkboundary correpondence (with V. Mathai). Advances in Theoretical and Mathematical Physics, 23(3) 803840 (2019) [1712.02952]
 Topological characterization of classical waves: the topological origin of magnetostatic surface spin waves (with K. Yamamoto, P. Pirro, K.W. Kim, K. EverschorSitte, E. Saitoh). Physical Review Letters, 122 217201 (2019) [1905.07909]
 Crystallographic Tduality (with K. Gomi). Journal of Geometry and Physics, 139 5077 (2019) [1806.11385]
 Crystallographic bulkedge correspondence: glide reflections and twisted mod 2 indices (with K. Gomi). Letters in Mathematical Physics, 109(4) (857904) (2019) [1804.03945]
 Tduality simplifies bulkboundary correspondence: the noncommutative case (with K. Hannabuss and V. Mathai). Letters in Mathematical Physics 108(5) 11631201 (2018) [1603.00116]
 FuKaneMele monopoles in semimetals (with K. Sato and K. Gomi). Nuclear Physics B 923 107125 (2017) [1705.06657]
 Differential topology of semimetals (with V. Mathai). Communications in Mathematical Physics 355(2) 561602 (2017) [1611.08961]

Global topology of Weyl semimetals and Fermi arcs (with V. Mathai). Journal of Physics A: Mathematical and Theoretical (Letter) 50(11) 11LT01 (2017) [1607.02242] Publicity at JPhys+

Tduality simplifies bulkboundary correspondence: the parametrised case (with K. Hannabuss and V. Mathai). Advances in Theoretical and Mathematical Physics 20(5) 11931226 (2016) [1510.04785]

Tduality simplifies bulkboundary correspondence: some higher dimensional cases (with V. Mathai). Annales Henri Poincaré 17(12) 33993424 (2016) [1506.04492]

Tduality simplifies bulkboundary correspondence (with V. Mathai). Communications in Mathematical Physics 345(2) 675701 (2016) [1505.05250]

On the Ktheoretic classification of topological phases of matter. Annales Henri Poincaré 17(4) 757794 (2016) [1406.7366]

Tduality of topological insulators (with V. Mathai). Journal of Physics A: Mathematical and Theoretical (Fast Track Communication) 48 42FT02 (2015) [1503.01206] Publicity at IOPSCIENCE

Topological phases: isomorphism, homotopy and Ktheory. International Journal of Geometric Methods in Modern Physics. 12 1550098 (2015) [1412.4191]

Degree of Separability of Bipartite Quantum States. Physical Review A 82(1) 012332 (2010)

Optimal LewensteinSanpera Decomposition for twoqubit states using Semidefinite Programming (with B.G. Englert and P. Raynal). Physical Review A 80(5) 052313 (2009)
Refereed Conference Proceedings
 Tduality and Ktheory: a view from condensed matter physics. In: Noncommutative Geometry and Physics IV, proceedings for TFC thematic year 2015 on "Fundamental Problems in Quantum Physics: Strings, Black Holes and Quantum Information", pp. 279314 (2017)
 On the Ktheoretic classification of topological phases of matter (conspectus). In: Proceedings of Frontiers of Fundamental Physics 14 (2014)
Edited volume String geometries, dualities and topological matter (with V. Mathai, P. Hekmati, H. Bursztyn, P. Bouwknegt, D. Baraglia). Journal of Geometry and Physics 138 (2019)

Files
 Lecture notes on topological phases and Ktheory (updated 2 May 17)  Leiden_Lectures_2_May.pdf [492.2K] (application/pdf)
 CV  Oct 2019  CV__Oct_2019.pdf [103.8K] (application/pdf)
 Seoul lectures on Ktheory and Tduality of topological phases  Seoul_lectures.pdf [2.6MB] (application/pdf)
 Mathematics and topological materials  Introduction  HKU_MoM_talk.pdf [3.2MB] (application/pdf)
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Entry last updated: Thursday, 5 Dec 2019
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