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 currently an ARC DECRA Fellow (from 2017) at the Institute for Geometry and its Applications, University of Adelaide, specialising in mathematical physics.
I was awarded a University of Adelaide Research Fellowship for 2018.
From 20152017, 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, 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.
My current interests lie in discovering mathematical dualities from physical phenomena, and in reverse, understanding certain physical systems through dualities. 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.
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.
An exciting new result which I obtained in collaboration with theoretical and experimental physicists in spintronics, is the topological characterisation of the classical phenomenon of magnetostatic spin waves (MSSWs). The theoretical underpinning requires insights from the mathematics of topological semimetals, applied and interpreted in the context of classical mechanics and "boson diagonalisation".
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
Refereed papers
 Topological phases on the hyperbolic plane: fractional bulkboundary correpondence (with V. Mathai). Advances in Theoretical and Mathematical Physics (accepted for publication) [1712.02952]
 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 (published online) [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)
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)
Preprints 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) preprint
 Topological phases on the hyperbolic plane: fractional bulkboundary correpondence (with V. Mathai). [1712.02952]

Files
 Lecture notes on topological phases and Ktheory (updated 2 May 17)  Leiden_Lectures_2_May.pdf [492.2K] (application/pdf)
 CV  Jan 2019  CVJan19.pdf [97.2K] (application/pdf)
 Seoul lectures on Ktheory and Tduality of topological phases  Seoul_lectures.pdf [2.6MB] (application/pdf)
 Topological magnetostatic spin waves preprint  MSSW_preprint_Jan_19.pdf [4.3MB] (application/pdf)
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Entry last updated: Tuesday, 12 Feb 2019
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