Carving Niche Areas

About two weeks ago, I was invited to a workshop in IIUM called "Workshop on Dynamical Systems and Their Applications in Mathematical Physics, Engineering and Economics" on 15 May 2017. The event is held also in conjunction with the 70th birthday of Prof. Nasir Ganikhodjaev. The invitation was conveyed to me by Dr. Pah Chin Hee. The first thing that went through my mind when I first received the invitation, I'm not really an expert in dynamical systems and was thinking what should I talk about. I do have interest in symbolic dynamics on hyperbolic surfaces and toyed about the idea years before but I have no new results to actually present. Thus,  I fall back to what I know more about, namely quantization relating operator algebraic approach in the research of Prof. Ganikhodjaev and as for dynamical systems, I take it to be represented by the phase space as a (co-)tangent bundle to some configuration space. Most of the materials are just reviews but I include some outlook of the current research on hyperbolic surfaces. Here, I would like not to talk about what I present but the impression I get from the workshop on their research in CTS, IIUM.

Some pics first. Here is Prof. Ganikhodjaev during opening remarks:

Prof. Muhammad Ridza Wahiddin, Deputy rector during the opening:

The papers of Prof. Ganikhodjaev can be found here or here. Prof. Ganikhodjaev have many disciples here and this includes Pah Chin Hee and also Farrukh Mukhamedov who used to be in IIUM but has now left for Emirates.(see incomplete list here). During the workshop, I found many more of their younger staff working on areas of statistical mechanics and operator algebras. It strikes me that they have carved out a niche in these areas. These are actually important areas that go unappreciated particularly in Malaysia. Let me put the rest of the pics before I go on to discuss on what niche area that we have carved out over in UPM over the years.

The group photo:

Myself during own presentation and with the audience:

Prof. Ganikhodjaev donning an Uzbek traditional coat called chapan:

Back to us. Have we carved our own niche? Here, us means our theoretical physics group (perhaps later, I will also ask the same for the institute). What are we known for? I remember a respectable colleague once said that he wasn't sure what I am expert on. In a way, since I came back I have to admit I have been exploring areas. My own formal training is on quantization but I have taken courses in various areas of theoretical physics (including particle physics and general relativity) at Adelaide, Cambridge and Durham. So, I guess, I am pretty flexible in terms of the mathematical tools though my inclination has always been towards use of geometry and topology in physics. Much more generally, I have interest in seeing how abstract ideas get realised in physical systems and besides that I love to see far-flung ideas flock together. So this made me experiment more than others.

Fresh from PhD and back at UPM, I was suggested by a colleague to look into problems of condensed matter, which I did through quantum Hall effect. This led me to hyperbolic geometry, which we studied until now. Initially from the perspective on quantization (which I returned to in the talk at the workshop) but later on proceeded to computation of Maass cusp forms until now. The work could have been an opportunity to collaborate with number theorists. Occasionally, I entertain requests by students and back then many came to me with the interest in philosophy, general relativity & cosmology and Bohmian mechanics. I tried as much as I could to blend these into what I am interested in; at heart I am still very much on mathematical aspects of quantum theory. So when Prof. Kwek suggested I should go into quantum information in my years at ITMA, I readily take it up in areas closer to my interest (at the time I was interested in Kochen-Specker theorem/quantum contextuality; Toh took this up as a PhD student and he still continues in this area). Also the hope is that this area is more relevant to the institute that I was associated to back then (Institute of Advanced Technology). Presently, I am still interested in both quantum contextuality and quantum entanglement, seeking ways to understand it better (note this is much in the physics mode of doing things as opposed to an engineering one, which is dominant in quantum information).

When I join the Institute for Mathematical Research, I was looking for another area to start for which I saw complex networks to be one area that I thought could benefit the institute. Note that this area seems to mix up statistics, graph theory and computation with plenty of applications. I thought that this would be great for an institute that looks for interdisciplinary areas. For me, in a way it is sort of a mixed beginning, I was also looking for other areas in which hyperbolic geometry can be used and saw several papers of Tomasso Aste on hyeprbolic geometry of complex networks. Thus the venture into this. Presently Dr. Chan Kar Tim is taking this up more seriously than me, being computationally trained (through Maass cusp forms) and currently teaching statistical mechanics, complex networks could be a natural evolution of his expertise.

Another academic in our group is Dr. Nurisya Mohd Shah. She began as my M.Sc. student, working on energy eigenequations on hyperbolic surfaces (evolved from the quantization problem). Later I introduced her to the late Prof. Twareque Ali who was also working on quantization and then became his PhD student at Concordia. She worked in noncommutative quantum mechanics, much in the veins of quantization theory with connections to biorthogonal polynomials.

I still take up many students despite my administrative duties but I intend to slow down as I will be retiring soon. They are

• Zurita Ismail (M.Sc.) on UPM scientific collaboration networks
• Hafizuddin Mohd Taha (M.Sc.) on complex networks build from triangle groups
• Ganesh Subramaniam (M.Sc.) on Killing tensors on 5-dimensional space-time 
• Wan Dimashqi (M.Sc.) on discrete phase spaces and Spekkens toy model
• Nor Syazana (M.Sc.) on Maass cusp forms on asymmetric hyperbolic tori
• Siti Aqilah (M.Sc.) on categorical quantum mechanics
• Sarah Diyana (M.Sc.) on complex networks and stock manipulation
• Mohd Faudzi Umar (Ph.D.) on noncommutative quantum mechanics and canonical group quantization
• Umair Abdul Halim (Ph.D.) on symplectic topology on complex projective spaces
• Ahmad Hazazi (Ph.D.) on complex projective geometry and entanglement geometry
• M.A. Ahmed (Ph.D.) on lattice gauge theories and quantum information
• Choong Pak Shen (Ph.D.) on quantum marginal problem and entanglement

Many potential students still come to me but I have started to suggest my younger colleagues to them. I intend to stop taking students at some point in the future dependent on how things evolve. 

So what niche areas can we identify from the above? Lying deep at the core are mathematical structures of geometry, topology, group theory and graph theory. At the outer level one can identify quantum structures, complex networks and to a lesser extent mathematical aspects of relativity and cosmology. I hope that my younger colleagues will evolve this further. Another expansion that I'm still thinking about is the work with MICEMS, which at this stage is still very open (still in discussion).