(1) Brief Overview
My general area of research is in functional analysis and operator algebras, with special attention given to the operator algebraic approach to ``non-commutative [or quantum] geometry''. This is a relatively new field of study, but it has many interesting connections with various other areas of mathematics and physics.
In ordinary geometry/topology, the main object of study is a manifold [or a topological space] M. But the study of M is actually equivalent to the study of the (continuous) function space C0(M), which is a commutative ring or algebra. Non-commutative geometry is based on this latter point of view: We study geometric/topological questions on an arbitrary (i.e. no longer commutative) algebra A, which is a ``continuous function space on a non-commutative manifold''.
Although this paradigm does not necessarily have to be restricted to the case of continuous functions, working with continuous functions enables us to keep the full aspects of topology. In this sense, it is generally accepted that the correct framework to develop non-commutative topology or non-commutative differential geometry is the framework of C*-algebras. Furthermore, there are many situations in ordinary geometry and topology where the natural object of study is a ``singular space'' that often cannot be studied directly in purely topological terms, but where the C*-algebraic approach is helpful: Indeed, one can sometimes associate to the singular topological space X a non-commutative C*-algebra playing the exact role of C0(X), and the C*-algebra theory (including K-theory) provides a crucial tool in understanding the topological situation.
Transition from ordinary geometry to non-commutative geometry is broadly called the ``quantization''. In nature, this is essentially the same as the process in physics of jumping from a classical mechanical system to a quantum mechanical system. It is not always straightforward or easy to make this jump, but there is usually a strong correspondence relation between quantum (non-commutative geometric) objects and their classical (Poisson geometric) counterparts.
My research focus has been centered around these two main themes above: quantization and non-commutative geometry. In particular, I have been studying for some time the C*-algebraic (locally compact) quantum groups, trying to explore their roles as non-commutative geometric objects of study while viewing them as objects quantized from their classical counterparts called Poisson-Lie groups.
In the next a few years, I plan to continue my research in this direction, including the topics like: deformation quantization of Poisson manifolds; theory and construction of C*-algebraic, locally compact quantum groups; quantum group representation theory; and their implications to mathematical physics. As I conduct my research, I wish to emphasize and utilize the close interplay between Poisson geometry and the C*-algebra approach.
See below for more ...(2) A More Detailed Description of My Research
Interests and Plans (in PDF)
This is a more technical version, including a list of references.
(3) Publications and Preprints
Abstracts of my papers are available, as well as the PDF files of the
papers.
(4) Article Reviews I wrote for Mathematical Reviews
I wrote 14 reviews for Mathematical Reviews (from American Mathematical Society).
The main page of the above link is the MathSciNet,
which is actually the Mathematical Reviews on the web. I also wrote 8 reviews for
Zentralblatt Math.
(5) Operator Algebra Resources Page (maintained by N. Christopher Phillips)
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