My research area specialty can be described broadly as Functional Analysis. But primarily my work is focused on C*-Algebras (a branch of Operator Algebras), K-theory, and Noncommutative Geometry. Also, theoretical and mathematical Physics.

The field of Operator Algebras is a mathematical offshoot of Quantum Theory, which developed in the 1920's into an independent field in pure Mathematics. That is why there are connections between theoretical Physics and this area of Mathematics.

My research work centers on the structure of finite order automorphisms of the irrational rotation C*-algebra -- usually denoted by A_θ -- also called the noncommutative 2-torus. (It is sort of a model for a "fuzzy", "fractalized", or "quantized" surface of a doughnut.) This algebra is one of the most important algebras in the field of C*-algebras. It is known thru a lot of hard work that this family of algebras are all different (non-isomorphic) for different parameters θ in the interval [0, 1/2]. (Indeed, not all donuts are created equal!) There is an analogue to this from the theory of Riemann surfaces, specifically from complex 2-tori which depend on the angle between two generating lattice vectors (as conformal transformations preserve angles).

The algebra $A_{\mathrm{\theta <!--\; \#\; -->}}$ is defined by two unitary operators U and V that satisfy the (unitary) Heisenberg commutation relation

${\displaystyle VU=e^{2\mathrm{\pi <!--\; \#\; -->}i\mathrm{\theta <!--\; \#\; -->}}UV}$

This relation is just the complex exponential version of the Heisenberg commutation relation of quantum mechanics:

${\displaystyle qp-<!--\; -\; -->pq=i\mathrm{\hslash <!--\; \#\; -->}}$

by setting

${\displaystyle U=e^{iq},V=e^{ip}}$.

Generic operators in this algebra have the Fourier series form

${\displaystyle \underset{m,n}{\sum <!--\; \#\; -->}{c}_{mn}{U}^m{V}^n}$

(We call them generic because although not all operators in A_θ can be so expressed, these do form a dense set of elements in the algebra.)

I am interested in the inductive limit structure of symmetries of A_θ -- more specifically, the canonical ("natural") finite order automorphisms of this algebra, such as the flip symmetry, the Fourier transform, the hexic, and the cubic transform. These symmetries have respective orders 2, 4, 6, 3. (It is known that the orders 2, 3, 4, 6 are the only possible finite orders that a symmetry/automorphism of the algebra A_θ can have.)

In addition, I am interested in the approximate finite dimensional structure of their associated fixed point algebras. For example, the flip is the symmetry defined by

${\displaystyle U\to <!--\; \#\; -->U^{-<!--\; -\; -->1},V\to <!--\; \#\; -->V^{-<!--\; -\; -->1}}$

This symmetry is the analogue of the symmetry that transforms a single variable function f(x) into the function f(-x). The Fourier transform (the noncommutative version) is defined by

${\displaystyle U\to <!--\; \#\; -->V^{-<!--\; -\; -->1},V\to <!--\; \#\; -->U}$

This symmetry can be shown to arise from the classical Fourier transform

${\displaystyle \stackrel{^<!--\; ^\; -->}{f}(t)={\int <!--\; \#\; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \#\; -->}}^{\mathrm{\infty <!--\; \#\; -->}}f(x)e^{-<!--\; -\; -->2\mathrm{\pi <!--\; \#\; -->}ixt}dx}$

of a function f(x). (Clearly, if you square the Fourier transform you get the flip, as one would in the classical case.)

The Questions.

If you have a complicated algebraic structure, is it possible to "approximate" it by means of "simpler" structures? And if you have a "symmetry" (automorphism) of such an algebraic structure, could it be the "limit" of simpler symmetries acting on simpler structures (which approximate the more complicated one)?

The Attempt.

These are the questions that my research work tries to address -- particularly in the case where the complicated algebraic structure is the algebra A_θ.

Studying the structure of such symmetries involves approximating the rotation algebra A_θ by simpler basic building blocks that are invariant under the automorphism. The basic building blocks we have in mind here are composed of finite dimensional subalgebras direct summed with matrix algebras over the C*-algebra of all continuous functions on the unit circle (what we call a `circle algebra`) -- i.e., these building blocks have the form

${\displaystyle M_{n_1}(\mathbb{C})\oplus <!--\; \#\; -->\cdots <!--\; \#\; -->\oplus <!--\; \#\; -->M_{n_k}(\mathbb{C})\oplus <!--\; \#\; -->M_{m_1}(C(\mathbb{T}))\oplus <!--\; \#\; -->\cdots <!--\; \#\; -->\oplus <!--\; \#\; -->M_{m_{\mathrm{\ell <!--\; \#\; -->}}}(C(\mathbb{T}))}$

It turns out that the rotation algebra A_θ for irrational parameters θ is made up of such things! And the idea is that it is possible to choose these building blocks so that they are invariant under the symmetry (flip, Fourier transform, or the hexic/cubic transforms).

Other Interests.

I am also interested in connections between operator algebra theory and theoretical and mathematical physics. One such connection is stated in Hilbert's 6th Problem which poses the challenge of formalizing, mathematically, the foundations of Physics. It is still too early to tackle this question since Physics is still a growing field.