Yilin Ma

Room 2.62 Hanna Neumann Building 145, Mathematical Sciences Institute

The Australian National University, Science Rd, Acton, ACT 2601

yilin.ma@anu.edu.au

The Calderón problem. In its simplest formulation, the problem was solved in 1987 by John Sylvester and Gunther Uhlmann through their groundbreaking introduction of complex geometric optics (CGO) solutions. While not purely a microlocal concept, CGO solutions are closely tied to the ideas and techniques of microlocal analysis. The solutions of inverse problems like the Calderón problem represent some of the most significant triumphs of microlocal analysis in recent times.

Welcome to my academic website! I am a PhD student at the Mathematical Sciences Institute of the Australian National University, under the supervision of Professor Andrew Hassell.

My research focuses on microlocal analysis.

Feel free to take a look at my CV here.

I was suggested to upload a picture of myself but was too shy to do so.

What is microlocal analysis?

Microlocal analysis is an interdisciplinary field of mathematics. From my very personal perspective, it can be viewed as a collection of very useful tools, which are based on some fundamental principles of nature, in particular quantum mechanics.

Mathematicians frequently adopt ideas from physics and apply them to problems in mathematics. This is especially prominent in the analysis of partial differential equations (PDEs), where many important questions arise in physical contexts. Indeed, consider a linear partial differential operator

$P: L^{2}( \mathbb{R}^{n} ) \rightarrow L^{2}( \mathbb{R}^{n} ).$

Such an operator can naturally be interpreted as a ”quantum observable” of some “classical observable” $p(z, \zeta)$ . In the physical setting, one then refers to $P$ as the “quantization” of $p$ .

In fact, the following intuitive principle of quantizing a classical dynamical system on the flat space has become well-known since the Schrödinger equation was written down. In a system with $n$ degrees of freedom, the phase space is given by $\mathbb{R}^{n} \times \mathbb{R}^{n}$ . The classical observables in this system are functions, say $p( z, \zeta )$ , where $z = ( z_{1} , ..., z_{n} )$ and $\zeta = ( \zeta_{1}, ... , \zeta_{n} )$ , both belong ing to $\mathbb{R}^{n}$ , which are respectively the position and momentum variables. In the corresponding quantum mechanical system, the Hilbert space of square integrable functions $L^{2}( \mathbb{R}^{n} )$ are the quantum states. One can then define quantizations for the position and momentum variables directly by setting

$\displaystyle \mathrm{Op}_{h}( z_{j} ) u:= z_{j} u, \ \mathrm{Op}_{h}( \zeta_{j} )u : = hD_{z_{j}}u, \ j =1,...,n,$

for $u \in L^{2}(\mathbb{R}^{n})$ , where $D_{z_{j}} : = - i \partial_{z_{j}}$ and $h$ is the Planck constant. A quantum observable $P$ , corresponding to the classical observable $p$ , is then defined by replacing each $z_j$ by $\mathrm{Op}_{h} ( z_j )$ and $\zeta_{j}$ by $\mathrm{Op}_{h} (\zeta_j)$ .

Thus, for example, if $P$ is the linear differential operator

$\displaystyle P = \sum_{\alpha \in \mathbb{N}^{n}_{0}} a_{\alpha}(z) h^{|\alpha|} D_{z}^{\alpha},$

then the above algorithm would suggest that $P$ is given by quantizing the classical observable

$\displaystyle p = \sum_{\alpha \in \mathbb{N}^{n}_{0}} a_{\alpha}(z) \zeta^{\alpha}.$

A straightforward mathematical generalization of this would be to define, for each classical observable $p$ , that the corresponding quantum observable be $P = \mathrm{Op}_{h}(p)$ , where

$\displaystyle \mathrm{Op}_{h}(p) u : = \frac{1}{(2 \pi h)^{n}} \int_{\mathbb{R}^{2n}} e^{ \frac{i}{h} (z-z') \cdot \zeta } p(z,\zeta) u(z') dz' d\zeta.$

The converse, i.e., to get from quantum observables to classical observables, is trickier, since it is well-known that the aforementioned quantization procedure is not the only one. However, assuming that one has a way to do this, then the study of $P$ (quantum observable) can be reduced to the study of Hamiltonian dynamic of $p$ (classical observables). This provides a powerful framework for the study of PDEs.

Publications: You can find my publication list here.

Talks:

The Calderón problem with unbounded potentials in two dimensions. (Slides)
Semilinear Calderón problems on complex manifolds. (Slides)