For reference onlymost material is beyond the scope

Mathematical Society, Providence, RI, 2013. (for reference only—most material is beyond the scope of
the course). MR 3136492
Inverse Problems for Boundary Value Problems
Lecturer: L. Tzou
This course forms part of the mathematical foundation behind Gunther Uhlmann’s Sydney
Ideas Lecture "Inverse Problems and Harry Potter’s Cloak" on March 26th 2018. Students
interested in this course are encouraged to attend and enjoy the public lecture.
Can we recover the conductivity of an object by making only voltage-to-current mea-
surements on the boundary? Also, can there be portions which are invisible from these
measurements?
In order to answer this question posed by Calderón one needs to have a deep understanding
of analysis and geometry. This course will use these questions as motivation to introduce
some of the important tools in modern PDE - microlocal analysis, Carleman estimates,
and semiclassical analysis. We will see how these tools can be put together to answer
Calderón’s question.
At the end of this course the student will have the tools to read recently published research
papers on this subject (e.g. see some of the references) with the help of the instructor.
Assessment: 80% Assignments, 20% Final Exam
Prerequisite: We will use some ideas (e.g. Hahn-Banach, Riesz Representation, and
B.L.T. Thm) from Functional Analysis in Semester I. Some basic knowledge of Fourier
transform will also be helpful.
Resources:
[1] John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value
problem , Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380
[2] Leo Tzou, The reflection principle and Calderón problems with partial data , Math. Ann. 369 (2017),
no. 1-2, 913–956. MR 3694665
[3] Colin Guillarmou and Leo Tzou, Calderón inverse problem with partial data on Riemann surfaces , Duke
Math. J. 158 (2011), no. 1, 83–120. MR 2794369

18 CHAPTER . COURSE DESCRIPTIONS

Samuel Johnson
Chapter 4
The Essay
4.1 Introduction
q The essay project has several objectives. First and foremost, it is intended to provide an
essentially open-ended framework whereby you may pursue, develop and discover your
interests in mathematics unencumbered by syllabus and the prospect of eventual written
examination. Basic to this process is the use of the library and communication with others,
most especially your supervisor. The writing of the essay is a most valuable part of the
project. The very act of writing is an invaluable aid to comprehension. A good essay
should be carefully organised, clear, readable by others, laid out well, properly referenced
and convey the essential ideas. Attainment of such writing skills is of great benefit whether
or not you elect to stay in mathematics.
One point should, perhaps, be emphasized: the essay project is not generally intended to
be a contribution to original research; however, the essay must clearly demonstrate that you
understand and have mastered the material. Originality in presentation or view in the essay
is required.
4.2 Choosing a Supervisor and topic