Harnack inequalities on graphs

The Department of Mathematics at the University of Manchester seeks to find a PhD student in Probability Theory to work under the supervision of Dr Sebastian Andres on a research project on random walks in random environment and related probabilistic or analytic objects. 

Since the 1960’s Harnack inequalities have had a big influence on the interplay of probability theory and partial differential equations, which forms a fascinating part of mathematics, see the book by Bass for more details on this. Harnack inequalities provide control on the oscillations and imply Hölder regularity of harmonic functions. They also play an important role in probability theory because the transition probabilities (or heat kernels) of Markov processes such as random walks and diffusions are fundamental solutions of the associated heat equation. In this project we will study Harnack inequalities on graphs and the associated random walks. One goal will be to transfer the so-called Krylov-Safonov technique into the discrete setting of a graph to deduce parabolic Harnack inequalities for certain discrete divergence-form operators with possibly unbounded coefficients. Possible directions for research include operators for random walks with general speed measure or for random walks with long-range jumps.

The successful candidate will have completed (or nearly completed) an MSc degree in Mathematics (or closely related field) and will have demonstrated a capacity to produce mathematical research. Priority will be given to candidates with an advanced skill set in probability theory, PDE analysis or other related fields.

Shortlisted candidates will be invited to be interviewed by a small panel via Zoom, which will take place shortly after the application deadline. During the interview we seek to understand your motivation, aptitude and present knowledge.We will ask questions to discover these things, including some technical mathematical questions about topics you have covered.

You can apply online via the following link:

https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/08191/phd-probability/

You need to submit:

1) a CV (including contact details of two academic references),

2) copy of MSc degree (or equivalent, or evidence of the expected date of obtaining) in Mathematics (or closely related field),

3) transcripts of grades in English,

4) a personal statement (of maximal one page) indicating interest in this research topic; here candidates can point out anything they would like us to know which is not covered by the other documents;

5) copy of your passport if a visa will be needed to study in the UK,

6) evidence of English language ability if English is not the first language (can be obtained at a later stage). 

For further enquiries, contact Dr Sebastian Andres: [Email Address Removed]

Literature:

- R. Bass. Diffusions and elliptic operators, Springer 1998