PhD Studentship: Microlocal analysis and its applications in quantum resonances, partial differential equations with analytic or Gevrey regularity, and several complex variables
University of Birmingham, United Kingdom
- Institution
- University of Birmingham
- Country
- United Kingdom
- Position type
- PhD
- Subject area
- Physics & Astronomy
- Location
- Birmingham, England, United Kingdom
- Funding
- Fully funded position
Want to be the strongest applicant for this position?
Book a free 15-minute call with our team. We will look at your profile against what University of Birmingham is actually looking for, and tell you honestly where you stand and what to strengthen before you apply.
Book a free call about this position →No cost, no obligation. Mentors who have secured funded PhD positions themselves.
We mentor STEM applicants only. If your research aspirations are outside STEM subjects, you're very welcome to use this board and apply directly; we just aren't the right people to mentor you.
About this position
Microlocal analysis explores the classical/quantum correspondence in partial differential equations (PDEs). For example, the concept of symbol quantisation in the theory of pseudodifferential operators links classical observables (functions on phase space) with quantum operators. Microlocal analysis also provides essential tools to rigorously define, compute, and analyse resonances, which describe the oscillation and decay of waves in systems where energy can escape to infinity, replacing eigenvalues and eigenfunction expansions in non-compact domains.
The consideration of analytic or Gevrey regularity in the study of PDEs has attracted considerable attention in recent years. The aim of this project is to develop microlocal analysis techniques in analytic and Gevrey settings. The analytic scheme often yields exponentially accurate approximation or controls in PDEs. Gevrey regularity allows one to go beyond the smooth theory and obtain quantitative remainder estimates, while retaining useful tools such as cutoffs and partitions of unity that are unavailable in the analytic setting. We expect to apply these techniques to geometry, quantum mechanics, and other fields in mathematical physics.
The project will be supervised by Dr Haoren Xiong (h.xiong@bham.ac.uk).
Funding notes:
Funding is available through the School of Mathematics for a suitably strong candidate. The scholarship will cover tuition fees, training support, and a stipend at standard rates for 3-3.5 years.
Candidates are encouraged to make an informal inquiry with Dr Haoren Xiong.
References:
Xiong, H., Pseudodifferential operators with formal Gevrey symbols and symbolic calculus, arXiv: 2603.07693.
Xiong, H., Boundary spectral estimates for semiclassical Gevrey operators. J. Spectr. Theory 15 (2025), no. 4, pp. 1503–1522.
Xiong, H. and Xu, H., Semiclassical asymptotics for Bergman projections with Gevrey weights, arXiv: 2403.14157.
Xiong, H., Generic simplicity of resonances in obstacle scattering, Trans. Amer. Math. Soc. 376 (2023), 4301-4319.
Apply directly to University of Birmingham
Emerging Scholars Council is not the employer and does not recruit for this position. It is advertised by University of Birmingham, and your application goes to them. We help students prepare and strengthen their applications.
Go to the official application page →Contact listed in the advert
Some positions — especially in Germany, Austria and Switzerland — are filled by writing directly to the supervisor rather than through an online form. These addresses were taken from the advert text, so check them against the official listing before you write.
A cold email to a supervisor is the highest-leverage thing you can send — and the easiest to get wrong. Ask us to look at yours before you hit send.