Quantization on Lie groups

The proposed research will concentrate on the development of the non-commutative quantization theory with further applications to areas such as phase space analysis, time-frequency analysis, and the theory of partial differential equations.

There are many important examples motivating the great need in the proposed analysis. The importance of nilpotent Lie groups has been realised since a long time in general harmonic analysis as well as in problems involving partial differential operators on manifolds. Such questions go back to the celebrated Hörmander's sum of the square theorem, to the Rothschild-Stein lifting theorems, the Folland-Stein work on the Hardy spaces on homogeneous Lie groups, and Beals-Greiner calculus on the Heisenberg manifolds. We are interested in developing a new approach to pseudo-differential operators in the nilpotent and other non-commutative settings, to make advances in a general theory, but keeping in mind all the particular important motivating examples of groups and of PDEs, with applications to the time-frequency analysis. The problem of the quantization of operators in the non-commutative setting is long-standing and notoriously difficult in the area of partial differential equations.

One of the aims of this proposal is to build up on the recent advances in the quantization theory on compact Lie groups as well as on recent works on the quantization of symbols on the Heisenberg group with further applications to problems on Heisenberg manifolds, most notoriously ones of the subelliptic estimates and of the index theorems. From this point of view the recently introduced concepts such as those of difference operators linking the general quantization on manifolds to the Coifman-Weiss theory of Calderón-Zygmund operators, the emerging techniques of symbolic quantization provide for a possibility to making a new attempt at tackling these problems. However, there are certainly many interesting obstacles one needs to overcome to carry out this program, e.g. using an appropriate C*-algebra language for the Fourier analysis in the locally compact setting, development of appropriate Sobolev spaces taking into account the group structure in the non-stratified setting or in the case of differential operators of general orders, thus finding ways to linking several areas of analysis in the non-commutative setting.

Consequently, having constructed the satisfactory symbolic calculus of operators, we plan to apply this to the problems in the theory of partial differential equations, which is one of the most important objectives for such analysis. This will include symbolic expressions for propagators of evolution partial differential equations allowing for deriving necessary estimates for them (energy, Strichartz, smoothing), establishing global lower bounds for operators elliptic or hypoelliptic with respect to the group structure, as well as to long-standing global solvability problems for vector fields on spheres through the group action (Greenfield-Wallach and Katok conjectures).

This is important, challenging and timely research with deep implications in the theories of non-commutative operator analysis and partial differential equations, as well as their relations to other areas and applications.

As it is often the case with pure mathematics, the main impact will be academic. However, the range of the academic beneficiaries will be potentially very wide as the area of the operator analysis and its applications to partial differential equations influences advances in a variety of subjects. As it is written more specifically in the "Academic Beneficiaries" section, the expected impact on mathematics (and possibly on theoretical physics) is expected to be substantial. Besides these, there is a link to engineering through the planned work in the direction of the time-frequency analysis, and thus part of our research will be applicable there. The impact to this end is specified in more detail in the Pathways to Impact supplement to this application.

In addition to the academic aspects, in order to maximise the impact and exploitation of the EPSRC investment in this research, and to increase the knowledge transfer, we plan to organise an intensive workshop/conference devoted to the topic of the grant. A high-profile meeting would be extremely useful, to communicate the obtained results to the leading experts in the field of the phase space analysis, the main topic of the EPSRC grant, to colleagues working in their applications, to discuss the achievements and future developments, thus also increasing the long-term influence of the conducted research. Inviting the leading mathematicians from a variety of countries working in the field will certainly significantly contribute to the worldwide academic advancement of the area highlighting in a unique way the results obtained during our project. Communicating the research findings in an especially designed meeting to an internationally wide-spread selection of world leaders in the field would be an ideal way to facilitate and to maximise the knowledge transfer related to this research. A minicourse given by the PI/RA planned in the framework of the meeting will contribute to the training of highly skilled researchers and the participation of PhD students and young postdocs will be very useful for improving teaching and learning. The meeting will also serve as an excellent way of identifying further research areas that would be influenced by the conducted research in a longer run. Consequently, we will consider editing and producing a volume of research papers originating from the meeting to increase its visibility and impact.