EPSRC Reference: 
EP/K039407/1 
Title: 
Quantization on Lie groups 
Principal Investigator: 
Ruzhansky, Professor M 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Dept of Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research 
Starts: 
01 August 2013 
Ends: 
31 August 2016 
Value (£): 
315,524

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The proposed research will concentrate on the development of the noncommutative quantization theory with further applications to areas such as phase space analysis, timefrequency 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 RothschildStein lifting theorems, the FollandStein work on the Hardy spaces on homogeneous Lie groups, and BealsGreiner calculus on the Heisenberg manifolds. We are interested in developing a new approach to pseudodifferential operators in the nilpotent and other noncommutative 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 timefrequency analysis. The problem of the quantization of operators in the noncommutative setting is longstanding 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 CoifmanWeiss theory of CalderónZygmund 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 t 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 nonstratified setting or in the case of differential operators of general orders, thus finding ways to linking several areas of analysis in the noncommutative 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 longstanding global solvability problems for vector fields on spheres through the group action (GreenfieldWallach and Katok conjectures).
This is important, challenging and timely research with deep implications in the theories of noncommutative operator analysis and partial differential equations, as well as their relations to other areas and applications.

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