Microlocal Day #4

Noncommutative Harmonic Analysis and Probability

Imperial College London, UK

5 March 2014 (Wednesday)

 

Organisers

Veronique Fischer

Michael Ruzhansky

 

(Imperial College London)

 In the framework of LMS Joint Research GroupNoncommutative Harmonic Analysis and Probability

 

Conference Venue: Room 642 for 12am-2pm, and Clore Lecture Theatre for 2pm-6pm, Huxley Building, Imperial College London

Address :     Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

 

The Microlocal Day is an occasional event devoted to intensive series of lectures or talks on different aspects of the microlocal analysis and related topics. The program includes research lectures as well as survey lectures aimed at researchers and PhD students interested in the subject. All are welcome to attend.

Previous events: Microlocal Day #1, Microlocal Day #2, Microlocal Day #3.

Speakers

Schedule

Wednesday, 5 March, 12am—5pm, Imperial College London

 

Room 642

 

 

Room Clore Lecture Theatre

 

 

 

                                                                             

For further information please contact 
Michael Ruzhansky at this e-mail address 
 

Suggestion of hotels in the area (Earl’s Court Station, 15 mins walk to Imperial College)

 

Merlyn Court Hotel
Maranton House Hotel
Barkston Gardens
City Hotel Kensington
For other hotels see here
 

How to get to the Department of Mathematics, Imperial College London

 

Travel to the tube station Gloucester Road (District, Circle, and Piccadilly Lines).

When you exit the station, turn left along Gloucester Road, crossing Cromwell Road 50 meters from the exit.

After 4-5 minutes walk along Gloucester Road, turn right to Queen's Gate Terrace.

This is a short road leading directly to the entrance of the Huxley Building, at 180 Queen's Gate. We are on floor 6.

 

Abstracts

 

Paula Cerejeiras (University of Aveiro, Portugal) Non-commutative harmonic analysis for difference-difference operators

 

In many applications it would be advantageous to have a function theory based on reflection groups instead of on rotation groups, for example, in the analysis of quantum many-body systems of Calogero–Moser–Sutherland type or the study of crystallographic structures. This is the starting point for differential-difference operators, or Dunkl operators. In this rather recent theory differential operators are replaced by a commuting algebra of differential-difference operators which are invariant under the action of some finite reflection group. Their importance is based on the fact that they allow an analysis of the waves of certain quantum harmonic oscillators linked to specific physical models. Some promising results for solving differential-difference equations were obtained by an explicit construction of spherical Dunkl wavelets. In this talk we present a construction of Dunkl-monogenic and Dunkl-harmonic functions starting from holomorphic functions in the plane, based on Vekua-type systems. Additionally, we prove a version of Fueter’s theorem in the case of finite reflection groups and present some examples as well as giving a construction method of Dunkl-wavelets.



Yasuo Chiba (Tokyo University of Technology, Japan) Correspondence of some microlocal solutions for hyperbolic equations

 

In this talk, I show the properties of microlocal solutions of boundary value problems for degenerate hyperbolic operators. Here, 'microlocal' means hyperfunction (or microfunction) solutions. We use a transformation of coordinates with fractional power to construct them. We shall present the properties of solutions through this transformation as well as correspondences to the WKB solutions. 

 

 

Clotilde Fermanian-Kammerer (Université Paris Est, France) Long time analysis of solutions of completely integrable systems on the torus

 

In this talk, I will present a work in collaboration with Nalini Anantharaman and  Fabricio Macia devoted to the study of the behaviour in large time of solutions of semiclassical pde's of Schrödinger type on the torus. We will be interested in the weak limits of densities associated with the solutions, averaged on large intervals of time. We will shed light on the existence of a threshold above which every weak limit of the densities is  absolutely continuous with respect to Lebesgue measure. This result holds for a large class of completely integrable equations.  

 

 

Uwe Kähler (University of Aveiro, Portugal) Discrete Dirac operators and non-commutative harmonic analysis

 

In the last decades one can observe an increased interest in discrete structures, in particular in discrete Dirac operators. This lead also to the question of the corresponding discrete function theory, i.e. the theory of null-solutions of the discrete Dirac operators. But here a problem arises. While in the continuous case methods of non-commutative harmonic analysis where successfully employed, such methods run into the problem that in the discrete case there exist two partial difference operators instead of a single partial derivative. Although these difference operators commute the same cannot be said about the corresponding vector variable operators, i.e. the operators which form the Heisenberg algebra. This means that we cannot use the usual algebraic duality between polynomials of derivatives and polynomials of multiplication operators. In this talk we will present a possible way out by replacing the Heisenberg algebra with a Heisenberg-Sommen algebra and establish the principal tools for a discrete function theory.

 

Previously organised: Microlocal Day #1, #2, #3