Graduate Student Workshop on
Clifford Algebras and Inverse Problems
Tampere University of Technology, September 1–5, 2008

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Tampere University of Technology Tampere University of Technology

 

FinCAIP - Graduate Student Workshop on
Clifford Algebras and Inverse Problems

The workshop concentrates on the common aspects and possible combinations of Clifford algebras and inverse problems. Initial progress in the combination of these fields have started to appear recently, for example the article by Takanori Ide, Hiroshi Isozaki, Susumu Nakata, Samuli Siltanen and Gunther Uhlmann in Comm. Pure Appl. Math 60 (2007).

Clifford algebras are generalizations of complex numbers. They are generated by vector spaces equipped with associative (but not necessarily commutative) multiplication, allowing the representation of geometric objects such as Moebius transformations. Clifford algebras can be used for the linearization of differential operators (e.g. Laplace, Laplace-Beltrami, Navier-Stokes, Helmholz, Dirac, and Schrödinger operators) and for applications in electromagnetics, signal processing, and robotics. In particular, the Maxwell equations can be formulated as a single equation using Clifford algebras.

Inverse problems are related to the interpretation and analysis of indirect physical measurements. The applications of this active field of applied mathematics include medical imaging, geological prospecting, non-destructive testing, signal processing and financial option pricing. John Sylvester ja Gunther Uhlmann made a major breakthrough in the solution of inverse problems in 1987. They used the so-called complex geometric optics (CGO) solutions to the solution of Calderón's inverse conductivity problem. Later, CGO solutions have been applied to several other inverse problems as well, leading to novel medical imaging techniques tested on living patients.

The wildly successful CGO solutions are based on exponential asymptotic conditions. On the other hand, hyperbolic function theory (developed using Clifford algebras by Heinz Leutwiler and S-L Eriksson) leads to the definition of generalized exponential functions in higher dimensions. Exciting possibilities related to the combination of these results will be discussed in the course with particular emphasis on Maxwell's equations.

Proceedings

New Please see the instructions how to get your paper to the proceedings.

Participants

Participants

In front of Tietotalo building.

The invited lecturers are

  • Anatoliy Butkovskiy: "Some Methodological Questions of Control" (Institute of Control Sciences, Russia)
  • Paula Cerejeiras and Uwe Kähler: "Clifford analytic methods for inverse problems" (Universidade de Aveiro, Portugal)
  • Klaus Gürlebeck: "Holomorphic functions in the plane and n-dimensional space" (Bauhaus Universität Weimar, Germany)
  • Matti Lassas: TBA (University of Helsinki, Finland)
  • Heinz Leutwiler: Introduction to hyperbolic function theory Abstract (Universität Erlangen-Nürnberg, Germany)
  • Petri Ola: Inverse problems for time-harmonic Maxwell's equations: what is known, and what remains to be done. Abstract (University of Helsinki, Finland)
  • Samuli Siltanen: "Probing with complex spherical waves"

The participants abstracts

  • Daniel Alayon Solarz: "Hyperholomorphic functions in a quaternionic and Fueter operator framework" Abstract;
  • Hendrik De Bie: "Fourier and related integral transforms in superspace" Abstract;
  • Pasquale Candito: "Multiple solutions for a discrete nonlinear boundary value problem via critical point theory" Abstract;
  • Nelson Faustino: "Continuous and Discrete Clifford Analysis: Two sides of the same?" Abstract;
  • Juho Linna: "Probing for Electrical Inclusions" Abstract;
  • Heikki Orelma: "On modied Dirac Operators in Geometric Algebras" Abstract;
  • Ebert Svend: "Wavelets on the Three Dimensional Sphere" Abstract;
  • Leo Tzou: "Inverse Problem for the Pauli-Dirac Operator"
    Abstract: The Pauli-Dirac equation models the quantum mechanical state of spin 1/2 particles. In this talk we will discuss inverse boundary value problems associated to this model and the method for solving these problems. In particular, we will see how clifford algebra can be combined with micro-local techniques to determine the coefficients of the equation with measurements taken only on part of the boundary.
  • Lauri Ylinen: "An Inverse Boundary Value Problem for Dirac Operators with Right-Multiplying Potentials Abstract

How to participate?

The registration has ended.

The 7'th international conference on Function spaces, Differential Operators and Nonlinear Analysis

The 7'th international conference on Function spaces, Differential Operators and Nonlinear Analysis will be organized between August 25 - 29, 2008 in Helsinki, Finland.

The conferences are placed one after the other to able the participation to both conferences.

www.helsinki.fi/~hasto/fsdona/index.html

FinCAIP Contact Information

Organizing committee.

Tampere

Last update 17.9.2008 TS