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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, LaplaceBeltrami, NavierStokes, 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, nondestructive 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 socalled 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 SL 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
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 ndimensional
space" (Bauhaus Universität Weimar, Germany)
 Matti Lassas: TBA (University of Helsinki, Finland)
 Heinz Leutwiler: Introduction to hyperbolic function theory Abstract (Universität ErlangenNürnberg, Germany)
 Petri Ola: Inverse problems for timeharmonic 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 PauliDirac Operator"
Abstract: The PauliDirac 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 microlocal 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 RightMultiplying 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.
