Tractable analytic models of multiple-pinhole tomographs have not been developed. Such models when developed will be related to the attenuated Radon transform, the model of an ideal emission tomograph, but the complexities of multiple-pinhole systems make specification of the related transforms difficult. To date, results obtained on the quest to characterize multiple-pinhole tomographs have been obtained through analysis of matrix models generated via computer simulation. Numerical singular-value decomposition (SVD) (generalized eigenvalue decomposition) has been the primary tool of the analysis.

Conventional transmission computed tomography (CT) is an overdetermined problem. Multiple-pinhole tomography is an underdetermined problem. It is this mathematical context--that of non-square matrices--that suggests the use of SVD analysis. The mathematical context follows from the physical reality that available detector resolution does not match desired object-image reconstructed resolution.

Physically, a multiple-pinhole tomograph only acquires some of the information that defines an object being imaged--some object information is not measured and thus is lost to the imaging process. One aspect of design of multiple-pinhole tomographs is determination of the number and locations of pinholes that lead to acquisition of the object information that is "most desired". Mathematically, the goal is to map from the object to the "measurement" space the information that is the most useful information and to map to the "null" space that which is the least useful--for the diagnostic imaging task being performed.

SVD analysis has proved useful in the characterization of many imaging systems. This has been true for the systems of interest here as well and investigations of various system matrices has led to the asking of the following question. Can the relationship between the singular values and singular vectors of a complex pinhole tomograph, one with a multiple-pinhole coded aperture, and those of a simple pinhole tomograph, one with a single-pinhole aperture, be characterized explicitly, given the simple system is a building block of the complex system? Presented will be a discussion of multiple-pinhole tomographs being constructed, numerical results that lead to the asking of the just stated question, and results obtained to date relevant to the answering of the question.

What is the shape of the length-minimizing configuration?

The problem above is a natural geometric optimization problem; given a space curve with an embedded tubular neighborhood of fixed radius, find the shortest curve which can be obtained by any isotopy of the original curve and tubular neighborhood which keeps the tube embedded.

It is known that solutions to these problems always exist, and that they have a certain amount of regularity. For some simple links, such as the classical Gehring link problem, we know explicit examples of ropelength minimizing curves.

Going further requires the development of a general theory of ropelength criticality, and something analogous to an Euler-Lagrange equation for critical configurations. In this talk, we'll use ideas from the theory of frameworks to construct a criterion for ropelength criticality.

We'll then use our picture to come up with some surprising explicit solutions to the problem above.

In my talk, which is accessible to graduate students, I will review the mathematical problems of classical electromagnetic theory and their solution within the classical Born--Infeld program. I will also present the first successful steps from there into the quantum world.