Appropriate use of mathematics in science and technology: not only from the standpoint of efficiency and potential profit, but also from the point-of-view of the people who will be part of the system. I am especially interested in projects which have a good synergy with Georgetown University. This includes medicine, biotechnology, business, government, law, foreign service and international affairs. My own favorite would be a ``War against PFFD '' involving a combination of doctors, scientists, engineers and mathematicians. Indeed, I hope to interest suitable parties in such a project.
I am a mathematician specializing in graph theory This includes issues of design and organization in many large-scale systems, and corresponding problems of computing logistically efficient routing. Some of my work has been applied to such issues.
However, for the last ten years, I have also been working on neural network theory. Geometry is the mathematical tool which my co-researchers and I have found useful to describe the nonlinear approximation capabilities of neural networks (feedforward-type, usually with just a single hidden layer). Thus, our group is only studying ``toy'' networks which are sufficiently tractable for analysis.
A goal of further research is to combine insights from both these areas. If the structure of a situation can be encoded in a succinct graph (a tree is especially nice), then the resource requirement is sharply reduced. So the net could learn more quickly. Graph theory, in the form of geometric combinatorics, might facilitate a heuristic computation of the network parameters - a neural implementation of the Finite Elements Method, based on integral representation theorems of the sort proved by our research group. This allows, in principle, the construction of much more robust neural networks, able to handle serious applications of neural networks - e.g., in intelligent prostheses.
In considering the introduction of neural network technology into various modern disciplines such as sensor engineering or financial analysis, it may be useful to have domain knowledge regarding issues of implementation. I have some familiarity regarding photonics, telecommunications, biophysics, parallel computer architectures, evolutionary algorithms and heuristics, quantum computing, visual perception, neuroscience, multimedia, and medical imaging.
My experience includes network facility capacity planning, vehicle traffic control, spectral analysis of ``free radical'' compounds, reliability and availability analysis, digital cartography, computer-aided design, industrial lighting, and software automation.
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