|
Update: I have moved to the
University of Wisconsin-Madison,
where I will begin graduate school in the fall of 2007. My new page can
be found at
http://cs.wisc.edu/~mfredrik.
I am currently a senior Computer Science (B.S.) and
Mathematics (B.A.) student at Duquesne University. I am originally from
Sioux City, IA and now reside in Squirrel Hill, a neighborhood in the east
end of Pittsburgh.
For the past two years, I have worked as an
undergraduate research assistant to
Dr. Eric Rawdon, a Math
professor in my department. His work is in Physical and Computational Knot
Theory. To date, I have worked on three significant problems for Dr. Rawdon
-
Crossing Conversion:
Many researchers find themselves working in
knot theory for one reason or another. However, certain types of
analysis are difficult given only raw vertex data describing a
knot. It is useful to store this data in a more convenient form,
e.g. as a graph which describes the crossings in a knot and their
relation to one another. There is not a single, robust software
solution to this problem which is up-to-date and easy to use. I
have written a complete software package that performs this task
reliably not only for arbitrarily complex knots and links but also
for links with an arbitrary number of components. The output is
both compact and verbose, making this a very useful tool. A public
release of the package will be made available soon.
-
Knot/Link Complexity Reduction:
Oftentimes knot or link data comes from noisy or random sources.
In these cases, much of the time the link contains superfluous
complexities that can be removed while maintaining knot type. It
is desirable to remove these complexities for many reasons,
including polynomial invariant computations and similar
processing. I have written the first piece of software which
implements Reidemeister I and II as well as generalized
Reidemeister I and II reductions reliably for both knots and
links. On average, this removes 80% of the crossings on
two-component links generated by random walk. Reduction
performance for links of arbitrary component number has not yet
been extensively evaluated, but will be in the future. At the
moment, source code for this project is not yet available but it
can be obtained in binary format for both Win32 and Unix platforms
by email.
-
Improving
HOMFLY-PT Calculation: The HOMFLY
polynomial for knots and links is a very powerful invariant that
is practical for common use. However, software to calculate the
polynomial is either very outdated (cannot compute on complex
knots) or simply unreliable. I am currently working on a solution
that can compute the HOMFLY polynomial for arbitrarily
large/complex knots and links and does so quickly enough to remain
a practical solution for such knots. This is a very difficult
problem; calculation is
NP-Hard.
Hopefully a beta release of the software can be made public soon.
|
As well as my work for Dr. Rawdon, I have recently
started working with Dr. Patrick Juola (with plenty of help from Dr. Don
Simon!) on a computer security problem. We are currently examining the
possibility of applying certain machine learning techniques to the problem
of discovering the execution of malicious code on Windows NT-based
systems. I will provide more details on this project in the future.
|