Brothers Chudnovsky: Expert Q&A
 Posted 07.01.05
 NOVA scienceNOW
In July 2005, Tom Morgan answered questions about the Chudnovskys' lab and his work there.
Tom Morgan
Tom Morgan is a doctoral candidate in mathematics at Brooklyn Polytechnic University's Institute for Mathematics and Advanced Supercomputing, studying under the Chudnovskys. Full Bio
Tom Morgan is a doctoral candidate in mathematics at Brooklyn Polytechnic University's Institute for Mathematics and Advanced Supercomputing, studying under the Chudnovskys. Tom is pursuing a Ph.D. in math after a 30year career in information technology, most recently as chief technologist for a Fortune 500 company.
Q: What kind of supercomputer did you build in the lab? Is it a cluster of PCs? Do you use any kind of software to distribute the computations among nodes? ChinLong Shu, Austin, Texas
Tom Morgan: Yes, the supercomputer is a cluster of large PCs, connected by a gigabit Ethernet.
Computations are distributed around the nodes either by direct TCP/IP network calls, or by using a public domain messaging package called Parallel Virtual Machine (PVM):
Q: Can you tell us about some of the kinds of mathematical problems you and the Chudnovskys are most interested in and tend to work on in your lab? Kabir, Detroit, Michigan
Morgan: The Chudnovskys head the "Institute for Mathematics and Advanced Supercomputing" (IMAS). Their current projects emphasize the supercomputing area. Detailed design is under way by the Chudnovskys and their students on the floating point and the whole C64 chip, which will be processor for the Cyclops (also known as BlueGene/C) petaflop supercomputer. You can read more about the BlueGene/C project, for example, at:
en.wikipedia.org/wiki/Blue_Gene#Blue_Gene.2FC
and at links from that page.
Q: How did you discern from the vectors' positions that the tapestry had moved by itself rather than being bumped during photography or some other human error? Brian Booth, Hartford, Connecticut
Morgan: It's really not possible to be sure, but there are at least two good reasons to think so.
 Museum curatorial practice requires that objects like the tapestries only be handled by curators and enormous care was taken to construct rigs to keep the cameras away from the tapestry.
 The patterns of motion of the fibers appear in the images of all of the tapestries, and show a very complex set of miniscule motions throughout the tapestry. These kinds of motions don't look like responses to a small number of external events.
Q: My family name is Chudnovsky. I just returned from visiting my father and grandparent's birthplace, the village of Chudnov in Ukraine. I found family archival records there. What an experience! Where are the Chudnovskys from exactly? Do they know where their grandparents where born? Mel Chudnov, West Bloomfield, Michigan
Morgan: Dear Mel,
Nice to hear from you. The family as far as we know is not from Chudnov but other parts of Ukraine. We do love all Chudnovskys.—Gregory and David
Q: It says in your bio that you had a whole different career before entering your Ph.D. program in math. What decided you on this new path and where are you hoping it will take you? Nora Mailman, New York, New York
Morgan: It is actually a return to formal study of math after a long period. I began graduate study in math right after college; then life happened and I went to work in software development for industry.
A chance to resume study came up and I decided to take it. For me, the study of math is its own end. It is a topic that has fascinated me for as long as I can remember.
Q: Does your lab write its own applications for the supercomputer? R. B. Williams, Detroit, Michigan
Morgan: The applications that we run are a combination of locally written codes, libraries, and packages. The most interesting applications, like the codes used to process the unicorn tapestry were written here.
We use the Mathematica environment extensively to create and design algorithms, and then program in a mixture of C, C++, Fortran, and assorted scripting languages to do the largescale computations.
Q: Can you explain a little bit about the warping equations you used to solve the tapestry conundrum and why those were the equations to use? Anonymous, Burke, Virginia
Morgan: There were two main processes used to align the image tiles. The first process used the mathematical operation of "correlation" to locate corresponding feature points in the areas of overlap of the image.
The second process, the warping itself, used the locations of the feature points to construct a Delaunay triangulation for the overlap area. The essential idea here is that having located the feature points that correspond to each other, you know how to move these points to line up correctly, but you need to figure out how to move the rest of the area of the image. A Delaunay triangulation extends the matching points to a set of triangles that cover the overlapping area. You can then use the known moves of the triangle corners to interpolate the moves needed to bring all of the interior points of the triangles into good alignment.
There's a formal description of the Delaunay triangulation here:
mathworld.wolfram.com/DelaunayTriangulation.html
and a more informal description and extensive links here:
Q: Is your lab already working on analyzing the Vermeer painting that's shown in the program? Have you encountered any interesting problems with the analysis so far? Sharon Colmbs, Darien, Connecticut
Morgan: We haven't started work on this project yet. It is a fascinating idea, but no serious work has been done yet.
Q: Have you started your dissertation yet? If so, what is the topic? If not, what topics do you have in mind? Graham Allen, Cooperstown, New York
Morgan: I am working on my thesis now. The working title for the thesis is "Sparseness in frequency and space—spectral properties of objects, sparse in space or time."
Roughly, the idea is to study a pair of contrasting basis for function spaces, one basis, which is concentrated in the space, the second, which is diffuse, the operator that effects the change in basis and the properties of certain sets which are remarkable with respect to the basis pair.
An example would be the normal basis, the basis of sines and cosines and the discrete Fourier transform that relates them. For this combination, there are curious sets which are (nearly) "concentrated" in both bases—which is a very special property. The study is largely abstract, but there are potential applications to data reconstruction and to data concealment.
Q: Hi Tom,
After watching the tapestry episode I came up with three suggestions for preventing the problem your team faced in the future when other antique tapestries need photographing. What do you think?
 After the tapestries have come down and been treated in the wet lab, they should be laid out in a constant climate controlled area to finish drying and settling before scanning takes place.
 To keep the tapestries immobile perhaps they could be "sandwiched" between sheets of glass. With some pressure applied to the glass sheets, movement of the fabric could be minimized.
 Perhaps some form of laser scanning could take place after treatment and just before sandwiching. The laser scanning could count and map the ridges, valleys, and colors in the fabric, which would ultimately give the reassembly algorithms more accurate reference points to work with.
Bob MacDonald, Toronto, Ontario
Morgan: #1 is a really good idea.
#2 would probably terrify the curators.
#3 is also on the money. We would definitely like to recover the 3D structure of the object; it's likely this could be done with a combination photographic technique, with reference points established by a method like the one outlined, or by projection of a reference grid onto the object.
Q: Are the Chudnovsky brothers interested in only applied problems, or do they ever work on open problems in a pure subject (for example, algebraic number theory)? Matt, Brewerton, New York
Morgan: Their interests are very wide, but in fact they are best known for their work in pure mathematics, in number theory. You can look at some of their work in Proceedings of New York Number Theory Seminar, which they cochair. Look at "Number Theory, New York Seminar" (Springer Verlag) titles or their other books.
Related Links

Profile: Brothers Chudnovsky
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Meet the Brothers Chudnovsky
Mathematicians Gregory and David Chudnovsky point out how people use number theory every day.

Mathematical Siblings
Gregory and David Chudnovsky, who share a lab at the Brooklyn Polytechnic University, discuss the beauty of math.