Georgia Tech Undergraduate Mathematics

What's Going On?----2002 Edition (2001 Edition is here.)

This file is being updated!



Brandon Meredith ; Claire Conner ; Patty Pichardo; Blair Dowling; Lauren Hansen; REUs; Elizabeth Sanders ; Clark Alexander; Matthew Fisher ; Keith Crunk; Erika Norenberg; Michael Abraham; Jeremy Barrett; David Eger; Anup Rao; Joe Montgomery; ACE Lab; Malika Hines




Brandon Meredith writes: Right now I am studying in France at a university called l'Universite de Technologie de Compiegne (UTC). This is my second semester studying here, and right now I'm looking for an internship in France for summer and fall. It's very interesting being here since the French seem to have a very different outlook on mathematics (among other things). The most interesting mathematics that I am doing here is an individual research course in the field of operations research that I am taking with Prof. Villon. France ain't so bad either. (I'll bet! ;-) --ML) 

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Patty Pichardo is graduating Spring 2002. She is continuing graduate studies at Georgia Tech. In fact, she has a VIGRE/GT Traineeship and Goizueta Fellowship, which will be quite a nice step up from being an undergrad TA. Especially 'cuz she won't have to be a TA next year. 

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Blair Dowling writes: 

I'm spending this summer with the College of Computing's Study Abroad Program in Barcelona, Spain, taking two computer sciences classes, and an architecture course. I'm planning on continuing my research with Dr. Tetali into next year, and am doing a senior research project with Dr. Randall, of the CS department next fall. I'm currently planning to continue with my involvement in both the Putnam Examination, and Pi Mu Epsilon throughout next school year, though since I'm graduating in May 2003, I'd love to have some younger students become more involved next year.

Blair also reports being sad that she won't TA in the fall, because of a heavy course load, 


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Lauren Hansen is graduating in the Spring of 2002, and is starting in the Master's Degree program in Mathematics of Finance at Columbia University, NYC.

This area of math/finance was largely created in the 1980s and has quickly become an important, even essential, aspect of many modern corporations. 


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Claire Conner (dmath, graduating Spring 2003) is completing her last Co-op work assignment during Spring 2002 with New Energy Associates, a firm developing financial software for energy markets. (This is a very difficult and complicated task.) Claire was also Co-op representative to the Georgia Tech Student Government and Co-op Club president in academic year 2001-2002. She has a strong interest in cryptography, which she is currently studying in her Math Senior Project and CS Theory 2 classes. 

Editor's Note: The software problem that Claire is working on is indeed one very challenging problem. Financial Mathematics was first developed for the stock market and other highly liquid markets. The complicated control and optimization mathematics developed for these markets is now taught in specialized masters degree programs, including GT. 


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REU participants are to include at least 12 undergrads: Ryan Hynd, David Eger, Joseph Montgomery, Jeremy Barrett, David Skoog, Boris Kerzhner, Ganesh Sandramoorthi, Jeffrey Elms, Roberto Lopez, Erika Norenberg, and Andy Wand. 

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Elizabeth Sanders is cooping this summer and heading off in the fall to Moscow State University to participate in the "Math in Moscow" semester. She is in addition minoring in International Affairs. This semester will be an outstanding complement to her undergraduate education. 

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Clark Alexander writes: "I will for certain be travelling to University of Maryland this summer to do research with the experimental geometry lab. The project will involve hyperbolic geometry and something about 3-manifolds." 

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Boris Kerzhner will be in the REU program, with a project directed by Prasad Tetali. It concerns applications of graph theory in randomized coding."

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Michael Abraham will be in the REU program, working with Professor Mucha and the School of Math Bewoulf cluster. The "Small World Networks." Using the Beowoulf Cluster, develop codes to study the connectviity of such networks, for comparison with theories in development by Professor Mucha. Michael will be developing his parallel programing skills. 

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Jeremy Barrett's REU program, directed by Professor Yu, calls for a study of a question that combines notions of algebra, topology and combinatorics. The first step is to reading up on groups acting on low dimensinal topological spaces, by the (no unfortunately infamous) Dunwoody. 

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David Eger sends his description of an REU into Aritifical Intelligence and Theorem Proving. As usual, he is well underway already! 

(1) How can we use formal logic to codify typical mathematics in a machine verifiable form? 

This exploration would start with filling in the holes in my knowledge of predicate calculus and using the program metamath (http://www.metamath.org/) to codify some basic proofs of theorems from Abstract Algebra. The basic pretense behind this project is that several large works (e.g. the Classification of the Simple Groups) are simply too large for any person to verify by hand and be absolutely sure he has not overlooked something. Codifying proofs and theorems in a database that is verifiable by a computer may offer us a valuable tool.

Also, I would like to see if I can codify some basic problems from combinatorics and the proofs their results. It always seemed rather bogus to me that someone would ask "Suppose you have a bus with one driver and fifteen students. There are twenty seats and one for the bus driver. How many valid sittings are there?" And with some hand waving one points out how many there must be. Perhaps this is the best we can do. But perhaps there is a proper natural symbolic representation we could have for this sort of problem. I could then contrast these attempts with the original problem and its solution. Does presenting a formal proof detract from understanding? Can we have our cake and eat it too: can we both have logic-level proof AND understanding? If so how? People tend to think in geometric manners; what geometric representations can be represented as alternatives to predicate calculus, and can such pictures be treated formally in provable, verifiable ways?

(2) Once we have a structure for representing logically our mathematics, can we use a computer to discern important properties and patterns about our mathematical objects, and if so how?

That is, in Abstract Algebra we have defined certain properties of operators which we find important in some way - commutativity, associativity, alternate associativity - from which many other "nice" properties follows. Are there less obvious properties from we might find in sets with operators that give us nice properties? What are the patterns that we see if we look at a selection of quasigroups? Can we use monte carlo methods to look for patterns which we might then put forward as hypotheses to then attempt to prove? Might we find things as important as Sylow's theorems in this manner? It's much of a "pi in the sky" question, but it's one that has perked my interest from time to time.

Alternatively I could try to construct a Theorem Proving System, which given certain truths, could try to deduce useful theorems. Embedding heuristics for "useful" could be quite a challenge.

(3) What do we want to do with computation? Why is it seen as such a fluid thing - practicing computer scientists commonly eschew the formal methods that seem to me essential? How could formal methods be applied to complex software systems, and what are the limits to such applications and why?

I would start this with a survey of texts both on Software proof systems such as Zed, some remedial reading on lambda calculus and functional programming, and a couple of texts on programming as a mathematical art, specifically, I'd like to explore Dijkstra's "A Discipline of Programming" and "Predicate Calculus and Program Semantics".

(3a) From this starting point I could draw material to try tackling what I believe to be an NP-Complete problem - a variant of the classical SAT problem called "Paint by Number", a pencil puzzle game which appeared in GAMES magazine in the 1990s. This I believe should take me on a journey through enumeration methods, some combinatorics, perhaps some graph theory, and in general should give me a good amount of ground to explore.

(3b) I could alternatively survey various software systems and with each ask the question "What elements of this system are (not) provably correct and why?" Which elements have simply ill-defined requirements; which are impossible to do correctly; which are trying to correct in some way for a break of an assumption of the programming model (out of memory conditions and other errors) and are these attempts misled in their nature, or useful?

Suppose for instance we are examining the halting problem and have written a program that determines whether a program will halt. Obviously such a program will not work for every program. But then, perhaps our program, by looking for certain signatures, will work, but only in a restricted subspace of the space of all programs. People often, I think, get caught in the rut that simply because the general case is impossible that the whole endeavour is hopeless.

Perhaps I will find that the vast majority of software has nothing to do with reality, since pre- and post- conditions are so rarely stated properly. The problem then may become, "What is the proper context within which we can look at the mathematically verifiable properties of our program?" 


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Anup Rao is a double major in D-Math and CS. He received the Senior Prize in 2002. He writes:

I will be a graduate student here (masters) at Georgia Tech, with the college of computing, in the Fall. My long term plans are to get a Phd in theoretical computer science. My math background is a definite plus (and almost a prerequisite) for work in this area.

Over the summer I'm going to work with Dr. Ding at the CoC on some topics in coding theory and cryptography. 


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The topic of Joe Montgomery's REU is " Classification of varieties with degenerate Gauss maps."

Given a surface in Euclidean three space, one can define its Gauss map, taking a point to its normal line translated to the origin, or equivalently, a point to its tangent space in the Grassmanian. In the analytic category, surfaces with degenerate Gauss maps (that is, where the image is one-dimensional) have been completely classified, they are either cones over a point or the union of tangent lines to a curve. The same question in higher dimensions and codimensions is open. Montgomery will work on this open question, building on the classification results of Griffiths-Harris, and Akivis-Goldberg for complex subvarieties of projective space with degenerate Gauss maps. 


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Matthew Fisher is earning degrees in Math and CS. He writes: "After graduation, I will be taking a full time job with National Instruments in Austin, TX. I will be working in Image Algorithms (machine vision) Division."

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Keith Crunk, Amath major, is graduating and taking a position with Mirant Corporation here in Atlanta. He had Cooped with Mirant, then dropped the Coop in order to finish his degree more quickly.

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Erika Norenberg is also in the REU, with a topic of "Morse theory and the topology of algebraic varieties."

The classical Lefshetz theorem implies that much of the topology of a smooth hypersurface in projective space is inherited from the ambient projective space. One of its standard proofs uses classical Morse theory, where the topology of a manifold is studied via critical points of a sufficiently generic function on it. A more general theorem, due to W. Barth, was proved in the early 1970's stating that smooth varieties of small codimension also inherit much of their topology from the ambient projective space. Barth's proof is rather complicated, but recently there is a new proof, due to Schoen and Wolfson, based on ideas of Gromov, based on Morse theory in infinite dimensions. Norenberg will work through Milnor's classic book on Morse theory and the Schoen-Wolfson paper. If time allows, she will study additional recent work generalizing these results and calculate some new examples. 


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The ACE lab has three students and two professors involved in the REU program. The students are Ryan Hynd, Jeffery Elms, and Roberto Lopez. The supervising professors are Professors McCuan and Pelesko.

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Mailika Hines wanted to teach high school for a bit. She writes: "I will be teaching high school math with Dekalb County Schools. Although I do not have any teaching experience or certification, they hired me right on the spot at a teacher job fair in February."

Malika Hines graduated with honors, and also writes that she may be back in touch with questions about graduate school.