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VIGRE seminar, fall 2001: Gröbner Basis Theory and Applications

Elizabeth Arnold
Students enrolled
Erik Baumgarten, Kelli Carlson, Amy Collins, Cody Patterson (undergraduate mathematics students); Woonjung Choi, Teresa Guagliardo, Robert Main (graduate mathematics students)
Gröbner basis theory and the broader field of Computational Commutative Algebra and Algebraic Geometry are young, fast-growing, exciting fields of research. The applications of Gröbner bases are wide and varying. These include, but are not limited to: Commutative algebra and algebraic geometry, computer science, engineering, statistics, graph theory, combinatorics, robotics, wavelets, partial differential equations, automated theorem proving, etc. We explored some elementary applications of Gröbner bases including Elimination, Polynomial maps, Varieties, Field extensions, Graph coloring and Integer programming. Students did computations with a computer algebra system such as Maple or CoCoA. The first part of the seminar involved lectures on the basics of Gröbner basis theory and included a guest lecture by the founder of Gröbner basis theory and inventor of Buchberger's algorithm for computing Gröbner bases, Bruno Buchberger, from the Research Institute for Symbolic Computation in Linz, Austria. The second part involved presentations by the students on different applications of Gröbner bases. The third part consisted of lectures on more in-depth aspects of Gröbner basis theory, with written assignments requiring some research into the current literature. At the end of the seminar students presented their work on these problems.