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VIGRE seminar, Fall 2002: Introduction to Computational Algebraic Geometry

Maurice Rojas
Students enrolled:
Maria Kobiela and Ryan Thomas (undergraduates). Marvin Decker, Jyh-Ming Lien, Suhir Pargaonkar, Swaminatha Sethuraman, Jody Wilson, Zhigang Zhang (graduate students).

Algebraic geometry was born over two millenia ago, with the aim of understanding how to solve systems of polynomial equations. Ironically, algebraic geometry courses nowadays rarely talk about the most efficient way to solve a polynomial system you may encounter in practice. The need for robust and efficient algorithms for algebraic geometry is now motivated by numerous applications: even a short list would have to include chemistry, signal processing, robotics, coding theory, optimization, mathematical biology, computer vision, game theory, and statistics.

This course takes a novel --- albeit 2000 year old --- approach by focusing on the mathematics most relevant to solving polynomial systems. The course is geared toward upper division students in mathematics, computer science, and engineering, but with plenty of topics that graduate students can pursue in greater depth. The emphasis will be on concrete constructions and algorithms, so a strong algebra background is not required --- any necessary algebra will be developed as needed.