Description

Invariant Theory has gone through many ups and down throughout its two centuries history. Computation was the main focus until Hilbert, who was the first to prove that the invariants of the classical groups are finitely generated. These works feature such results as the Nullstellensatz and the Basis Theorem, and were a crucial step in the development of modern Commutative Algebra and Algebraic Geometry. After leaving the spotlight for a few decades, Invariant Theory came back in force in the middle of the twentieth century and greatly benefited from the advances made by Commutative Algebra and Algebraic Geometry.

Apart from being a source of interesting and difficult questions on its own as well as an invaluable source of examples for testing all sorts of mathematical theories, Invariant Theory has a wide range of applications, such as: Coding Theory, Equivariant Dynamical Systems, Material Sciences, Computer Vision, Galois Group Computations, Systems of Algebraic Equations with Symmetries, Graph Theory, Combinatorics, …

After spending the first term learning the basics of Invariant theory using the text of Neusel as a starting point. Each student will select a particular application to explore further. Learning more advanced topics in Invariant Theory as needed.

Resources

[1] Harm Derksen and Gregor Kemper, Computational invariant theory. Invariant Theory and Algebraic Transformation Groups, I. Encyclopaedia of Mathematical Sciences, 130. Springer-Verlag, Berlin, 2002.

[2] Mara Neusel, Invariant theory. Student Mathematical Library, 36. American Mathematical Society, Providence, RI, 2007.

[3] Bernd Sturmfels, Algorithms in invariant theory. Second edition. Texts and Monographs in Symbolic Computation. SpringerWienNewYork, Vienna, 2008.