CAREER: Invariant Theory, Algorithms and Applications

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DMS-0349019Harm DerksenThis is a CAREER award investigating Invariant Theory, Algorithms, and Applications. The PI will build up a research group working on algorithms in algebra. He will work on algorithms for various difficult problems in algebra and the connection between complexitytheory and Invariant Theory. The PI will work with Jessica Sidman on Castelnuovo-Mumford regularity and applications to various conjectures. The PI will work with Jerzy Weyman on quiver representations, invariant theory and applications. They also plan to write an introductory book on quiver representations. A graduatestudent will work on the connection between semi-invariants for quivers, combinatorics and exact sequences of abelian p-groups. In addition, the PI will also work on the new notion of "black box algebras". These algebras can be used to describe classical invariant theory. There are many interesting applications, such as PI-theory, and universal formulas for decompositions of tensor products of representations of simple lie algebras. Courses will be developed in the area of algorithms in algebra.In Invariant Theory one studies algebraic quantities which remain unchanged after certain transformations. For example, consider a 3-dimensional space. Consider a point (x,y,z) under rotation around the z-axis. Two fundamental invariants are the distance to the z-axis, and the z-coordinate. These quantities remain unchanged after any rotation around the z-axis. Every algebraic invariant can be expressed in terms of these two fundamental invariants. A famous theorem of Hilbert from 1890 tells us that given a collection of transformations (satisfying certain hypotheses) of a space of arbitrary dimension, there exists a finite set of fundamental invariants such that every invariantcan be expressed in these fundamental invariants. This proposalsupports the study of applications of Invariant Theory to several areasin mathematics. It also supports research on algorithms forvarious problems in Invariant Theory.