In Invariant Theory, one studies algebraic expressions which remain the same under spatial symmetries. Such algebraic expressions are called invariants. For example, the distance to the origin does not change under the rotation of the plane around the origin. The distance to the origin is an invariant. The PI will study a new approach to classical invariant theory. In collaboration with mathematicians and engineers at UIUC, the PI will study applications of theory of subspace arrangements to computer vision and image compression.
The PI will work on various other topics in commutative algebra, algebraic combinatorics, number theory. Cluster algebras were introduced by Fomin and Zelevinsky. These algebras have deep connections with the theory of quiver representations. The PI will explore this further in collaboration with Weyman and Zelevinsky. To a matroid or polymatroid one can associate various quasi-symmetric functions. The PI will study universal properties of such invariants.The PI will collaborate with Masser to find an effective version of the Mordell-Lang problem in positive characteristic.
Invariant Theory An invariant is a quantity that remains invariant under symmetry or certain equivalences. For example, If we rotate a point (x,y) in the plane around the origin, then the quantity x2+y2, the square of the distance to the origin, does not change. Several results in this project are related to invariants. universal invariants In earlier work the PI introduced functions on graphs that are invariant under graph isomorphism. Together with Alex Fink, he proved that these invariants are universal with respect to a certain "valuative property". This means that every graph invariant with the valuative property can be expressed in terms of the invariants defined by the PI. polynomial size invariants In his thesis, the graduate student Harlan Kadish showed that for a fixed given group of symmetries of a higher dimensional space, there exists a small list of fundamental constructible invariant functions of small size, such that every invariant constructible function can be expressed in these fundamental invariants. cluster algebras and quivers with potentials In the theory of cluster algebras certain properties, such as a rational function being a Laurent polynomial, are being preserved under certain operations called mutations. Quivers (=directed graphs) with potentials are related to cluster algebras and also String Theory in Theoretical Physics. In joint work with Jerzy Weyman and Andrei Zelevinsky, the PI showed that various quantities for quivers with potentials remain invariant under mutations. These results were used to prove some open conjectures in the theory of cluster algebras. The thesis of Jiarui Fei is also related to the representations of quivers with potentials and representations generally finite dimensional algebras. complexity of the graph isomorphism problem It is an open question in complexity theory, whether there exists an algorithm that can decide in polynomial time whether two graphs are isomorphic. The PI introduced an algorithm that can solve the isomorphism problem for for large classes of graphs in polynomial time. tensors One-dimensional arrays are vectors, two-dimensional arrays are matrices and higher dimensional analogs are called tensors. A fundamental problem is to write a given tensors as a sum of simple tensors. There are many practical applications, such as in psychometrics, chemometrics and machine learning. Kruskal gave a certain inequality under which the decomposition of a tensor as a sum of the fewest possible number of simple tensors is unique. The PI showed that this inequality cannot be weakened. Also, the PI described a connection between the tensor decomposition problem and the problem of recovering missing data in a matrix. recurrence sequences A recurrence sequence is a sequence that is given by a simple relation. For example, the Fibonacci sequence is given by F0=1, F1=1 and the recurrence relation Fn+1=Fn+Fn-1. The famous Skolem-Mahler-Lech theorem states that the set {n| Fn=0} is a union of a finite set and finitely many arithmetic infinite progressions. In earlier work, the PI give an analogous result for sequences that lie in a field of positive characteristic. In joint work with David Masser, the PI generalized this result in various directions.
