Ziel
Research activities are related to the following topics.
Jacobian Conjecture for polynomial morphisms of affine spaces;
Algebraic group actions on affine space.
Dual varieties of orbits and generalised discriminants;
Standard monomial theory and geometry of Schubert varieties;
Constructive methods for invariant theory and representations;
Quantum groups with applying methods of algebraic transformation groups;
Representation and invariant theories of some remarkable classes of linear actions, and applications;
Algebraic geometric properties of algebraic group actions.
Objectives and expected results:
1. Investigating two representation theoretic approaches to the Jacobian Conjecture and related problems with a view towards proving Mathieu conjecture (at least for some groups) and the corresponding identity for the covariants;
2. Investigating nonreductive algebraic group actions on affine space with a view towards finding answers (presumably, negative) to the basic problems (e.g. cancellation problem);
3. Describing dual varieties of a class of orbits of linear actions and developing a generalization of the theory of discriminants of linear bundles over flag varieties;
4. Investigating some problems related to the general construction of a standard monomial theory for Schubert varieties and involved quantum groups at root of unity, with application to constructing deformations of Schubert varieties for all simple groups;
5. Developing methods of constructive Invariant theory with a view towards finding the improved general degree bounds, the relevant computer algorithms and implementations, and the specific explicit descriptions for binary forms, cyclic and symmetric groups;
6. Obtaining a canonical presentation by generators and relations for the affine coordinate algebra of a connected reductive algebraic group with a view towards finding its quantization in arbitrary characteristic;
7. Developing "quantized shuffle approach" to quantized Kac-Moody algebras and constructing PBW type bases for finite dimensional enveloping algebras at root of unity;
8. Investigating orbits and some problems of harmonic analysis for dual pairs, certain classes of prehomogeneous vector spaces, and gradings of Lie algebras;
9. Finding new stability criteria for some nonlinear actions;
10. Obtaining the combinatorial counterparts of the basic algebraic geometric properties for some natural classes of algebraic transformation spaces;
11. Classifying linear actions of connected simple algebraic groups such that the algebra of invariants is a complete intersection;
12. Proving equality of essential dimensions of an algebraic group and its Levi subgroup.
Programm/Programme
Aufforderung zur Vorschlagseinreichung
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4051 Basel
Schweiz