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Research Interests Please download my complete research statement (last updated 9 December 2006) in any of these formats: .dvi, .ps, .pdf. Preprints and Publications: Affine quasihomogeneous varieties of a reductive group, preprint. Note on the stability of principal bundles (with Donghoon Hyeon), Proc. Amer. Math. Soc. 132 (2004), 2205-2213. Equivariant Embeddings of Algebraic Groups, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2004. Research Related Awards: Irving Reiner Memorial Award in Algebra, Department of Mathematics, University of Illinois at Urbana-Champaign, 2002-2003 Undergraduate Research Award, Department of Mathematics and Statistics, Western Michigan University, 1995-1996 NSF Research Experience for Undergraduates, Department of Mathematics, Texas Christian University, Summer 1995 Past Research: Group theory - undergraduate REU research Representation theory - representations of Ga in positive characteristic Spherical varieties - computing cohomology and Picard groups for a family of complete spherical varieties Invariant theory - stability of principal bundles on Riemann surfaces, alternatives to GIT quotients using Hilbert schemes Current Research: Quasihomogeneous varieties for algebraic groups: Classification of affine group embeddings Cohomology for projective group embeddings Classifying affine quasihomogeneous G-varieties Geometric representation theory: Nilpotent cone of a symmetric space Desingularizations of nilpotent orbit closures Cohomology of Frobenius kernels and support varieties Future Research: relating the group embedding problem with representation theory relative cohomology of group embeddings with respect to the embedded group - generalizing results of Borel-Serre representation theory of algebraic groups, Lie algebras, Hecke algebras, etc. using algebraic and geometric methods noncommutative algebraic geometry, group actions on noncommutative spaces, noncommutative invariant theory classical invariant theory real algebraic geometry and its applications computer vision

Past Research:

As an undergraduate, I conducted research in graph theory and combinatorics at Western Michigan University under the supervision of Professor Yousef Alavi.  I had the pleasure of meeting and discussing my work with Paul Erdos, who annually visited the department to work with my advisor and others in the department.  I participated in the NSF Research Experience for Undergraduates at the Department of Mathematics, Texas Christian University, during the summer of 1995.  My research there studied the solvability of various types of sliding piece puzzles using group theory.  I presented my results at the Pi Mu Epsilon Student Seminar during the Joint AMS-MAA Meeting at the University of Washington, Seattle, in August 1996.

When I came to the University of Illinois for graduate school, I initially planned to study the representation theory of finite and algebraic groups (especially representations of G_a in characteristic p > 0).   This in turn led me to study multiplicity-free spaces, and hence spherical varieties.  In particular, I was interested in generalizing the computation of the cohomology groups of toric varieties and flag varieties to spherical varieties in general.  In the process, I investigated a special family of spherical varieties of the form (G x G) *_{B x B^-} Z = [ (G x G) x Z ] / (B x B^-), where G is a reductive algebraic group, B, B^- are opposite Borel subgroups of G and Z is a toric variety for the torus B \cap B^-.  After the successful completion of this task (computing the Picard group and cohomology groups of line bundles over such varieties), I sought to generalize these results to varieties of the form G *_B Y, where G and B are as before but Y is an embedding of the Borel subgroup B.  Such varieties are not spherical G-varieties, but they are compactifications of G whenever Y is a compactification of B.  Conversely, whenever X is a compactification of G, there is an associated compactification Y of B (take the closure of B in X) and we may then construct a new compactification G *_B Y of G and relate it to the original compactification X.

During this time, I collaborated with a fellow student, D. Hyeon (now an Assistant Professor of Mathematics at Northern Illinois University), to write the paper Note on the stability of principal bundles, which has been published in the Proceedings of the American Mathematical Society, 132 (2004), 2205-2213.  In this paper, we compare various definitions of stability for principal bundles and show that over a Riemann surface of genus > 2, there exist SL(2)-bundles that are Ad-stable.  A principal G-bundle E is said to be Ad-stable if the associated vector bundle Ad(E) = (E x g)/G is stable, where g denotes the Lie algebra of G on which G acts by the adjoint representation.  D. Hyeon previously used this notion of stability in the construction of a quasi-projective moduli scheme for principal bundles over a projective scheme of arbitrary dimension in his paper "Principal bundles over a projective scheme", Trans. AMS, 354 (2002), 1899-1908.

I have done work on a notion of stability and semi-stability for so-called Hilbert quotients.  Given a variety X with an action of an algebraic group G, the closures of G-orbits in X are parametrized by the Hilbert scheme of X (assuming it exists; e.g., X is projective).  For some sufficiently small open subset U of X, mapping a point x to the closure of its G-orbit, Gx, defines a morphism U --> Hilb(X/k).  The image of the closure of this morphism is independent of U and is called the Hilbert quotient of X by G.  My work was to find conditions describing the maximal open set U of X for which there is a quotient morphism from U to the Hilbert quotient similar to those Mumford employs to define the subsets of stable and semi-stable points of X with respect to the group G.

I participated in the Geometry and Algebra of Computer Vision RAP (Research Among Peers seminar) while I was a graduate student.   In this seminar, I presented work on the applications of invariant theory to computer vision and assisted Robert Fossum's undergraduate student as she studied computer vision under his supervision as part of the McNair Mentoring Program.

Current Research:

Presently I am working on the classification of compactifications of algebraic groups   -   varieties containing an open subset isomorphic to an algebraic group G such that the action of G on itself by left translations extends to an action on the whole variety.  The so-called "wonderful compactification" of a reductive group is a special case.  (For instance, the "wonderful compactification" of PGL_n is P(M_n), the projective space associated to the vector space of all n x n matrices, for all n > 1.)   The classification of "quasihomogeneous varieties" (i.e., G-varieties that possess an open G-orbit) by colored fans employed in the study of spherical varieties cannot be used for compactifications in general.   In place of this classification, developed by Luna and Vust in 1983, I am developing an alternative classification to distinguish distinct group compactifications as well as to describe, in an algebraic manner, how they can be constructed.

One-parameter subgroups play a critical role in the affine case, which is similar to their application to instability problems in geometric invariant theory. I am studying this connection as well as broadening my classification from the case of affine embeddings of linear algebraic groups to non-affine compactifications. Finally, I am following my classification strategy to also classify affine embeddings of homogeneous spaces G/K, where K is a reductive subgroup of G.

I am also presently engaged in a joint research project with Terrell Hodge, Western Michigan University, investigating orbits and orbit closures in the nullcone and restricted nullcone of Lie algebras in positive characteristic p less than the Coxeter number h.

Future Research:

In the future, I am interested in extending my classification to embeddings of homogeneous spaces in hopes of eliminating the complexity restrictions in the original work in this subject by Luna and Vust (1983).    Furthermore, I intend to study the group embedding problem in characteristic p > 0 and consider if and when group compactifications are Frobenius split.   I want to explore the connection between the group embedding problem and moduli of principal bundles as well as the representations of the group associated to its embeddings.   In this context, I wish to study representations of algebraic groups, and apply these results to the representation theory of Lie algebras, Hecke algebras, etc.   I also plan on investigating problems in classical, algebraic, and geometric invariant theory related to my past research involving G_a and Hilbert quotients.