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My research has been in the area of singularity theory and its application to nonlinear problems. It has specifically concerned :
1) the relation between smooth and topological stability of mappings
2) establishing the basic theorems of singularity theory (unfolding, determinacy theorems, and infinitesimal characterizations of local stability) for equivalences preserving additional structures for smooth, real analytic, or holomorphic mappings
3) establishing topological analogues of these theorems with applications to topological stability and equisingularity
4) applications to invariants and classification for bifurcation theory
5) local structure of nonlinear Fredholm operators
6) singular Milnor fibers and their applications for nonisolated complete intersections, including discriminants and nonlinear arrangements of hypersurfaces
7) determining the freeness of discriminants and bifurcations sets for versal unfoldings for general equivalence groups
8) solvable group representations yielding free divisors, and their application to determining the vanishing topology of nonisolated matrix singularities
9) Topology of exceptional orbit hypersurfaces of prehomogeneous spaces, including the varieties of m x m singular matrices, which are general, symmetric or skew-symmetric
10) Characteristic Cohomology of matrix singularities
11)singularity theory for solutions to PDE's with applications to computer medical imaging
12) scale-based methods for computer imaging
13) local and relative geometry of objects and their boundaries from medial data; global geometry via skeletal and medial integrals, and characterizing complexity of 3D regions via graph structures
14) the analysis of multi-object configurations via medial/skeletal linking structures capturing both shape and geometry of individual objects and the positiona lgeometry of the configuration
15) determining the evolving self-intersections of evolving surfaces defined by splines and the application to computing medial axes for regions defined by splines
16) using singularity theory for mappings on special semianalytic stratifications to determinine the local structure in natural images allowing shade/shadow, geometric features, and apparent contours, for both stable views and transitions under viewer movement.
1) the relation between smooth and topological stability of mappings
2) establishing the basic theorems of singularity theory (unfolding, determinacy theorems, and infinitesimal characterizations of local stability) for equivalences preserving additional structures for smooth, real analytic, or holomorphic mappings
3) establishing topological analogues of these theorems with applications to topological stability and equisingularity
4) applications to invariants and classification for bifurcation theory
5) local structure of nonlinear Fredholm operators
6) singular Milnor fibers and their applications for nonisolated complete intersections, including discriminants and nonlinear arrangements of hypersurfaces
7) determining the freeness of discriminants and bifurcations sets for versal unfoldings for general equivalence groups
8) solvable group representations yielding free divisors, and their application to determining the vanishing topology of nonisolated matrix singularities
9) Topology of exceptional orbit hypersurfaces of prehomogeneous spaces, including the varieties of m x m singular matrices, which are general, symmetric or skew-symmetric
10) Characteristic Cohomology of matrix singularities
11)singularity theory for solutions to PDE's with applications to computer medical imaging
12) scale-based methods for computer imaging
13) local and relative geometry of objects and their boundaries from medial data; global geometry via skeletal and medial integrals, and characterizing complexity of 3D regions via graph structures
14) the analysis of multi-object configurations via medial/skeletal linking structures capturing both shape and geometry of individual objects and the positiona lgeometry of the configuration
15) determining the evolving self-intersections of evolving surfaces defined by splines and the application to computing medial axes for regions defined by splines
16) using singularity theory for mappings on special semianalytic stratifications to determinine the local structure in natural images allowing shade/shadow, geometric features, and apparent contours, for both stable views and transitions under viewer movement.
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Frontiers in Computer Science (2022)
Bulletin of the American Mathematical Societyno. 3 (2022): 453-469
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Medical image analysis (2021): 102020-102020
Riemannian Geometric Statistics in Medical Image Analysispp.233-271, (2020)
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