B.A. Dartmouth 1967
Diploma in Advanced Math Oxford 1969
Ph.D. Harvard 1972 (Adviser : John Mather)
My office is in Phillips Hall 300. My office phone number is: (919) 962-9617.
You can also email me at email@example.com.
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 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)singularity theory for solutions to PDE's with applications to computer medical imaging
11) scale-based methods for computer imaging
12) 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
13) 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
14)  determining the evolving self-intersections of evolving surfaces defined by splines and the application to computing medial axes for regions defined by splines
15)  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.
joint with P.Giblin and G. Haslinger, "Characterizing Stable Local Features of Illuminated Surfaces and Their Generic Transitions from Viewer Movement" 215 pages, being submitted for publication
"Scale-based Geometry for Nondifferentiable Functions, Measures, and Distributions", parts I-III, preliminary version
joint with E. Gasparovic, "Modeling Multi-object Configurations via Medial/Skeletal Linking Structures" submitted for publ.
joint with E. Gasparovic, "Shape and Positional Geometry of Multi-Object Configurations" preprint to be submitted for publ.
joint with J.S. Marron, "Backwards Principal Component Analysis and Principal Nested Relations" Jour. Math. Imag. Vision 50 (2014) 107-114
joint with S. Pizer,D. Goswami, J. Vicory, R. Chaudhuri, X. Zhao, S. Huckemann, S. Marron, "Nested Sphere Statistics of Skeletal Models" Innovations for Shape Analysis, Eds. M. Breuss, A. Bruckstein, and P. Maragos, Mathematics and Visualization series, Springer-Verlag (2013) 93-115.
joint with P.Giblin and G. Haslinger,"Local Image Features Resulting from 3-Dimensional Geometric Features, Illumination, and Movement:I" IJCV vol 82(1) (2009) 25-47
joint with P.Giblin and G. Haslinger,"Local Image Features Resulting From 3-Dimensional Geometric Features, Illumination, and Movement: II" SIAM Jour. Imaging Sciences vol 4. no 1 (2011) 386-412
Addenda to "Local Image Features
Resulting From 3-DimensionalGeometric Features, Illumination, and Movement: II"
Visibility Diagrams for Corner Transitions; Animated Corner Transitions
joint with S. Musuvathy and E. Cohen,"Computing the Medial Axis of Generic 3D Regions Bounded by B-spline Surfaces" Computer-Aided Design 43(11) (2011) 1485-1495
joint with S. Musuvathy, J-K. Seong, and E. Cohen,"Principal Curvature Ridges and Geometrically Salient Regions of Parametric B-spline Surfaces", Computer- Aided Design 43(7) (2011) 756-770
joint with S. Musuvathy, J-K. Seong, and E. Cohen,"Tracing Ridges on B-spline Surfaces", Proc. 2009 SIAM/ACM Conf. Geom. and Phys. Modeling, ACM (2009) 55-66
"Geometry and Medial Structure", chapter of book, Eds. S. Pizer and K. Siddiqi, "Medial Representations: Mathematics, Algorithms, and Applications", Vol. 37 Springer/Kluwer series Comp. Imag. and Vision, Springer-Verlag 2008
"Tree Structure for Contractible Regions in R^3", IJCV 74 no. 2 (2007) 103-116,
joint with Qiong Han, D. Merck, J. Levy, C. Villarruel, E. L. Chaney, and S. M. Pizer"Geometrically Proper Models in Statistical Training", Proc. IPMI, LNCS 4584 (2007) 751-762
joint with T. Terriberry, S. Joshi, G. Gerig, and S. M. Pizer, "Population Based Fitting of Medial Shape Models with Correspondence Optimization", Proc. IPMI, LNCS 4584 (2007) 700-712
joint with X. Chen, E. Cohen, and R. Riesenfeld, "Theoretically based Algorithms for Robustly Tracking Intersection Curves of Deforming Surfaces", Computer-Aided Design 39 No. 5 (2007) 389-397
joint with Q. Han and S. Pizer "Interpolation in Discrete Single Figure Medial Objects", Proc. MMBIA '06 (2006) 85 - 92
joint with X. Chen, E. Cohen, and R. Riesenfeld, "Theoretically based Algorithms for Robustly Tracking Intersection Curves of Deforming Surfaces", Proc. GMP '06 Springer LNCS 4077 (2006) 101-114.
"Determining the Geometry of Boundaries of Objects from Medial Data ", IJCV 63 no. 1 (2005) 45-64
joint with S. Pizer, K. Siddiqi, G. Szekely, and S. Zucker, "Multiscale Medial Loci and Their Properties", IJCV 55 no. 2-3 (2003), 155-179
joint with P. Dimitrov and K. Siddiqi, "Flux Invariants for Shape", Proc. CVPR (2003) 1063-1069
"Ridges and Cores for Two Dimensional Images", Jour. Math. Image and Vision 10, (1999) 163-174.
"Local Morse theory for Gaussian blurred functions", in "Gaussian Scale Space Theory ", Ed. J. Sporring et al., Kluwer Series in Comp. Science and Vision, Vol. 8, Kluwer Acad. Publ., (1997) 147-163.
"Semi-Coherence for SemiAnalytic Sets and Stratifications and Singularity Theory of Mappings on Stratifications" Proc. CIRM Conf. Geom. and Top. of Sing. Spaces, Jour. of Singularities vol 13 (2015) 42-56joint with B. Pike, "Solvable Groups, Free Divisors, and Nonisolated Matrix Singularites I: Towers of Free Divisors" Annales Inst. Fourier 65 no.3 (2015) 1251-1300. joint with B. Pike, "Solvable Groups, Free Divisors, and Nonisolated Matrix Singularites II: Vanishing Topology" Geometry and Topology 18 no. 2 (2014) 911-962.
joint with B. Pike, "Solvable Group Representations and Free Divisors whose Complements are K(\pi, 1)'s" Proc. Valencia Conf. Generic Geom.in Top. and its Appl.159 (2012) 437-449
"Swept Regions and Surfaces: Modeling and Volumetric Properties", Conf. Comput. Alg. Geom. in honor of Andre Galligo, Theor. Comp. Science 392 (2008) 66-91
"Global Geometry of Regions and Boundaries via Skeletal and Medial Integrals",Comm. Anal. and Geom.15 No.2 (2007) 307-358
Editor (joint with Jean Paul Brasselet, Le Dung Trang, and Mutsuo Oka)"Singularities in Geometry and Topology",Proc. 2005 ICTP Trieste Singularity Summer School and Workshop, World Scientific Publ. (2007)
"Global Medial Structure of Regions in R^3", Geometry and Topology Vol 10 (2006) 2385- 2429
"On the Legacy of Free Divisors III : Functions and Divisors on Complete Intersections", Oxford Quart. Jour. Math. 57 (2006) 49-79
"On the Smoothness and Geometry of Boundaries Associated to Skeletal Structures II: Geometry in the Blum Case ", Compositio Mathematica 140 no. 6 (2004) 1657-1674
"On the Legacy of Free Divisors II : Free* Divisors and Complete Intersections", Moscow Math. Jour. 3, Special Issue in honor of V. I. Arnol'd (2003) 361-395
"On the Smoothness and Geometry of Boundaries Associated to Skeletal Structures I: Sufficient Conditions for Smoothness ", Annales Inst. Fourier 53 (2003), 1941-1985
"On the Freeness of Equisingular Deformations of Plane Curve Singularities", Arrangements in Boston: Special Issue, Eds. D Cohen and A. Suciu, Top. and its Appl. 118 (2002), 31-43
"Nonlinear Sections of Nonisolated Complete Intersections", New Developments in Singularity Theory, Eds. D. Siersma et al, NATO Conf. Series vol 21, Kluwer Pub.l (2001), 405-445
""On the Number of Bounding Cycles for Nonlinear Arrangements", Arrangements-Tokyo-1998, Adv. Stud. Pure Math 27 (2000) 51-72.
"Critical points of Affine Multiforms on the Complements of Arrangements", Singularity Theory, Ed J.W.Bruce and D. Mond. Lond.Math. Soc. Lecture Notes, 263 (1999) Cambridge Univ. Press, 25-53.
"A Global Weighted Version of Bezout's Theorem ", "Arnol'd Fest", Fields Communications Series 24 (1999) 115-129
"On the Legacy of Free Divisors: Discriminants and Morse Type Singularities", Amer. Jour. Math. 120, 1998, 453-492.
"Generic Structure of Two Dimensional Images under Gaussian Blurring", SIAM Jour. Appl. Math 59, 1998, 97-138.
"Singularities with Scale Threshold and Discrete Functions Exhibiting Generic Properties", Proc. Int. Workshop on Real and Complex Singularities, Sao Carlos, Maria Ruas Ed., Math. Contemp. vol 12, 1997, 45-65
" Generic properties of solutions to partial differential equations", Arch. Rat. Mech. Anal. 140 (1997), 353-403.
" Singular Milnor fibers and higher multiplicities for nonisolated complete intersections", Int'l Sem. Sing. and Complex Geometry, AMS-IP Studies Adv. Math., Vol. 5 (1997), 28-53.
"Higher multiplicities and almost free divisors and complete intersections", Memoirs of the AMS, vol.123, no. 589 (1996)
"Applications of singularity theory to the solutions of nonlinear equations ", in Topological Nonlinear Analysis : Degree, Singularity and Variations, ed. M. Matzeu and A. Vignoli, Prog. Nonlinear Differential Equatiions and Applications, Vol.15 (1995), Birkhauser, 178-302.
" A Bezout theorem for determinantal modules ", Compositio Math. 98 (1995), 117-139.
Here is a Complete publication list
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Last modified: 1 September 2015