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0 -12 Td (In the paper 'Topology of negatively curved real algebraic surfaces in R^3', which is joint work with C. Connell, we find quartic examples of) Tj
0 -12 Td (smooth embedded negatively curved surfaces in R^3 homeomorphic to singly and doubly punctured tori, and a triply punctured sphere. These) Tj
0 -12 Td (constitute explicit solutions to Hadamard's problem on constructing complete surfaces with negative curvature and Euler characteristic in R^3.) Tj
0 -12 Td (Further we show that our solutions have the optimal degree of algebraic complexity via a topological classification for smooth cubic surfaces) Tj
0 -12 Td (with a negatively curved component in R^3: any such component must either be topologically a plane or an annulus. In particular, there exists) Tj
0 -12 Td (no cubic solutions to Hadamard's problem.\r) Tj
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-520 723 Td (\r) Tj
0 -12 Td (In the paper on 'Topology of Riemannian submanifolds with prescribed boundary, which is joint work with Stephanie Alexander and Jeremy) Tj
0 -12 Td (Wong, We prove that a smooth compact immersed submanifold of codimension 2 in R^n, n>2, bounds at most finitely many topologically) Tj
0 -12 Td (distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous) Tj
0 -12 Td (results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness) Tj
0 -12 Td (theorems may not hold if the codimenion is too high, or the prescribed boundary is not sufficiently regular.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In the paper on 'Relative isometric embedding of Riemannian manifolds, which is joint work with Rober Greene, we prove the existence of) Tj
0 -12 Td (C^1 isometric embeddings, and smooth approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed) Tj
0 -12 Td (values in a neighborhood of a point.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In the paper 'A Riemannian four vertex theorem for surfaces with boundary' PI proves that every metric of constant curvature on a compact) Tj
0 -12 Td (surface M with boundary bdM induces at least four vertices, i.e., local extrema of geodesic curvature, on a connected component of bdM, if,) Tj
0 -12 Td (and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of) Tj
0 -12 Td (constant curvature on M which induce only two critical points of geodesic curvature on each component of bdM. With few exceptions, these) Tj
0 -12 Td (metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces. \r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Another paper on four vertex theorems is 'Vertices of closed curves in Riemannian surfaces', were the PI uncovers some connections between) Tj
0 -12 Td (the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed) Tj
0 -12 Td (curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve) Tj
0 -12 Td (has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve) Tj
0 -12 Td (bounding a compact immersed surface has more than two vertices. \r) Tj
/SerifB 10 Tf
0 -35 Td (Training and Development:) Tj
/Serif 10 Tf
0 -14 Td (PhD Student:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Ioulia Tiourina completed her PhD thesis in the Summer of 2005 on skew surfaces in R^4.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Jim Krysiak is currently working on Alexnadrov's conjecture.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Postdoc:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Gordanna Stoyanovic, 2007-2009 at Georgia Tech, who worked on the skew subamanifolds and embeddings.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Teaching Assistants:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Erik Van Erp has gained more experience in discussing differential geometry problems with undergraduate students. He assisted the PI in) Tj
0 -12 Td (teaching an undergraduate class on differential geometry.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Robert Yuncken assisted the PI in teaching an undegraduate differential geometry class, as a part of PSU's MASS program, in the Fall of '04.) Tj
0 -12 Td (He attended all lectures and wrote solutions to homework problems.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (REU and Undegraduate Students:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (During Penn State's REU program in the Summer of '04 the PI helped supervise or provide support for the research of several students:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Casey Bush \(PSU\) and Toan Phan \(Bucknell\) worked jointly on the problem of rigidity for closed surfaces.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Zachary McCoy \(Yale\) and James Krysiak \(Alfred University\) worked on Alexandrov's conjecture on the diameter of convex surfaces.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Dmitry Vaintrob \(University of Oregon\) worked on the Bicycle problem.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Christopher Cutler \(Berkeley\) worked on outer billiards.\r) Tj
0 -12 Td (\r) Tj
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-520 723 Td (In the Fall of '04, the PI taught the class on Differential Geometry during the Penn State's MASS program and supervised the following) Tj
0 -12 Td (research projects:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Jenna Hammang \(Valparaiso U.\), Distortion of Curves.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Rob King \(Clark U.\), Geometry of minimal surfaces bounded by convex planar curves.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Sarah Mall \(St. Edwards's U.\), the DNA inequality.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Casey Bush \(PSU\), Schwartz-Child solution in General Relativity.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (James Krysiak \(Alfred U.\), Alexandrov's conjecture on degenerate surfaces and surfaces of revolution.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Zachary McCoy \(Yale\), Alexandrov's conjecture for convex polytopes with few sides.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Van Cyr \(U. Buffalo\), the bicycle problem.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (At Georgia Tech, the PI supervised the research of AJ Friend on the unfolding problem for convex polytopes.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Further, he supervised two REU students in the Summer of 2006: Brian Nakamura and Bobby Demarco.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Currently, the PI is supervising a PhD student, Jim Krysiak, at Georgia Tech.) Tj
/SerifB 10 Tf
0 -35 Td (Outreach Activities:) Tj
/Serif 10 Tf
0 -14 Td (The PI presented a talk on 'Solution to the shadow problem' in the MASS colloquium at Penn State University in the Fall of 2003. MASS is a) Tj
0 -12 Td (semester long program which brings talented undergraduates to Penn State each year for intensive study of Mathematics.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In the Summer of 2004, the PI served as an advisor to several REU students, including one from high school, and also gave a talk at Penn States) Tj
0 -12 Td (MASSfest conference for REU students.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In the Fall of 2004, the PI taught a class on differential geomety as a part of Penn State's MASS program.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Since arriving at Georgia Tech, in the Fall of 2005, the PI has been organizing a series of mathematical talks aimed at a general audience,) Tj
0 -12 Td (especially the undergraduate students, the first three speakers in this series have been Tom Banchoff and John Sullivan, and John Morgan. Each) Tj
0 -12 Td (event attracted about 100 audience members from the entire campus.) Tj
/SerifB 10 Tf
250 -33 Td (Journal Publications) Tj
/Serif 10 Tf
-250 -29 Td (A. Abrams, J. Cantarella, J. Fu, M. Ghomi, and R. Howard, "Circles minimize most knot energies", Topology, p. 381, vol. 42, \(2003\).) Tj
0 -12 Td (Published, ) Tj
0 -28 Td (S. Alexander and M. Ghomi, "The convex hull property and topology of hypersurfaces with nonnegative curvature", Advances in) Tj
0 -12 Td (Mathematics, p. 324, vol. 180, \(2003\). Published, ) Tj
0 -28 Td (M. Ghomi, "A smooth convex loop with vanishing projections", Topology, p. 245, vol. 43, \(2004\). Published, ) Tj
0 -27 Td (M. Ghomi, "Optimal smoothing for convex polytopes", Bulletin of London Mathematical Society, p. 483, vol. 36, \(2004\). Published, ) Tj
0 -27 Td (M. Ghomi, "Shortest periodic billiard trajectories", Geometric and Functional Analysis \(GAFA\), p. 295, vol. 14, \(2004\). Published, ) Tj
0 -27 Td (S. Alexander and M. Ghomi, "The convex hull property of noncompact hypersurfaces with positive curvature", American Journal of) Tj
0 -12 Td (Mathematics, p. 891-897, vol. 126, \(2004\). Published, ) Tj
0 -28 Td (M. Ghomi, "Tangent bundle embeddings of manifolds in Euclidean space", Comment. Math. Helv., p. 259, vol. 81, \(2006\). Published, ) Tj
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-520 723 Td (M. Ghomi and S. Tabachnikov, "h-Principles for hypersurfaces with Prescribed principal curvatures and directions", Tran. Amer. Math. Soc., p.) Tj
0 -12 Td (129, vol. 358, \(2006\). Published, ) Tj
0 -28 Td (M. Ghomi and S. Tabachnikov, "Totally skew embedding of manifolds", Math. Z., p. , vol. 258, \(2008\). Published, ) Tj
0 -27 Td (M. Ghomi, "Nonexistence of skew loops on ellipsoids", Proc. Amer. Math. Soc, p. 3687, vol. 133, \(2005\). Published, ) Tj
0 -27 Td (J. Choe, M. Ghomi, and M. Ritore, "Total positive curvature of hypersurfaces with convex boundary", J. Differential Geom., p. 129, vol. 72,) Tj
0 -12 Td (\(2006\). Published, ) Tj
0 -28 Td (J. Choe, M. Ghomi, M. Ritore, "Relative isoperimetric inequality outside convex domains in R^n", Calc. Var. Partial Differential Equations, p.) Tj
0 -12 Td (42, vol. 29, \(2007\). Published, ) Tj
0 -28 Td (M. Ghomi, "Topology of Surfaces with Connected Shades", Asian J. Math, p. , vol. , \( \). Accepted, ) Tj
0 -27 Td (M. Ghomi, "h-Principles for curves and knots of constant curvature", Geom. Dedicata, p. , vol. , \(2008\). Accepted, ) Tj
0 -27 Td (C. Connell, and M. Ghomi, "Topology of negatively curved real algebraic surfaces in R^3", J. Reine Angew. Math., p. 1, vol. 624, \(2008\).) Tj
0 -12 Td (Published, ) Tj
0 -28 Td (S. Alexander; M. Ghomi; J. Wong, "Topology of Riemannian submanifolds with prescribed boundary", Preprint, p. , vol. , \(2008\). Preprint, ) Tj
0 -27 Td (M. Ghomi; R. Greene, "Relative isometric embeddings of Riemannian manifolds", To appear in Transactions of American Mathematical) Tj
0 -12 Td (Society, p. , vol. , \(2008\). Accepted, ) Tj
0 -28 Td (Ghomi, Mohammad, "A Riemannian four vertex theorem for surfaces with boundary", Submitted to Journal of Differential Geometry, p. , vol. ,) Tj
0 -12 Td (\(2009\). Submitted, ) Tj
0 -28 Td (Ghomi, Mohammad, "Vertices of closed curves in Riemannian surfaces", Submitted to Journal of Topology, p. , vol. , \(2009\). Published, ) Tj
/SerifB 10 Tf
205 -43 Td (Books or Other One-time Publications) Tj
45 -45 Td (Web/Internet Site) Tj
-250 -30 Td (URL\(s\):) Tj
/Serif 10 Tf
0 -14 Td (www.math.gatech.edu/~ghomi) Tj
/SerifB 10 Tf
0 -13 Td (Description:) Tj
/Serif 10 Tf
0 -14 Td (This site contains the preprints for the PI's papers as well as links to websites which the PI set up for his graduate and undergraduate classes.) Tj
0 -12 Td (These websites contain lecture notes and Mathematica software packages written by the PI.) Tj
/SerifB 10 Tf
240 -29 Td (Other Specific Products) Tj
-240 -30 Td (Product Type:) Tj
0 -15 Td (Software \(or netware\) ) Tj
0 -15 Td (Product Description:) Tj
/Serif 10 Tf
0 -14 Td (Mathematica software packages for exploring differential geometry of curves and surfaces. There are three main ones on planar curves, space) Tj
0 -12 Td (curves, and surfaces. The one on planar curves is close to its final form, and the rest will be refined further. Each notebook contains dozens of) Tj
0 -12 Td (programs for computing various geometric quantities or generating graphs.) Tj
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0 -14 Td (Sharing Information:) Tj
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-520 723 Td (These software packages are currently available freely on line at:\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (www.math.gatech.edu/~ghomi) Tj
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250 -25 Td (Contributions) Tj
-250 -15 Td (Contributions within Discipline: ) Tj
/Serif 10 Tf
0 -14 Td (The PI's work on 'Prescribing principal curvatures and directions of hypersurfaces with boundary' uses the h-principle of Gromov -- a) Tj
0 -12 Td (relatively recent development in differential geometry and topology -- to generalize some theorems of Gluck and Pan on the structure of) Tj
0 -12 Td (positively curved surfaces with boundary in Euclidean space. It is hoped that this result will not only encourage further study of the h-principle,) Tj
0 -12 Td (but also prompt more people to search for applications of recent techniques to classical problems in differential geometry.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (The work on 'Skew submanifolds' is concerned with a Whitney type topological problem \(finding the least codimension for an embedding of a) Tj
0 -12 Td (manifold in Euclidean space\) in the setting of affine or projective geometry \(via the requirement that the embedding have no pairs of parallel or) Tj
0 -12 Td (intersecting tangent lines\). This is a new area of research initiated by the PI and his collaborators, Bruce Solomon and Serge Tabachnikov. This) Tj
0 -12 Td (field contains many interesing open problems which provide opportunity for interaction between topology, differential geometry, and algebraic) Tj
0 -12 Td (geometry.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (The two papers which have been written in collaboration with J. Choe and M. Ritore are concerned with a generalization of the classical) Tj
0 -12 Td (isoperimetric inequality, which states that the sphere is the surface of least area bounding a given volume. Here we obtain an analogous result) Tj
0 -12 Td (for surfaces which trap a given volume outside a convex body. Study of isoperimetric inequalities has been a very active area of research in) Tj
0 -12 Td (geometric analysis ever since this field has existed, and has been directly or indirectly responsible for a development of many techniqes which) Tj
0 -12 Td (have found application elsewhere. \r) Tj
0 -12 Td (\r) Tj
0 -12 Td (PI's work on shades on illuminated surfaces answers a fundamental question in this are: when does a surface have connected shades? This field) Tj
0 -12 Td (is of interest in computer vision where 'shape from shading' problems are studies, and also has applications in geometric variational problems;) Tj
0 -12 Td (specifically, the stability of contant mean curvature surfaces, or soap films.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (Another work on h-principle in differential geometry is on knots of constant curvature where the PI proves that any curve in Euclidean space) Tj
0 -12 Td (may by c^1-approximated by a curve of constant curvature. Thus one could say that there exists no 'visual difference' between an arbitrary) Tj
0 -12 Td (smooth curve in R^3 and one whose curvature is constant. More importantly, the PI obtains a sharp estimate for the curvature of the) Tj
0 -12 Td (approximating curve. \r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In the joint work with Chris Connell, the PI explores the relation between differential geometry and classical algebraic geometry by studying) Tj
0 -12 Td (Hadamard's problem on constructing complete surfaces of negative curvature in R^3 with nontrivial topology. In particular we construct) Tj
0 -12 Td (explicit examples of quartic surfaces with negative curvature and Euler characteristic and show that there exist no such cubic surfaces.\r) Tj
0 -12 Td (\r) Tj
0 -12 Td (In joint work with Stephanie Alexander and Jeremy Wong we use methods from the theorem of Alexandrov spaces with curvauture bounded) Tj
0 -12 Td (below to answer affirmatively a question of Guan and Spruck on finiteness of topological types of positively curved hypersurfaces in Euclidean) Tj
0 -12 Td (space bounded by a given boundary. This question is related to a problem if Yau: when does a curve in Euclidean 3-space bound a surface of) Tj
0 -12 Td (positive curvature. Further, to prove our main results here, we develop a relative version of Nash's isometric embedding theorem in joint work) Tj
0 -12 Td (with Robert Greene, which proved the existence of isometric embeddings with prescribed values in a neighborhood of a point. This result is) Tj
0 -12 Td (used to show that the finiteness theorem mentioned above does not hold when the codimension is too high.) Tj
/SerifB 10 Tf
0 -46 Td (Contributions to Other Disciplines: ) Tj
/Serif 10 Tf
0 -14 Td (The PI's work on Shades or Shadows on illuminated surfaces are of interest in computer vision, where the 'shape from shading' problems are) Tj
0 -12 Td (studied.) Tj
/SerifB 10 Tf
0 -19 Td (Contributions to Human Resource Development: ) Tj
/Serif 10 Tf
0 -14 Td (As described elsewhere in this report, the PI has directed the research of graduate, undergraduate, and high school students.) Tj
/SerifB 10 Tf
0 -18 Td (Contributions to Resources for Research and Education: ) Tj
/Serif 10 Tf
0 -14 Td (The PI has developed several Mathematica software packages for studying curves and surfaces. These are freely available on the PI's website.) Tj
/SerifB 10 Tf
0 -18 Td (Contributions Beyond Science and Engineering: ) Tj
/Serif 10 Tf
0 -14 Td ( ) Tj
/SerifB 10 Tf
250 -28 Td (Conference Proceedings) Tj
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