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Given n general points p_1, p_2, \\ldots , p_n \\in \\mathbb P^r, it is natural to ask when there exists a curve C \\subset \\mathbb P^r, of degree d and genus g, passing through p_1, p_2, \\ldots , p_n. In this paper, the authors give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N_C of a general nonspecial curve of degree d and genus g in \\mathbb P^r (with d \\geq g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H^0(N_C(-D)) = 0 or H^1(N_C(-D)) = 0), with exactly three exceptions.

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian These results are obtained by interpreting the Schubert bases of Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.

This volume covers a broad range of subjects in modern geometry and related branches of mathematics, physics and computer science. Most of the papers show new, interesting results in Riemannian geometry, homotopy theory, theory of Lie groups and Lie algebras, topological analysis, integrable systems, quantum groups, and noncommutative geometry. There are also papers giving overviews of the recent achievements in some special topics, such as the Willmore conjecture, geodesic mappings, Weyl's tube formula, and integrable geodesic flows. This book provides a great chance for interchanging new results and ideas in multidisciplinary studies.The proceedings have been selected for coverage in:• Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)• CC Proceedings — Engineering & Physical Sciences

This volume covers a broad range of subjects in modern geometry and related branches of mathematics, physics and computer science. Most of the papers show new, interesting results in Riemannian geometry, homotopy theory, theory of Lie groups and Lie algebras, topological analysis, integrable systems, quantum groups, and noncommutative geometry. There are also papers giving overviews of the recent achievements in some special topics, such as the Willmore conjecture, geodesic mappings, Weyl's tube formula, and integrable geodesic flows. This book provides a great chance for interchanging new results and ideas in multidisciplinary studies.

This book is the sequel to Geometric Transformation I, which appeared in this series in 1962. Part 1 treats length-preserving transformation (called isometries); this volume treats shape-preserving transformations (called similarities); and Part III treats affine and protective transformations. These classes of transformation play a fundamental role in the group-theoretic approach to geometry.As in the previous volume, the treatment is direct and simple. The introduction of each new idea is supplemented by problems whose solutions employ the idea just presented, and whose detailed solutions are given in the second half of the book.

The familiar plane geometry of high school - figures composed of lines and circles - takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24. Part I treats the geometry of rigid motions of the plane (isometries); Part II treats the geometry of shape-preserving transformations of the plane (similarities); Part III treats the geometry of transformations of the plane that map lines to lines (affine and projective transformations) and introduces the Klein model of non-Euclidean geometry. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincaré model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in high-school geometry and trigonometry. Numerous exercises lead the reader to a mastery of the methods and concepts. The second half of the book contains detailed solutions of all the problems.

The authors provide a complete classification of globally generated vector bundles with first Chern class $c_1 \\leq 5$ one the projective plane and with $c_1 \\leq 4$ on the projective $n$-space for $n \\geq 3$. This reproves and extends, in a systematic manner, previous results obtained for $c_1 \\leq 2$ by Sierra and Ugaglia [J. Pure Appl. Algebra 213 (2009), 2141-2146], and for $c_1 = 3$ by Anghel and Manolache [Math. Nachr. 286 (2013), 1407-1423] and, independently, by Sierra and Ugaglia [J. Pure Appl. Algebra 218 (2014), 174-180]. It turns out that the case $c_1 = 4$ is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (conveniently twisted). The authors also propose a conjecture concerning the classification of globally generated vector bundles with $c_1 \\leq n - 1$ on the projective $n$-space. They verify the conjecture for $n \\leq 5$.