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1	On Operads / Über Operaden Brinkmeier, Michael 18 May 2001 (has links) This Thesis consists of four independent parts. In the first part I prove that the delooping, i.e.the classifying space, of a grouplike monoid is an $H$-space if and only if its multiplication is a homotopy homomorphism. This is an extension and clarification of a result of Sugawara. Furthermore I prove that the Moore loop space functor and the construction of the classifying space induce an adjunction on the corresponding homotopy categories. In the second part I extend a result of G. Dunn, by proving that the tensorproduct $C_{n_1}\otimes\dots \otimes C_{n_j}$ of little cube operads is a topologically equivalent suboperad of $C_{n_1 \dots n_j}$. In the third part I describe operads as algebras over a certain colored operad. By application of results of Boardman and Vogt I describe a model of the homotopy category of topological operads and algebras over them, as well as a notion of lax operads, i.e. operads whose axioms are weakened up to coherent homotopies. Here the W-construction, a functorial cofibrant replacement for a topological operad, plays a central role. As one application I construct a model for the homotopy category of topological categories. C. Berger claimed to have constructed an operad structure on the permutohedras, whose associated monad is exactly the Milgram-construction of the free two-fold loop space. In the fourth part I prove that this statement is not correct. Operads Little Cubes Tensorproduct of Operads H-spaces Strongly Homotopy Commutative Homotopy Homomorphism Permutohedron 55P48 55P35 55P47 55P45 27 - Mathematik ddc:510
2	Cremona Symmetry in Gromov-Witten Theory / Cremona Symmetry in Gromov-Witten Theory Gholampour, Amin, Karp, Dagan, Payne, Sam 25 September 2017 (has links) We establish the existence of a symmetry within the Gromov-Witten theory of CPn and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants. / En este trabajo establecemos la existencia de una simetra en el marco de la teora de Gromov-Witten para CPn y su explosion a lo largo de puntos. La naturaleza de esta simetra queda codicada en la transformacion de Cremona y su resolucion en una variedad torica del permutoedro. Esta simetra expresa algunos invariantes difciles de calcular junto con otros que no lo son tanto. Nos centramos en implicaciones enumerativas; en particular esta tecnica ofrece una prueba enuna lnea de la unicidad de la curva racional normal. Nuestro metodo involucra un estudio de la geometra torica del permutoedro, as como el de la degeneracion de los invariantes de Gromov-Witten. Gromov-Witten Theory Enumerative Geometry Stationary Invariants Cremona Transform Projective Space Permutohedron Permutohedral Toric Variety Losev-Manin Space Msc(2010): 14n35 14e05 14m25 Teoría de Gromov-Witten Geometría Enumerativa Invariantes Estacionarios Transformación de Cremona Espacio Proyectivo Permutoedro Variadad Tórica Permutoedral Espacio de Losev-Manin Msc(2010): 14n35 14e05 14m25

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On Operads / Über Operaden

Cremona Symmetry in Gromov-Witten Theory / Cremona Symmetry in Gromov-Witten Theory