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1	Graph Cohomology Lin, Matthew 01 January 2016 (has links) What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the cohomology of the associated variety directly in terms of the graph G itself. 05C99 Graph theory 05E99 Algebraic combinatorics 14M25 Toric varieties Algebra Discrete Mathematics and Combinatorics
2	Gromov-Witten Theory of Blowups of Toric Threefolds Ranganathan, Dhruv 31 May 2012 (has links) We use toric symmetry and blowups to study relationships in the Gromov-Witten theories of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. These two spaces are birationally equivalent via the common blowup space, the permutohedral variety. We prove an equivalence of certain invariants on blowups at only points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$ by showing that these invariants descend from the blowup. Further, the permutohedral variety has nontrivial automorphisms of its cohomology coming from toric symmetry. These symmetries can be forced to descend to the blowups at just points of $\mathbb{P}^3$ and $\mathbb{P}^1\!\times\!\mathbb{P}^1\!\times\!\mathbb{P}^1$. Enumerative consequences are discussed. 14M25 Toric varieties Newton polyhedra 83E30 String and superstring theories
3	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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