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Organizers: Henrique Sa Earp (henrique@cantab.net) and Thomas Köppe (t.koeppe@ed.ac.uk)
The Second Latin Congress on Symmetries in Geometry and Physics will take place in Curitiba, Brazil in December 2010. Apart from the full-length research talks, we are also interested in shorter talks, around 20 minutes in length, by young researchers who have recently finished or are in the process of completing their Ph.D.
If you are interested in joining the conference, please get in touch with one of the organizers!
I will briefly introduce crystallographic groups and then talk about their application to flat manifolds.
I will present elementary examples of Lefschetz Fibrations. I will first focus on Lefschetz Pencils then derive Lefschetz Fibrations. I will also discuss an example of monodromy.
Continuing in the setting of the talk by my supervisor Wendy Lowen, I will present our findings on the compatibility between deformations and geometric helices. We will use this compatibility to reconstruct abelian deformations from derived deformations in some cases.
I will review two constructions of mirror-dual partners for Fano varieties (joint works with A. Bondal, T. Coates and A. Corti). Both constructions have their roots in a work of Eguchi, Hori and Xiong: one (Minkowski ansatz) is a recipe for cooking mirrors using toric degenerations, another exploits descent to a variety of lines to produce a mirror of a minuscule homogeneous variety. The Minkowski ansatz is general enough to provide an answer for all Fano surfaces and threefolds, but in its current state it is merely a phenomenology. Minuscule descent is very refined, but highlights some geometry that probably has to be used in general.
We find solutions of a system of coupled non-linear differential equations, known as the Strominger system, which occur as consistency conditions in heterotic string theory. We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3)-holonomy, then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain. This is joint work with Bjorn Andreas.
In this talk I present results from and beyond my Ph.D. thesis, much of which is joint work with my supervisor Elizabeth Gasparim. We consider Calabi-Yau threefolds which are the total space of certain bundles on CP1 and describe the moduli of vector bundles on these spaces. A priori one expects this to be a rather unwieldy stack-type object due to the non-compactness of the space, but we construct numerical invariants of vector bundles on these spaces derived from sheaf-cohomological constructions which give us a “nice” decomposition of the moduli and a notion of stability. We obtain dimension counts and hint at isomorphisms between the moduli of bundles on different spaces.
In this talk we discuss the relation between multiplicative 2-forms on Lie groupoids and linear 2-forms on Lie algebroids yielding a new simpler proof of the integration of Dirac structures. If time permits, I will also explain how graded symplectic supermanifolds fit into this framework. The talk will be based on a joint work with H. Bursztyn and A. Cabrera.
I will present three interesting questions in 7-dimensional gauge theory, which lead to the construction of G2-instantons on suitable bundles over A. Kovalev’s compact G2-manifolds (2002). First, how the Hermite-Yang-Mills problem over asymptotically cylindrical CY3 × S1 amounts essentially to “half” of the answer, since these spaces are the building blocks of Kovalev’s construction, via a “twisted gluing” procedure. Second, how the boundary conditions of this PDE take us straight into algebraic geometry, obtaining asymptotically stable bundles as cohomologies of linear monads. Third and finally, the G2-instanton gluing problem itself.
It is well-known that the suspension of the Hopf map η: S3 → S2 generates the stable group π1(S), which is isomorphic to the group of homotopy 8-spheres, θ8 ≅ Z2. We show how to construct an exotic 7-sphere with an action of SO(3) < G2 < SO(8) and an equivariant generator of π6(S3). Furthermore, through this construction and the symmetries of the fifth suspension of η, we relate in an explicit way π1(S) and θ8.
We also show that any homotopy 15-sphere that can be realized as a linear bundle over the standard 8-sphere can also be realized as a linear bundle over the exotic 8-sphere. No other 15-spheres are linear bundles over this exotic sphere.
Analogously, we relate π10(S7) ≅ Z3 to a subgroup of index 2 in θ10 ≅ Z6. The symmetries of such elements of θ10 provide a counterexample to a conjecture about the degrees of symmetry of exotic spheres.
All our constructions are described by simple formulas.
More details will follow soon, watch this space.
Last updated: November 23, 2010