Past Junior Geometry and Topology Seminar

15 May 2008
12:00
Dirk Schlueter
Abstract
Hilbert schemes classify subschemes of a given projective variety / scheme. They are special cases of Quot schemes which are moduli spaces for quotients of a fixed coherent sheaf. Hilb and Quot are among the first examples of moduli spaces in algebraic geometry, and they are crucial for solving many other moduli problems. I will try to give you a flavour of the subject by sketching the construction of Hilb and Quot and by discussing the role they play in applications, in particular moduli spaces of stable curves and moduli spaces of stable sheaves.
  • Junior Geometry and Topology Seminar
14 February 2008
11:00
João Costa
Abstract
The usual procedure to obtain uniqueness theorems for black hole space-times ("No Hair" Theorems) requires the construction of global coordinates for the domain of outer communications (intuitively: the region outside the black hole). Besides an heuristic argument by Carter and a few other failed attempts the existence of such a (global) coordinate system as been neglected, becoming a quite hairy hypothesis. After a review of the basic aspects of causal theory and a brief discussion of the definition of black-hole we will show how to construct such coordinates focusing on the non-negativity of the "area function".
  • Junior Geometry and Topology Seminar
7 February 2008
11:00
Martinus Kool
Abstract
Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric variety X. This combinatorial description can be used to construct moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles on X by Payne and Perling). We study how the moduli spaces of stable equivariant sheaves on X can be used to explicitly compute the fixed point locus of the moduli space of all stable sheaves on X, i.e. the subscheme of invariant stable sheaves on X.
  • Junior Geometry and Topology Seminar
31 January 2008
11:00
Oscar Randal-Williams
Abstract
For continuous maps $f: S^{2n-1} \to S^n$ one can define an integer-valued invariant, the so-called Hopf invariant. The problem of determining for which $n$ there are maps having Hopf invariant one can be related to many problems in topology and geometry, such as which spheres are parallelisable, which spheres are H-spaces (that is, have a product), and what are the division algebras over $\mathbb{R}$. The best way to solve this problem is using complex K-theory and Adams operations. I will show how all the above problems are related, give an introduction to complex K-theory and it's operations, and show how to use it to solve this problem.
  • Junior Geometry and Topology Seminar

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