# Past Junior Geometry and Topology Seminar

28 February 2008
11:00
Ana Ferreira
Abstract
• Junior Geometry and Topology Seminar
21 February 2008
11:00
Steven Rayan
Abstract
• 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
24 January 2008
11:00
Jeff Giansiracusa
Abstract
The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.
• Junior Geometry and Topology Seminar
22 November 2007
11:00
George Raptis
Abstract
Will discuss several constructions of the Grothendieck group in different contexts together with Wall's solution of the problem of determining homotopy types of finite CW complexes as a motivating application.
• Junior Geometry and Topology Seminar
15 November 2007
11:00
George Walker
Abstract
Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$, $$Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right)$$ is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$. I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.
• Junior Geometry and Topology Seminar