Past Seminars › Seminar Series

Junior Geometry and Topology Seminar (past)

Thu, 20/11/2008
12:00 		Alan Thompson (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The construction of ample <2>-polarised K3-fibrations
Fibrations are a valuable tool in the study of the geometry of higher dimensional algebraic varieties. By expressing a higher dimensional variety as a fibration by lower dimensional varieties, we can deduce much about its properties. Whilst the theory of elliptic fibrations is very well developed, fibrations by higher dimensional varieties, especially K3 surfaces, are only just beginning to be studied. In this talk I study a special case of the K3-fibration, where the general fibres admit a <2>-polarisation and the base of the fibration is a nonsingular curve.
Thu, 13/11/2008
12:00 		Spiro Karigiannis (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to G_2 geometry (Part II)
I will give a survey-type introduction to manifolds equipped with $ G_2 $ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk is the second of two parts.
Thu, 06/11/2008
12:00 		Spiro Karigiannis (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to G_2 geometry (Part I)
I will give a survey-type introduction to manifolds equipped with $ G_2 $ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk will be in two parts.
Thu, 16/10/2008
12:00 		Oscar Randal-Williams (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Descent in algebra, geometry, and topology
Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data. In algebraic geometry, faithfully flat descent says that if $ X\rightarrow Y $ is a faithfully flat morphism of schemes, then giving a sheaf on $ Y $ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $ X $, satisfying certain compatibility conditions. Translated to algebra, it says that if $ S\rightarrow R $ is a faithfully flat morphism of rings, then giving an $ S $-module is the same as giving a certain simplical module over a simplicial ring constructed from $ R $. In topology, given an etale cover $ X\rightarrow Y $ one can recover $ Y $ (at least up to homotopy equivalence) from a simplical space constructed from $ X $.
Thu, 12/06/2008
12:00 		Alan Thompson (University of Oxford) Junior Geometry and Topology Seminar Add to calendar L3
An Introduction to the Birational Classification of Surfaces
The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.
Thu, 05/06/2008
12:00 		Johannes Ebert (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
An introduction to Chern-Weil theory
Thu, 22/05/2008
12:00 		George Raptis (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Models for homotopy and concreteness
Thu, 15/05/2008
12:00 		Dirk Schlueter (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Moduli problems in algebraic geometry - an introduction through Hilbert and Quot schemes
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.
Thu, 08/05/2008
12:00 		Owen Cotton-Barratt (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
An introduction to Geometric Group Theory
Thu, 06/03/2008
11:00 		Johannes Ebert (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Stacks and classifying spaces
Thu, 21/02/2008
11:00 		Steven Rayan (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The Birkhoff-Grothendieck Theorem
Thu, 14/02/2008
11:00 		João Costa (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Global coordinates for the domain of outer communications of axisymmetric and stationary black hole space-times.
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".
Thu, 07/02/2008
11:00 		Martinus Kool (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Moduli of Equivariant and Invariant Sheaves on Toric Varieties
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.
Thu, 31/01/2008
11:00 		Oscar Randal-Williams (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The Hopf invariant 1 problem
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.
Thu, 24/01/2008
11:00 		Jeff Giansiracusa (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR2
The Nielsen realisation problem and K3 surfaces
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.
Thu, 17/01/2008
11:00 		George Raptis (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to simple-homotopy theory

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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