Forthcoming Seminars › Seminar Series

Junior Geometry and Topology Seminar

Thu, 11/10/2007
12:00 		Oscar Randal-Williams (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The Poincaré - Hopf index theorem
We will prove an extended Poincaré - Hopf theorem, identifying several invariants of a manifold $ M $. These are its Euler characteristic $ \chi(M) $, the sum $ \sum_{x_i} ind_V(x_i) $ of indices at zeroes of a vector field $ V $ on $ M $, the self-intersection number $ \Delta \cap \Delta $ of the diagonal $ \Delta \subset M \times M $ and finally the integral $ \int_M e(TM) $ of the Euler class of the tangent bundle.
Thu, 18/10/2007
12:00 		David Baraglia (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Cartan connections and parabolic geometries
Klein's famous lecture proposes that to study geometry we study homogeneous spaces ie study transformation groups acting on a space. E. Cartan found a generalization now known as "Cartan geometries", these are a curved generalization of homogeneous spaces, eg Riemannian manifolds are Cartan geometries modeled on {Euclidean group}/{orthogonal group}. Topics for my talk will be Cartan geometries / Cartan connections Parabolic geometries - a special class of Cartan geometries Examples - depending on how much time but I will probably explain conformal geometry as a parabolic geometry
Thu, 25/10/2007
12:00 		Mitul Shah (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Why did Lie Invent Lie Groups?
This talk will be about the systematic simplification of differential equations. After giving a geometric reformulation of the concept of a differential equation using prolongations, I will show how we can prolong group actions relatively easily at the level of Lie algebras. I will then discuss group-invariant solutions. The key example will be the heat equation.
Thu, 01/11/2007
11:00 		Liam Wall (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Hyperbolic 3-manifolds
In this talk I will introduce hyperbolic 3-manifolds, state some major conjectures about them, and discuss some group-theoretic properties of their fundamental groups.
Thu, 08/11/2007
11:00 		Alexander Coward (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Hyperbolic 3-manifolds II
Thu, 15/11/2007
11:00 		George Walker (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Exposition on point counting using rigid cohomology
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.
Thu, 22/11/2007
11:00 		George Raptis (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Grothendieck groups and Wall's finiteness obstruction
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.
Thu, 29/11/2007
11:00 		Johannes Ebert (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The Poincaré conjecture in high dimensions

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

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