Past 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
25 October 2007
12:00
Abstract
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.
  • Junior Geometry and Topology Seminar
18 October 2007
12:00
David Baraglia
Abstract
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
  • Junior Geometry and Topology Seminar
11 October 2007
12:00
Oscar Randal-Williams
Abstract
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.
  • Junior Geometry and Topology Seminar

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