Past Junior Geometry and Topology Seminar

25 November 2010
13:00
Robert Clancy
Abstract
All of Joyce's constructions of compact manifolds with special holonomy are in some sense generalisations of the Kummer construction of a K3 surface. We will begin by reviewing manifolds with special holonomy and the Kummer construction. We will then describe Joyce's constructions of compact manifolds with holonomy G_2 and Spin(7).
  • Junior Geometry and Topology Seminar
18 November 2010
13:00
Abstract
<p><span style="font-family: tahoma; font-size: x-small;">I will begin by defining the space of algebraic metrics in a particular Kahler class and recalling the Tian-Ruan-Zelditch result saying that they are dense in the space of all Kahler metrics in this class.&nbsp; I will then discuss the relationship between some special algebraic metrics called&nbsp;'balanced metrics' and distinguished Kahler metrics (Extremal metrics, cscK, Kahler-Ricci solitons...). Finally I will talk about some numerical algorithms due to Simon Donaldson for finding explicit examples of these balanced metrics (possibly with some pictures).</span></p>
  • Junior Geometry and Topology Seminar
11 November 2010
13:00
Christopher Hopper
Abstract
<p>The maximum principle is one of the main tools use to understand the behaviour of solutions to the Ricci flow. It is a very powerful tool that can be used to show that pointwise inequalities on the initial data of parabolic PDE are preserved by the evolution. A particular weak maximum principle for vector bundles will be discussed with references to Hamilton's seminal work [J. Differential Geom. 17 (1982), no. 2, 255–306; MR664497] on 3-manifolds with positive Ricci curvature and his follow up paper [J. Differential Geom. 24 (1986), no. 2, 153–179; MR0862046] that extends to 4-manifolds with various curvature assumptions.</p>
  • Junior Geometry and Topology Seminar
4 November 2010
13:00
Markus Röser
Abstract
In the first part of this talk we introduce hypersymplectic manifolds and compare various aspects of their geometry with related notions in hyperkähler geometry. In particular, we explain the hypersymplectic quotient construction. Since many examples of hyperkähler structures arise from Yang-Mills moduli spaces via the hyperkähler quotient construction, we discuss the gauge theoretic equations for a (twisted) harmonic map from a Riemann surface into a compact Lie group. They can be viewed as the zero condition for a hypersymplectic moment map in an infinite-dimensional setup.
  • Junior Geometry and Topology Seminar
28 October 2010
13:00
Maria Buzano
Abstract
We will recall basic definitions and facts about homogeneous Riemannian manifolds and we will discuss the Einstein condition on this kind of spaces. In particular, we will talk about non existence results of invariant Einstein metrics. Finally, we will talk briefly about the Ricci flow equation in the homogeneous setting.
  • Junior Geometry and Topology Seminar
21 October 2010
13:00
Alan Thompson
Abstract
A K3 surface of degree two can be seen as a double cover of the complex projective plane, ramified over a nonsingular sextic curve. In this talk we explore two different methods for constructing explicit projective models of threefolds admitting a fibration by such surfaces, and discuss their relative merits.
  • Junior Geometry and Topology Seminar
14 October 2010
12:00
Abstract
The theory of C*-algebras provides a good realisation of noncommutative topology. There is a dictionary relating commutative C*-algebras with locally compact spaces, which can be used to import topological concepts into the C*-world. This philosophy fails in the case of homotopy, where a more sophisticated definition has to be given, leading to the notion of asymptotic morphisms. As a by-product one obtains a generalisation of Borsuk's shape theory and a universal boundary map for cohomology theories of C*-algebras.
  • Junior Geometry and Topology Seminar

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