# Past Junior Geometry and Topology Seminar

13 May 2010
12:00
Vicky Hoskins
Abstract
A moduli problem in algebraic geometry is essentially a classification problem, I will introduce this notion and define what it means for a scheme to be a fine (or coarse) moduli space. Then as an example I will discuss the classification of coherent sheaves on a complex projective scheme up to isomorphism using a method due to Alvarez-Consul and King. The key idea is to 'embed' the moduli problem of sheaves into the moduli problem of quiver representations in the category of vector spaces and then use King's moduli spaces for quiver representations. Finally if time permits I will discuss recent work of Alvarez-Consul on moduli of quiver sheaves; that is, representations of quivers in the category of coherent sheaves.
• Junior Geometry and Topology Seminar
6 May 2010
12:00
Markus Roeser
Abstract
A Hyperkähler manifold is a riemannian manifold carrying three complex structures which behave like quaternions such that the metric is Kähler with respect to each of them. This means in particular that the manifold is a symplectic manifold in many different ways. In analogy to the Marsden-Weinstein reduction on a symplectic manifold, there is also a quotient construction for group actions that preserve the Hyperkähler structure and admit a moment map. In fact most known (non-compact) examples of hyperkähler manifolds arise in this way from an appropriate group action on a quaternionic vector space. In the first half of the talk I will give the definition of a hyperkähler manifold and explain the hyperkähler quotient construction. As an important application I will discuss the moduli space of solutions to the gauge-theoretic "Self-duality equations on a Riemann surface", the space of Higgs bundles, and explain how it can be viewed as a hyperkähler quotient in an infinite-dimensional setting.
• Junior Geometry and Topology Seminar
29 April 2010
12:00
Maria Buzano
Abstract
The aim of this talk is to get a feel for the Ricci flow. The Ricci flow was introduced by Hamilton in 1982 and was later used by Perelman to prove the Poincaré conjecture. We will introduce the notions of Ricci flow and Ricci soliton, giving simple examples in low dimension. We will also discuss briefly other types of geometric flows one can consider.
• Junior Geometry and Topology Seminar
11 March 2010
12:00
Abstract
• Junior Geometry and Topology Seminar
4 March 2010
12:00
Michael Groechenig
Abstract
Descent theory is the art of gluing local data together to global data. Beside of being an invaluable tool for the working geometer, the descent philosophy has changed our perception of space and topology. In this talk I will introduce the audience to the basic results of scheme and descent theory and explain how those can be applied to concrete examples.
• Junior Geometry and Topology Seminar
25 February 2010
12:00
Jessica Banks
Abstract
In 2008, Juhasz published the following result, which was proved using sutured Floer homology. Let $K$ be a prime, alternating knot. Let $a$ be the leading coefficient of the Alexander polynomial of $K$. If $|a|<4$, then $K$ has a unique minimal genus Seifert surface. We present a new, more direct, proof of this result that works by counting trees in digraphs with certain properties. We also give a finiteness result for these digraphs.
• Junior Geometry and Topology Seminar
18 February 2010
12:00
to
18 March 2010
13:00
Laura Schaposnik
Abstract
We will consider the monodromy action on mod 2 cohomology for SL(2) Hitchin systems. We will study Copeland's approach to the subject and use his results to compute the monodromy action on mod 2 cohomology. An interpretation of our results in terms of geometric properties of fixed points of a natural involution on the moduli space is given.
• Junior Geometry and Topology Seminar
11 February 2010
12:00
Hwasung Mars Lee
Abstract
We will present a physical motivation of the SYZ conjecture and try to understand the conjecture via calibrated geometry. We will define calibrated submanifolds, and also give sketch proofs of some properties of the moduli space of special Lagrangian submanifolds. The talk will be elementary and accessible to a broad audience.
• Junior Geometry and Topology Seminar
4 February 2010
14:00
Jorge Vitoria
Abstract
• Junior Geometry and Topology Seminar