# Past 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
4 February 2010
12:00
Imran Qureshi
Abstract
Many interesting classes of projective varieties can be studied in terms of their graded rings. For weighted projective varieties, this has been done in the past in relatively low codimension. Let $G$ be a simple and simply connected Lie group and $P$ be a parabolic subgroup of $G$, then homogeneous space $G/P$ is a projective subvariety of $\mathbb{P}(V)$ for some\\ $G$-representation $V$. I will describe weighted projective analogues of these spaces and give the corresponding Hilbert series formula for this construction. I will also show how one may use such spaces as ambient spaces to construct weighted projective varieties of higher codimension.
• Junior Geometry and Topology Seminar
28 January 2010
13:15
Steven Rayan
Abstract
After reviewing the salient details from last week's seminar, I will construct an explicit example of a spectral curve, using co-Higgs bundles of rank 2. The role of the spectral curve in understanding the moduli space will be made clear by appealing to the Hitchin fibration, and from there inferences (some of them very concrete) can be made about the structure of the moduli space. I will make some conjectures about the higher-dimensional picture, and also try to show how spectral varieties might live in that picture.
• Junior Geometry and Topology Seminar
21 January 2010
13:30
Steven Rayan
Abstract
PLEASE NOTE THE CHANGE OF TIME FOR THIS WEEK: 13.30 instead of 12. In the first of two talks, I will simultaneously introduce the notion of a co-Higgs vector bundle and the notion of the spectral curve associated to a compact Riemann surface equipped with a vector bundle and some extra data. I will try to put these ideas into both a historical context and a contemporary one. As we delve deeper, the emphasis will be on using spectral curves to better understand a particular moduli space.
• Junior Geometry and Topology Seminar
10 December 2009
12:00
Abstract
We will begin by reviewing the construction of the symplectic quotient and the definition of the Kirwan map. Then we will give an overview of Kirwan's original proof of the surjectivity of this map and some generalizations of this result. Finally we will talk about the techniques that are being developed to construct right inverses for the Kirwan map.
• Junior Geometry and Topology Seminar
3 December 2009
12:00
Martijn Kool
Abstract
Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. Using torus localisation, one can then compute topological invariants of M. We consider the case X=S is a toric surface and compute generating functions of Euler characteristics of M. In case of torsion free sheaves, one can study wall-crossing phenomena and in case of pure dimension 1 sheaves one can verify, in examples, a conjecture of Katz relating Donaldson--Thomas invariants and Gopakumar--Vafa invariants.
• Junior Geometry and Topology Seminar
26 November 2009
12:00
Ana Ferreira
Abstract
We will present a self-contained introduction to gauge theory, self-duality and instanton moduli spaces. We will analyze in detail the situation of charge 1 instantons for the 4-sphere when the gauge group is SU(2). Time permitting, we will also mention the ADHM construction for k-instantons.
• Junior Geometry and Topology Seminar