Past Seminars › Seminar Series

Junior Geometry and Topology Seminar (past)

Thu, 29/04/2010
12:00 		Maria Buzano (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
An introduction to the Ricci flow
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.
Thu, 04/03/2010
12:00 		Michael Groechenig (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to descent theory
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.
Thu, 25/02/2010
12:00 		Jessica Banks (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Knots, graphs, and the Alexander polynomial
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.
Thu, 18/02/2010
12:00 		Laura Schaposnik (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Monodromy of Higgs bundles
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.
Thu, 11/02/2010
12:00 		Hwasung Mars Lee (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
An overview of the SYZ conjecture and calibrated geometry
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.
Thu, 04/02/2010
15:45 		Yuhi Sekiya (Nagoya/Glasgow) Junior Geometry and Topology Seminar Add to calendar DH 2nd floor SR
(CALF) : Moduli spaces of Mckay quiver representations
Thu, 04/02/2010
12:00 		Imran Qureshi (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Weighted projective varieties in higher codimension
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.
Thu, 28/01/2010
13:15 		Steven Rayan (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Co-Higgs bundles II: fibrations and moduli spaces
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.
Thu, 21/01/2010
13:30 		Steven Rayan (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Co-Higgs bundles I: spectral curves
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.
Thu, 10/12/2009
12:00 		Andratx Bellmunt (Universitat de Barcelona / Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Right inverses of the Kirwan map
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.
Thu, 03/12/2009
12:00 		Martijn Kool (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Moduli Spaces of Sheaves on Toric Varieties
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.
Thu, 26/11/2009
12:00 		Ana Ferreira (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Introduction to self-duality and instantons
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.
Thu, 19/11/2009
12:00 		Richard Wade (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Graph Foldings and Free Groups
We describe John Stalling's method of studying finitely generated free groups via graphs and moves on graphs called folds. We will then discuss how the theory can be extended to study the automorphism group of a finitely generated free group.
Thu, 12/11/2009
12:00 		Tom Baird (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Group valued moment maps, Loop groups and Dirac structures
I will survey the theory of quasiHamiltonian spaces, a.k.a. group valued moment maps. In rough correspondence with historical development, I will first show how they emerge from the study of loop group representations, and then how they arise as a special case of "presymplectic realizations" in Dirac geometry.
Thu, 05/11/2009
12:00 		Peter Arndt (Göttingen / Cambridge) Junior Geometry and Topology Seminar Add to calendar SR1
Compactifying Spec $ \mathbb{Z} $
The spectrum of the integers is an affine scheme which number theorists would like to complete to a projective scheme, adding a point at infinity. We will list some reasons for wanting to do this, then gather some hints about what properties the completed object might have. In particular it seems that the desired object can only exist in some setting extending traditional algebraic geometry. We will then present the proposals of Durov and Shai Haran for such extended settings and the compactifications they construct. We will explain the close relationship between both and, if time remains, relate them to a third compactification in a third setting, proposed by Toen and Vaquie.
Thu, 29/10/2009
12:00 		George Raptis (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The problem of the homotopy invariance of configuration spaces
The talk is about the homotopy type of configuration spaces. Once upon a time there was a conjecture that it is a homotopy invariant of closed manifolds. I will discuss the strong evidence supporting this claim, together with its recent disproof by a counterexample. Then I will talk about the corrected version of the original conjecture.
Thu, 22/10/2009
12:00 		Alan Thompson (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Degenerations of <2>-polarised K3 surfaces
A <2>-polarised K3 surface admits an embedding into weighted projective space defined by its polarisation. Let X be a family of such surfaces, then one can construct a projective model W of X such that the map from X to W realises this embedding on the general fibre. This talk considers what happens to W when we allow the fibres of the family X to degenerate.
Thu, 15/10/2009
12:00 		Dirk Schlueter (Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The Hodge Conjecture

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

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