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Junior Geometry and Topology Seminar (past)

Thu, 31/05/2012
12:00 		Richard Manthorpe Junior Geometry and Topology Seminar Add to calendar L3
Diffeomorphism equivariance and the scanning map
Given a manifold $ M $ and a basepointed labelling space $ X $ the space of unordered finite configurations in $ M $ with labels in $ X $, $ C(M;X) $ is the space of finite unordered tuples of points in $ M $, each point with an associated point in $ X $. The space is topologised so that particles cannot collide. Given a compact submanifold $ M_0\subset M $ we define $ C(M,M_0;X) $ to be the space of unordered finite configuration in which points `vanish' in $ M_0 $. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $ \Sigma^nX $-bundle over $ M $. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on `little round $ \varepsilon $-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.
Thu, 24/05/2012
12:00 		Rosalinda Juer Junior Geometry and Topology Seminar Add to calendar L3
Unoriented cobordism categories and Klein TQFTs
The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $ \mathscr{K} $, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $ \mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}} $). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.
Thu, 17/05/2012
12:00 		Markus Röser Junior Geometry and Topology Seminar Add to calendar L3
Hyperkähler Metrics in Lie Theory

In this talk our aim is to explain why there exist hyperkähler metrics on the cotangent bundles and on coadjoint orbits of complex Lie groups. The key observation is that both the cotangent bundle of $G^\mathbb C$ and complex coadjoint orbits can be constructed as hyperkähler quotients in an infinite-dimensional setting: They may be identified with certain moduli spaces of solutions to Nahm's equations, which is a system of non-linear ODEs arising in gauge theory. 

In the first half we will describe the hyperkähler quotient construction, which can be viewed as a version of the Marsden-Weinstein symplectic quotient for complex symplectic manifolds. We will then introduce Nahm's equations and explain how their moduli spaces of solutions may be related to the above Lie theoretic objects.
Thu, 10/05/2012
12:00 		Laura Schaposnik Junior Geometry and Topology Seminar Add to calendar L3
Spectral data for the Hitchin fibration
We shall dedicate the first half of the talk to introduce classical Higgs bundles and describe the fibres of the corresponding Hitchin fibration in terms of spectral data. Then, we shall define principal Higgs bundles and look at some examples. Finally, we consider the particular case of $ SL(2,R) $, $ U(p,p) $ and $ Sp(2p,2p) $ Higgs bundles and study their spectral data. Time permitting, we shall look at different applications of our new methods.
Thu, 03/05/2012
12:00 		Henry Bradford Junior Geometry and Topology Seminar Add to calendar L3
Expander Graphs and Property $ \tau $
Expander graphs are sparse finite graphs with strong connectivity properties, on account of which they are much sought after in the construction of networks and in coding theory. Surprisingly, the first examples of large expander graphs came not from combinatorics, but from the representation theory of semisimple Lie groups. In this introductory talk, I will outline some of the history of the emergence of such examples from group theory, and give several applications of expander graphs to group theoretic problems.
Thu, 26/04/2012
12:00 		Alessandro Sisto Junior Geometry and Topology Seminar Add to calendar SR1
Teichmüller space: complex vs hyperbolic geometry
Complex structures on a closed surface of genus at least 2 are in one-to-one correspondence with hyperbolic metrics, so that there is a single space, Teichmüller space, parametrising all possible complex and hyperbolic structures on a given surface (up to isotopy). We will explore how complex and hyperbolic geometry interact in Teichmüller space.
Thu, 08/03/2012
13:00 		Markus Röser Junior Geometry and Topology Seminar Add to calendar L3
Twistor Geometry
Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $ \mathbb R^4 $ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $ \mathbb R^4 $ , or $ S^4 $, as the space of certain "real" lines in the (projective) Twistor space $ \mathbb{CP}^3 $. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $ S^4 $ as certain holomorphic vector bundles on $ \mathbb{CP}^3 $ due to Ward.
Thu, 01/03/2012
13:00 		Robert Clancy Junior Geometry and Topology Seminar Add to calendar L3
Applications of non-linear analysis to geometry
I will claim (and maybe show) that a lot of problems in differential geometry can be reformulated in terms of non-linear elliptic differential operators. After reviewing the theory of linear elliptic operators, I will show what can be said about the non-linear setting.
Thu, 23/02/2012
13:00 		Christian Paleani Junior Geometry and Topology Seminar Add to calendar SR2
Pseudo-Holomorphic Curves in Generalized Geometry
After giving a brief physical motivation I will define the notion of generalized pseudo-holomorphic curves, as well as tamed and compatible generalized complex structures. The latter can be used to give a generalization of an energy identity. Moreover, I will explain some aspects of the local and global theory of generalized pseudo-holomorphic curves.
Thu, 16/02/2012
13:00 		Roberto Rubio Junior Geometry and Topology Seminar Add to calendar SR2
Generalized Geometry - a starter course.
Basic and mild introduction to Generalized Geometry from the very beginning: the generalized tangent space, generalized metrics, generalized complex structures... All topped with some Lie type B flavour. Suitable for vegans. May contain traces of spinors.
Thu, 09/02/2012
13:00 		Hemanth Saratchandran Junior Geometry and Topology Seminar Add to calendar L3
Elliptic Curves and Cohomology Theories
I will give a brief introduction into how Elliptic curves can be used to define complex oriented cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.
Thu, 02/02/2012
13:00 		Chris Hopper Junior Geometry and Topology Seminar Add to calendar SR2
Monotonicity, variational methods and the Ricci flow
I will give an introduction to the variational characterisation of the Ricci flow that was first introduced by G. Perelman in his paper on "The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159. The first in a series of three papers on the geometrisation conjecture. The discussion will be restricted to sections 1 through 5 beginning first with the gradient flow formalism. Techniques from the Calculus of Variations will be emphasised, notably in proving the monotonicity of particular functionals. An overview of the local noncollapsing theorem (Perelman’s first breakthrough result) will be presented with refinements from Topping [Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055.]. Some remarks will also be made on connections to implicit structures seen in the physics literature, for instance of those seen in D. Friedan [Ann. Physics 163 (1985), no. 2, 318–419].
Thu, 26/01/2012
13:00 		Jakob Blaavand Junior Geometry and Topology Seminar Add to calendar SR2
Geometric Quantization - an Introduction
In this talk we will discuss geometric quantization. First of all we will discuss what it is, but shall also see that it has relations to many other parts of mathematics. Especially shall we see how the Hitchin connection in geometric quantization can give us representations of a certain group associated to a surface, the mapping class group. If time permits we will discuss some recent results about these groups and their representations, results that are essentially obtained from geometrically quantizing a moduli space of flat connections on a surface."
Fri, 20/01/2012
12:00 		Vittoria Bussi Junior Geometry and Topology Seminar Add to calendar L3
Derived Algebraic Geometry: a global picture II
This is the second of two talks about Derived Algebraic Geometry. We will go through the various geometries one can develop from the Homotopical Algebraic Geometry setting. We will review stack theory in the sense of Laumon and Moret-Bailly and higher stack theory by Simpson from a new and more general point of view, and this will culminate in Derived Algebraic Geometry. We will try to point out how some classical objects are actually secretly already in the realm of Derived Algebraic Geometry, and, once we acknowledge this new point of view, this makes us able to reinterpret, reformulate and generalize some classical aspects. Finally, we will describe more exotic geometries. In the last part of this talk, we will focus on two main examples, one addressed more to algebraic geometers and representation theorists and the second one to symplectic geometers.
Thu, 19/01/2012
12:00 		Vittoria Bussi Junior Geometry and Topology Seminar Add to calendar L3
Derived Algebraic Geometry: a global picture I
This is the first of two talks about Derived Algebraic Geometry. Due to the vastity of the theory, the talks are conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as it might reveal interesting new insights also on classical objects, rather than a detailed and precise exposition. We will start with an introduction on the very basic idea of this theory, and we will expose some motivations for introducing it. After a brief review on the existing literature and a speculation about homotopy theories and higher categorical structures, we will review the theory of dg-categories, model categories, S-categories and Segal categories. This is the technical part of the seminar and it will give us the tools to understand the basic setting of Topos theory and Homotopical Algebraic Geometry, whose applications will be exploited in the next talk.
Thu, 01/12/2011
12:00 		Martin Palmer Junior Geometry and Topology Seminar Add to calendar
Thom spectra and cobordism rings
After recalling some definitions and facts about spectra from the previous two "respectra" talks, I will explain what Thom spectra are, and give many examples. The cohomology theories associated to various different Thom spectra include complex cobordism, stable homotopy groups, ordinary mod-2 homology....... I will then talk about Thom's theorem: the ring of homotopy groups of a Thom spectrum is isomorphic to the corresponding cobordism ring. This allows one to use homotopy-theoretic methods (calculating the homotopy groups of a spectrum) to answer a geometric question (determining cobordism groups of manifolds with some specified structure). If time permits, I'll also describe the structure of some cobordism rings obtained in this way.
Thu, 24/11/2011
12:00 		John Calabrese Junior Geometry and Topology Seminar Add to calendar SR2
Respectra for Tensor Products
More perspectives on spectra.
Thu, 17/11/2011
12:00 		Michael Gröchenig Junior Geometry and Topology Seminar Add to calendar SR2
Perspectives on Spectra
This is the first in a series of $ \geq 2 $ talks about Stable Homotopy Theory. We will motivate the definition of spectra by the Brown Representability Theorem, which allows us to interpret a spectrum as a generalized cohomology theory. Along the way we recall basic notions from homotopy theory, such as suspension, loop spaces and smash products.
Thu, 10/11/2011
12:00 		Tim Adamo Junior Geometry and Topology Seminar Add to calendar SR2
Holomorphic analogues of Chern-Simons gauge theory and Wilson operators
Chern-Simons theory is topological gauge theory in three dimensions that contains an interesting class of operators called Wilson lines/loops, which have connections with both physics and pure mathematics. In particular, it has been shown that computations with Wilson operators in Chern-Simons theory reproduce knot invariants, and are also related to Gauss linking invariants. We will discuss the complex generalizations of these ideas, which are known as holomorphic Chern-Simons theory, Wilson operators, and linking, in the setting of Calabi-Yau three-folds. This will (hopefully) include a definition of all three of these holomorphic analogues as well as an investigation into how these ideas can be translated into simple homological algebra, allowing us to propose the existence of "homological Feynman rules" for computing things like Wilson operators in a holomorphic Chern-Simons theory. If time permits I may say something about physics too.
Thu, 03/11/2011
12:00 		Benjamin Volk Junior Geometry and Topology Seminar Add to calendar SR2
Some Remarks on d-manifolds and d-bordism
We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $ C^\infty $-rings, $ C^\infty $-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.

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

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