Forthcoming Seminars › Seminar Series

Junior Geometry and Topology Seminar

Thu, 19/01
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.
Fri, 20/01
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, 26/01
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."
Thu, 02/02
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, 09/02
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, 16/02
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, 23/02
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, 01/03
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, 08/03
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, 26/04
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, 03/05
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, 10/05
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, 17/05
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, 24/05
12:00 		Rosalinda Juer Junior Geometry and Topology Seminar Add to calendar L3
To Be Announced
Thu, 14/06
12:00 		Dawid Kielak Junior Geometry and Topology Seminar Add to calendar SR1
To Be Announced

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

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