Junior Geometry and Topology Seminar

Thu, 23/01
16:00
Elisabeth Grieger (King's College London) Junior Geometry and Topology Seminar Add to calendar C6
We give a short exposition on the zeta determinant for a Laplace - type operator on a closed Manifold as first described by Ray and Singer in their attempt to find an analytic counterpart to R-torsion.
Thu, 30/01
16:00
Alejandro Betancourt Junior Geometry and Topology Seminar Add to calendar C6
Ricci solitons were introduced by Richard Hamilton in the 80's and they are a generalization of the better know Einstein metrics. During this talk we will define the notion of Ricci soliton and I will try to convince you that these metrics arise "naturally" in a number of different settings. I will also present various examples and talk a bit about some symmetry properties that Ricci solitons have. Note: This talk is meant to be introductory and no prior knowledge about Einstein metrics will be assumed (or necessary).
Thu, 06/02
16:00
Lam Yan Junior Geometry and Topology Seminar Add to calendar C6
Quivers are directed graphs which can be thought of as "space" in noncommutative geometry. In this talk, we will try to establish a link between noncommutative geometry and its commutative counterpart. We will show how one can construct (differential graded) quivers which are "equivalent" (in the sense of derived category of representations) to vector bundles on smooth varieties.
Thu, 13/02
16:00
Cancelled Junior Geometry and Topology Seminar Add to calendar C6
Thu, 20/02
16:00
Roberto Rubio Junior Geometry and Topology Seminar Add to calendar C6
This talk will give an introduction to generalized complex geometry, where complex and symplectic structures are particular cases of the same structure, namely, a generalized complex structure. We will also talk about a sister theory, generalized complex geometry of type Bn, where generalized complex structures are defined for odd-dimensional manifolds as well as even-dimensional ones.
Thu, 27/02
16:00
Georgia Christodoulou Junior Geometry and Topology Seminar Add to calendar C6
We will talk about the Beilinson-Bernstein localization theorem, which is a major result in geometric representation theory. We will try to explain the main ideas behind the theorem and this will lead us to some geometric constructions that are used in order to produce representations. Finally we will see how the theorem is demonstrated in the specific case of the Lie algebra sl2
Fri, 28/02
14:30
Emily Cliff (Oxford University) Junior Geometry and Topology Seminar Add to calendar C5
A universal D-module of dimension n is a rule assigning to every family of smooth $ n $-dimensional varieties a family of D-modules, in a compatible way. This seems like a huge amount of data, but it turns out to be entirely determined by its value over a single formal disc. We begin by recalling (or perhaps introducing) the notion of a D-module, and proceed to define the category $ M_n $ of universal D-modules. Following Beilinson and Drinfeld we define the Gelfand-Kazhdan structure over a smooth variety (or family of varieties) of dimension $ n $, and use it to build examples of universal D-modules and to exhibit a correspondence between $ M_n $ and the category of modules over the group-scheme of continuous automorphisms of formal power series in $ n $ variables
Fri, 28/02
16:00
Carmelo Di Natale (Cambridge University) Junior Geometry and Topology Seminar Add to calendar L4

In the sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this talk we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks.

Thu, 06/03
16:00
Emanuele Ghedin Junior Geometry and Topology Seminar Add to calendar C6
Following last week's talk on Beilinson-Bernstein localisation theorem, we give basic notions in deformation quantisation explaining how this theorem can be interpreted as a quantised version of the Springer resolution. Having attended last week's talk will be useful but not necessary.
Thu, 13/03
16:00
Roland Grinis Junior Geometry and Topology Seminar Add to calendar C6
I plan to give a non technical introduction (i.e. no prerequisites required apart basic differential geometry) to some analytic aspects of the theory of harmonic maps between Riemannian manifolds, motivate it by briefly discussing some relations to other areas of geometry (like minimal submanifolds, string topology, symplectic geometry, stochastic geometry...), and finish by talking about the heat flow approach to the existence theory of harmonic maps with some open problems related to my research.
Thu, 01/05
16:00
Jakob Blaavand Junior Geometry and Topology Seminar Add to calendar C6

The first half of this talk will be an introduction to the wonderful world of Higgs bundles. The last half concerns Fourier--Mukai transforms, and we will discuss how to merge the two concepts by constructing a Fourier--Mukai transform for Higgs bundles. Finally we will discuss some properties of this transform. We will along the way discuss why you would want to transform Higgs bundles.

Thu, 08/05
16:00
Lucas Branco Junior Geometry and Topology Seminar Add to calendar C6
Since their introduction in the context of symplectic geometry, moment maps and symplectic quotients have been generalized in many different directions. In this talk I plan to give an introduction to the notions of hyperkähler moment map and hyperkähler quotient through two examples, apparently very different, but related by the so called ADHM construction of instantons; the moduli space of instantons and a space of complex matrices arising from monads.
Thu, 15/05
16:00
Cancelled Junior Geometry and Topology Seminar Add to calendar C6
Thu, 29/05
16:00
Thomas Wasserman ((Oxford University)) Junior Geometry and Topology Seminar Add to calendar C6
Topological insulators are a type of system in condensed matter physics that exhibit a robustness that physicists like to call topological. In this talk I will give a definition of a subclass of such systems: gapped, free fermions. We will look at how such systems, as shown by Kitaev, can be classified in terms of topological K-groups by using the Clifford module model for K-theory as introduced by Atiyah, Bott and Shapiro. I will be using results from Wednesday's JTGT, where I'll give a quick introduction to topological K-theory.
Thu, 05/06
16:00
Gerrit Goosen Junior Geometry and Topology Seminar Add to calendar C6
We give an overview of Kitaev's lattice model in the setting of an arbitrary finite group G (where $ G = Z_{2} $ is the famous Toric Code). We also exhibit the connection this model has with so-called 123-TQFTs (topological quantum field theories), making use of ideas coming from higher gauge theory and Hopf algebra representations.
Thu, 12/06
16:00
Omar Kidwai ((Oxford University)) Junior Geometry and Topology Seminar Add to calendar C6
Spectral networks are certain collections of paths on a Riemann surface, introduced by Gaiotto, Moore, and Neitzke to study BPS states in certain N=2 supersymmetric gauge theories. They are interesting geometric objects in their own right, with a number of mathematical applications. In this talk I will give an introduction to what a spectral network is, and describe the "abelianization map" which, given a spectral network, produces nice "spectral coordinates" on the appropriate moduli space of flat connections. I will show that coordinates obtained in this way include a variety of previously known special cases (Fock-Goncharov coordinates and Fenchel-Nielsen coordinates), and mention at least one reason why generalising them in this way is of interest.
Thu, 19/06
16:00
Brent Pym (Oxford University) Junior Geometry and Topology Seminar Add to calendar C6
Lie algebroids are geometric structures that interpolate between finite-dimensional Lie algebras and tangent bundles of manifolds. They give a useful language for describing geometric situations that have local symmetries. I will give an introduction to the basic theory of Lie algebroids, with examples drawn from foliations, principal bundles, group actions, Poisson brackets, and singular hypersurfaces.
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