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A learning seminar on topics at the beginning graduate level.
Time: Mondays, 4:00 pm - 5:00 pm Location: Lower Napier Building, Room LG11 This seminar is organised by David Baraglia, kindly contact me in order to be included in the mailing-list. 2019 ScheduleSpeaker: Matthias Ludewig Title:The Atiyah-Singer index theorem for the Dirac operator (part 1) (Notes attached) Abstract: We give an introduction to spin geometry and the Atiyah-Singer index theorem, which we prove using the heat kernel method and Getzler‘s rescaling trick. Speaker: Matthias Ludewig Title: The Atiyah-Singer index theorem for the Dirac operator (part 2) (Notes attached) Abstract: We give an introduction to spin geometry and the Atiyah-Singer index theorem, which we prove using the heat kernel method and Getzler‘s rescaling trick. Speaker: Matthias Ludewig Title: The Atiyah-Singer index theorem for the Dirac operator (part 3) (Notes attached) Abstract: We give an introduction to spin geometry and the Atiyah-Singer index theorem, which we prove using the heat kernel method and Getzler‘s rescaling trick. Speaker: Matthias Ludewig Title: The Atiyah-Singer index theorem for the Dirac operator (part 4) Abstract: We give an introduction to spin geometry and the Atiyah-Singer index theorem, which we prove using the heat kernel method and Getzler‘s rescaling trick. Speaker: Mathai Varghese Title: Spin, Spin-c manifolds and transgression formulae for Chern-Weil classes Abstract: I will cover some material omitted by Matthias Ludewig in his lectures. Speaker: David Baraglia Title: Introduction to Riemann surfaces (part 1) Abstract: I will give an introduction to Riemann surfaces including topics such as holomorphic vector bundles and the Riemann-Roch theorem. Prerequisites will be kept to a minimum. Speaker: David Baraglia Title: Introduction to Riemann surfaces (part 2) Abstract: I will continue to discuss holomorphic vector bundles on Riemann surfaces and related topics. Speaker: David Brook Title: An introduction to higher twisted K-theory Abstract: I will introduce the basic notions of topological K-theory allowing for a motivated discussion of higher twisted K-theory, including computations of higher twisted K-groups and a generalisation of the Atiyah-Hirzebruch spectral sequence to this setting. Speaker: Ahnaf Tahabub Title: Meissner Effect on Hyperbolic Space Abstract: Superconductors are characterised by perfect conductivity and perfect diamagnetism. The second property has strictly quantum origins and leads to the well known Meissner Effect - the phenomenon where an external magnetic field decays as one goes deep into a superconductor. This effect was originally modelled by the London Equations which have now been succceded by the Ginzburg-Landau model. The GL theory allows consideration of superconductors on Riemann surfaces i.e. spaces with different geometries to usual flat space. Remarkably the geoemetry impacts the rate of decay of the magnetic field and in this talk I shall compare the decay rates for flat space (genus =1) and that of hyperbolic space (genus >=2). This will be followed by a consideration of how other properties change due to this change in geometry e.g. maximum current, correlation length, flux lines. Finally we will take a brief leap into how this analysis can be extended to the modern BCS theory of superconductivity. Speaker: Nicholas McLean Title: Index Theory on Non-Compact Manifolds Abstract: For compact manifolds, it is well known that all elliptic differential operators are Fredholm if they are acting between the correct Sobolev spaces. However, for non-compact manifolds, the ellipticity of an operator is no longer sufficient to ensure said operator is Fredholm and moreover defining Sobolev spaces becomes more involved as it will depend on a few choices. In this talk, I aim to give a refresher on index theory for compact manifolds, highlight the subtleties which occur when trying to do the same construction for non-compact manifolds, and then give an overview of a method by Roe to circumvent the issues arising from non-compactness. Finally, I will talk about my own research which involves combining Roe's method with the equivariant index method to produce something with slightly more general than what is presented in Roe's paper. Speaker: Johnny Lim Title: Analytic Pontryagin Duality in K-theory Abstract: Let X be a smooth compact manifold. The Universal Coefficient Theorem in K-theory with coefficients in R/Z asserts that there is an isomorphism between the R/Z K-theory group K^i(X,R/Z) and the Hom group Hom(K_i(X),R/Z). We study an explicit analytic duality pairing in the even case which implements the isomorphism. We propose a model of the group K^0(X,R/Z). Together with the even Baum-Douglas geometric K-homology K_0(X), we formulate an analytic pairing comprises of the Dai-Zhang eta-invariant of a certain Dirac-type operator and a topological term. This analytic pairing is well-defined and non-degenerate, thus giving a robust R/Z invariant. As a motivation, we study two special cases of the analytic pairing in cohomology H^1(X,R/Z) and H^2(X,R/Z). Speaker: Matthias Fresacher Title: Properties of the Spectra of an Infinite Random Cayley Graph Abstract: Graphs can have a number of matrices associated to them like the adjacency matrix or transition matrix. When viewed as operators, one can construct the discrete Laplacian operator from these. The spectrum of the Laplacian is of interest, particularly in infinite dimensions. Specifically this talk will introduce the notion of a Cayley graph and examine the spectrum of the associated Laplacian. The extension to a magnetic Laplacian will be discussed as well as possible future paths of investigation such as randomness. Previous Years: |