2012 Strings Journal Club Seminars

A learning seminar on topics related to the geometric aspects of string theory. Several mathematical subjects, including topology, representation theory, and operator algebras are also discussed.

The notes of a few talks are available below.

Time: Tuesdays, 1:00 pm - 3:00 pm
Location: Ingkarni Wardli Building, Room 5.56

This seminar is organised by Pedram Hekmati, kindly contact me in order to be included in the mailing-list.

2012 Seminars

• January 24
  Speaker: S. Mahanta
  Title: Gromov-Witten Theory
  
  Abstract: I shall present my understanding of Gromov-Witten theory. This is a preparation for the upcoming IGA/AMSI workshop on Gauge-String Dualities.

• March 15 (Special time: Thursday 2-3 pm)
  Speaker: V. Mathai
  Title: Introduction to T-duality
  Venue: Room 5.57 Innova 21

• 14 August
  Speaker: David Baraglia
  Title: Constructions of hyperKähler manifolds
  
  Abstract: HyperKähler manifolds are Riemannian manifolds with holonomy group in Sp(n). Much rarer than Kähler manifolds, the list of known examples is short but includes some rich geometry such as K3 surfaces, gravitational instantons and moduli spaces of Higgs bundles on a Riemann surface. I will give an introduction to hyperKähler manifolds highlighting some examples and the techniques behind their construction.

• 21 August
  Speaker: David Baraglia
  Title: ALE spaces and the McKay correspondence
  
  Abstract: ALE hyperKähler manifolds lie at the intersection of a variety of topics spanning algebra topology and geometry. These aspects can be related to the simply laced Dynkin diagrams. I will give Kronheimer's construction of the ALE manifolds and explain some of the links to minimal resolutions, the McKay correspondence, quiver varieties and instantons.

• 28 August
  Speaker: Wolfgang Globke
  Title: Aspects of supermanifolds
  
  Abstract: Supermanifolds provide a geometrical framework for the supersymmetries arising in particle physics. Several different definitions have been given. I will introduce the definition of supermanifolds accredited to Berezin-Leites and Kostant, and discuss some of their simple properties.

• 4 September
  Speaker: David Roberts
  Title: Sheaf cohomology via abelian categories
  
  Abstract: One of the most powerful cohomology theories is sheaf cohomology, that is, cohomology with coefficients in a sheaf of modules on a space. It is defined using derived functors on abelian categories with enough injectives, in great generality, but can be computed in a number of concrete ways which are familiar from differential geometry. The aim of this talk to give an idea of the technical origins of sheaf cohomology and also to illustrate these concepts with down-to-earth examples. A basic familiarity with categories and functors will be assumed.

• 11 September
  Speaker: David Baraglia
  Title: What are derived categories and why do we need them?
  
  Abstract: Many instances of cohomology that arise in practice are obtained by constructing a chain complex and taking ordinary cohomology. I will show by examples why it is a bad idea to then discard the chain complex in favour of its cohomology groups. However, such chain complexes are often constructed in a non-canonical manner. Accounting for this non-uniqueness leads directly to derived categories.

• 18 September
  Speaker: Mathai Varghese
  Title: Superconnections
  
  Abstract: This will be an introduction to superconnections and some of their applications, including twisted analytic torsion and index theory.

• 25 September
  Speaker: Pedram Hekmati
  Title: Constrained quantisation and supermanifolds
  
  Abstract: I will explain some approaches to quantisation in the presence of symmetries, including BRST quantisation and the BV formalism, and emphasise the underlying supersymmetric structures.

• 2 October
  Speaker: David Baraglia
  Title: Some differential geometric constructions via supermanifolds
  
  Abstract: I will give examples of how the language of supermanifolds can be used to interpret some operations in differential geometry including Frolicher-Nijenhuis and Schouten brackets and the relation to complex and Poisson manifolds.

• 9 October
  Speaker: David Roberts
  Title: Elliptic cohomology - first steps
  
  Abstract: Elliptic cohomology is the name of a type of generalised cohomology theories which arose in the late 80s and early 90s. Since then these cohomology theories have been assembled into what is known as tmf, standing for 'topological modular forms'. In this talk I aim to explain how elliptic cohomology/-ies are constructed, how they are related to genera such as the Witten genus and hopefully some ideas about how tmf is defined. I will unfortunately not cover the deep links elliptic cohomology has to number theory and the stable homotopy groups of the spheres.

• 15-19 October
  IGA/AMSI Workshop: Geometry of Supermanifolds

• 23 October
  Speaker: Michael Albanese
  Title: An Introduction to Kähler Manifolds
  
  Abstract: After outlining some of the many equivalent ways to define Kähler manifolds, I will give some examples (and non-examples). Then I will move on to the Kähler identities and describe their implications at the level of cohomology.

• 30 October
  Speaker: Wolfgang Globke
  Title: Special Kähler Manifolds
  
  Abstract: I will give a short presentation of some elementary properties of special Kähler manifolds and describe their realisations as submanifolds of certain real and complex affine spaces.

• 6 November
  Speaker: Nicholas Buchdahl
  Title: Rudiments of twistor theory
  
  Abstract: Twistor theory has many different interpretations, some (ostensibly) bearing little relation others. My aim in this talk is to give you my own interpretation, as I learned it when I did my D.Phil. I make no claims to any expertise, not having worked in twistor theory since about that time.

• 13 November
  Speaker: Arman Taghavi-Chabert (Masaryk University Brno)
  Title: Pure spinors and curvature in higher dimensions
  
  Abstract: We give a generalisation of the Penrose-Petrov classification of the Weyl tensor from four to higher dimensions for pseudo-Riemannian manifolds of complex or split signature, based on the concept of principal pure spinors. We classify the various degrees of integrability of the totally null distribution associated to a pure spinor field. The relation between these classifications and solutions to spinorial differential equations such as the twistor equation is discussed. Finally, we apply these results to characterise the Riemannian extension of a projective structure.

• 27 November
  Speaker: Tyson Ritter
  Title: Introduction to toric varieties
  
  Abstract: Toric varieties form an important class of algebraic varieties, often occurring as the simplest non-trivial examples exhibiting interesting and important phenomena in algebraic geometry. A toric variety is completely determined by the associated combinatorial information of a fan of cones in a real vector space, making their theory explicitly computable in many instances. I will cover some introductory topics in the theory of toric varieties, giving the construction of affine and projective toric varieties from their combinatorial information, with several examples along the way.

• 4 December
  Speaker: Ryan Mickler
  Title: Bundle Gerbes in Geometric Quantization
  
  Abstract: I will talk about some ideas that relates bundle gerbes and geometric quantization.