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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 5.57 This seminar is organised by Pedram Hekmati, kindly contact me in order to be included in the mailing-list. Current Schedule Semester 2, 2013 ScheduleSpeaker: Antti Harju (University of Helsinki) Title: Twisted K-Theory and Families index Problem on Product Manifolds Abstract: Consider a product manifold $\mathbb{T} \times M$ with nontrivial twisted 3-cohomology classes. I will discuss gerbes, twisted K-theory elements and twisted index maps with twisting coming from a product class. This theory has been developed by Mathai-Melrose-Singer and Harju-Mickelsson. I will try to clarify the relationship between these two approaches. Title: Quantization and categorification 1-4 Abstract: Both "quantization" and "categorification" have influenced many recent developments in pure mathematics and modern mathematical physics, ranging from applications in knot theory to geometric representation theory. Yet, these deep and fundamental concepts can be explained in simple and concrete examples, which will be one of the main goals in my lectures. I will follow a "hands-on" approach, aimed at understanding explicit calculations in addition to learning the general theory. For example, we will see how a simple 19th century combinatorial problem can provide an answer to colored knot homology, and how the same answer can be reproduced from "quantization of algebraic curves," subject that has been very popular in recent years in physics as well as in pure math. Moreover, with the help of these illustrative examples I will try to explain that in many problems coming from the study of knots and 3-manifolds there is a deep connection between "categorification" and "quantization." We shall see how this connections leads to many new results and exciting conjectures. Speaker: David Roberts Title: An explicit String bundle Abstract: An n-manifold is String if it the classifying map for its frame bundle can be lifted to the 4-connected cover of BO(n). This cover corresponds abstractly to a 3-connected topological group with a homomorphism to O(n). However, the usual smooth constructions of this cover use categorified groups, or 2-groups; we thus have the Frechet-Lie strict 2-group String(n), described using the universal central extension of Spin(n). The lifted frame bundle is then a 2-bundle. We aim to describe, in gory detail, the lift of the structure group of FS^5 to Spin(5) (and in fact something a little smaller). Title: Survey of the Bost-Connes Problem Abstract: This talk will consist mostly of an introduction to the work on the Bost-Connes Problem. This problem has come about due to the work of Bost and Connes in 1995 as a by-product of their attempts to translate the Riemann Hypothesis into noncommutative geometric terms. Since then there has been much work since the realisation that solving this problem can have a significant impact on number theory. I will provide a survey of the work that has been done and include a brief survey of my own work in this direction. Title: Oka properties for Riemann surfaces Abstract: I will give a quick introduction to Oka theory before moving on to the restricted setting of maps between Riemann surfaces. Important to Oka theory are a number of flexibility properties of holomorphic maps, and my project has involved classifying the pairs of Riemann surfaces for which these properties hold. I will give the result, and talk about parts of the proof as time allows. Semester 1, 2013 ScheduleSpeaker: David Roberts Title: On the Baas-Dundas-Rognes approach to 2-vector bundles Abstract: Baas, Dundas and Rognes introduced 2-vector bundles as a categorification of vector bundles using the 2-vector spaces of Kapranov and Voevodsky. Bundle gerbes, when defined using line bundles, constitute the rank-one case. This talk aims to motivate and summarise the definition of BDR 2-vector bundles, and discuss some of the results proved, in particular that virtual 2-vector bundles represent elements in the algebraic K-theory of the K-theory spectrum. Speaker: David Baraglia Title: Rational homotopy theory and Kähler manifolds Abstract: A major theme in rational homotopy theory is the notion that differential forms can be used to detect homotopy theoretic invariants other than cohomology. I will introduce the theory and give an application - the formality theorem for compact Kähler manifolds. Speaker: Peter Hochs Title: Introduction to coarse geometry Abstract: When passing from a metric space to the underlying topological space, one retains the most essential information about its local structure and general shape. Information about "how big" the space is, with respect to the metric, is lost however. Coarse geometry complements topology, in the sense that the coarse space underlying a metric space still encodes the "size" of the space, although much topological information is lost. A key example is the Euclidean real line. Topologically, it is homeomorphic to the open unit interval, which is bounded. Coarsely, it is equivalent to the integers, which extend towards infinity in roughly the same way as the real numbers, but have very different topological properties. It turns out that coarse geometry has applications in several areas in geometry and topology, mainly of noncompact spaces. In this talk, I will introduce coarse geometry, and the natural notion of algebraic topology-type invariants of coarse spaces: Roe algebras and their K-theory. Speaker: Pawel Nurowski (Center for Theoretical Physics of the Polish Academy of Sciences) (Room 5.56) Title: Irreducible SO(3) geometry in dimension 5 Abstract: I will discuss a special Riemannian geometry (M^5, g, Y) in dimension 5 in which Y is a tensor that reduces the structure group of the tangent bundle from GL(5,R) to SO(3) sitting irreducibly in SO(5). Condition for compatibilty between g and Y will be given which guarantee that (M^5, g, Y) defines a unique metric connection with totally antisymmetric torsion. Examples of such geometries satisfying (sort of) Strominger equations from string theory will be given. IGA Workshop featuring lectures by Prof Jouko Mickelsson (University of Helsinki) Speaker: Christian Sämann (Heriot-Watt University) (Wednesday) Title: M2-brane models Abstract: I will talk about the physics of the M2-brane models proposed in 2008, starting from Basu-Harvey equation that led to their development. These models pass a number of important consistency checks, which I will briefly review. I will then discuss some recent developments, such as conformal deformations, quantum geometries arising in the context of these models and how the they are related to M5-brane models. Speaker: Tyson Ritter Title: Symplectic submanifolds and almost-complex geometry Abstract: I will explain Donaldson's construction of symplectic submanifolds of any even codimension within a given compact symplectic manifold V. The construction makes use of 'approximately holomorphic' sections of certain complex line bundles on V. Speaker: Pedram Hekmati (Thursday) Title: Aspects of Seiberg-Witten theory Abstract: In 1994, Seiberg and Witten introduced a new gauge theory capable of producing a system of invariants that revolutionised the study of 4-manifolds. Indeed, within a few months, several long-standing conjectures were proven using this new theory. This was the culmination of work started in 1983 with Donaldson's instanton invariants and there is a precise (conjectured) relationship between these invariants via S-duality. The aim of this talk is to introduce the Seiberg-Witten equations and review some of their applications. Speaker: David Roberts (Monday, Room 5.56) Title: Kapranov-Ganter on categorified supersymmetry Abstract: This talk is a summary of an approach to categorifying supersymmetry by M. Kapranov and N. Ganter. There are intriguing connections to low-dimensional truncations of the sphere spectrum and also classical superrepresentation theory. Speaker: Peter Hochs Title: Representations of compact Lie groups Abstract: I will review representation theory of compact Lie groups, leading up to the Weyl character formula. (Which will be discussed by Higson at the workshop in July.) The group SU(n) will be used as an example. If time permits, I will also discuss (a very small part of) representation theory of semisimple Lie groups. Speaker: Peter Hochs Title: C^*-algebras and K-theory Abstract: Since all commutative C^*-algebras are isomorphic to function algebras, noncommutative C^*-algebras may be viewed as function algebras on “noncommutative spaces”. In noncommutative geometry, one tries to generalise techniques from topology or geometry to such noncommutative algebras. This works very well in the case of K-theory. I will introduce C^*-algebras and K-theory, and in particular their links to representation theory of Lie groups. Speaker: Mathai Varghese Title: K-homology of C^*-algebras Abstract: I will motivate the definition of K-homology, give the definition and key properties. Then will give methods for calculation, variants of the definition such as geometric K-homology and extensions. Speaker: Laura Schaposnik (Ruprecht-Karls Universtät, Heidelberg) Title: The Hitchin fibration and real forms through spectral data Abstract: The talk will be dedicated to the study of the moduli space of G-Higgs bundles and the Hitchin fibration through spectral data. For this we shall recall general properties of G-Higgs bundles for G a real form of a complex Lie group, and then define the spectral data associated to them. By looking at some examples, we shall show some of the applications of this new geometric way of understanding the moduli space using spectral data. IGA/AMSI Workshop featuring lectures by Prof Nigel Higson (Penn State University) Previous Years:Last updated: 9 July 2013 |