|
| |
|
| |
|
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: Wednesdays, 1:00 pm - 3:00 pm Location: Ingkarni Wardli Building, Room 5.58 This seminar is organised by David Baraglia, kindly contact me in order to be included in the mailing-list. 2015 ScheduleSpeaker: Guo Thiang Title: K-theory applied to topological phases of matter Abstract: In physics, K-theory has traditionally been the preserve of string theorists, but ideas from real K-theory have recently found its way into condensed matter physics. There, the source of topology is the data of quantum mechanical dynamical symmetries, which is concisely encoded by a crossed product C*-algebra and may be twisted and graded. The most important examples are the Clifford algebras, which are closely connected to Bott periodicity in K-theory. The mathematical and physical notions will be introduced rather informally, and developed according to the interests of the audience. Speaker: Guo Thiang Title: K-theory applied to topological phases of matter II Abstract: In the first talk, I explained how compact unitary symmetries are described in terms of the K-theory of the group algebra, which classifies the symmetry-compatible "topological phases" in 0-dimensions. Next I will incorporate anti-unitary symmetries of time-reversal and particle-antiparticle conjugation, and explain how it leads to higher-K-theory groups through Clifford algebras. In non-zero dimensions with non-compact translational symmetries, the famous "topological insulators" make their appearance, and are classified by an appropriate K-theory group. These K-theory groups fit into a remarkable "Periodic Table". Speaker: Masoud Kamgarpour (1-2pm) Title: The Hitchin system and its generalisations Abstract: Discovered while considering dimensional reductions of the Yang-Mills equations, the Hitchin system has emerged as a fundamental object connecting deepest parts of pure mathematics and theoretical physics. After a leisurely introduction to Hitchin’s system, I will define twisted groups and propose a definition of the Hitchin map for these objects. Finally, I will report an on-going project whose aim is to describe the image of the twisted Hitchin maps. As we shall see, intricate combinatorial data attached to reductive groups come forth while studying this image for parahoric groups. Speaker: Andree Lischewski (2-3pm) Title: Supersymmetric M-theory backgrounds and their classification Abstract: I will sketch how the Killing spinor equation arises in the context of supergravity theories. Specialising to the 11-dimensional case, I review some results regarding the classification of M-theory backgrounds which admit Killing spinors, i.e. preserve some supersymmetry. Finally, I will elaborate on the notion of a Killing superalgebra naturally associated to a supergravity background. Speaker: Hang Wang Title: Classical mechanics and symplectic geometry Abstract: In the first talk of the series of lectures introducing geometric quantisation, we shall focus on the classical side. Starting by a motivating example Newton's law of motion, we introduce a mathematical framework (symplectic formulation) of classical mechanics and some basic concepts and facts in symplectic geometry. Speaker: Hang Wang Title: What is Quantisation? Abstract: Based on the analogy between classical mechanics and quantum mechanics of Schrödinger and Heisenberg, Dirac formulated a general quantum condition to pass from a given classical system to the corresponding quantum theory. This process is known as quantisation. In this introductory talk of geometric quantisation, we use an easy example to illustrate the analogy between classical and quantum mechanics, Dirac's rule for quantisation and the necessity to find an intrinsic and constructive description of the quantisation procedure. Speaker: Hang Wang Title: Prequantisation Abstract: A classical mechanical system is formulated as a symplectic manifold together with a Poisson algebra of smooth functions. In this talk, we shall learn how to construct a quantum system of Hilbert space carrying a faithful representation of classical observables for each quantisable symplectic manifold. This is the first step of geometric quantisation, known as prequantisation. Speaker: Hang Wang Title: Polarisation Abstract: Through pre-quantisation, classical observables admits a representation of Lie algebra in a pre-quantum Hilbert space, obtained from sections of a pre-quantum line bundle. It turns out that the pre-quantum Hilbert space is too large for the last principle of Dirac to be satisfied. By cutting down the pre-quantum Hilbert space, polarisations arise naturally in geometric quantisation. As a result, polarisation preserving observables represented in the polarised sections provide an accurate module for quantisation. We mainly focus on two examples (vertical polarisations and holomorphic polarisations). Speaker: Peter Hochs Title: Momentum maps and symplectic reduction Abstract: This is the first in a short series of talks on the `quantisation commutes with reduction' problem. I will start by outlining the general idea, and then define momentum maps. These are conserved quantities in classical mechanics associated to symmetries of a system. Marsden and Weinstein used these in 1974 to define symplectic reduction. This is a way to simplify the description of a classical mechanical system, by using the symmetry it has. Speaker: Peter Hochs Title: Quantising Kähler manifolds Abstract: This talk is about Guillemin and Sternberg’s result from 1982 that quantisation of compact Kähler manifolds commutes with reduction. I will state their result and some consequences, and go into the proof. This result inspired a search for natural generalisations, which I will discuss in the talk(s) after this one. Speaker: Peter Hochs Title: Quantisation and index theory Abstract: Last week I talked about Guillemin and Sternberg’s quantisation commutes with reduction result from 1982. Their quantisation is the space of holomorphic sections of a line bundle on a compact Kähler manifold. Via Dolbeault cohomology, this definition was generalised by Bott to the index of a Dirac operator. Meinrenken showed in 1998 that this version of quantisation still commutes with reduction. Last year, this was generalised from symplectic manifolds to Spin^c manifolds by Paradan and Vergne. In this way, geometric quantisation has evolved from a technique directly motivated by physics to a more general idea linking geometry to representation theory. Speaker: Guo Chuan Thiang Title: Real K-theory and applications in mathematical physics Abstract: Real K-theory has often been given the footnote treatment, having the same essential features as complex K-theory, but with significantly more complicated fine structure. The same can be said for real C*-algebras. Recent developments in physics, such as orientifold string theories and topological insulators, have provided genuine motivation to investigate and exploit this extra structure. I will give a crash course on real K-theory from the point of view of C*-algebras, relate it to the various topological K-theories, and give some examples arising from physics. Speaker: Guo Chuan Thiang Title: Real K-theory and applications in mathematical physics II Abstract: Building on last week's overview of real C*-algebras and group algebras, I will explain real operator K-theory from the point of view of C*-algebras, relate it to the various topological K-theories, and give some examples arising from physics. Speaker: Mathai Varghese Title: Spectral gap-labelling, applications to quantum Hall effect etc Abstract: I will introduce spectral gap-labelling for magnetic Schrodinger operators (and other elliptic operators), discuss an application to the QHE and to Guo Chuan's talk if time permits. Previous Years:Last updated: 17 March 2014 |