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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 David Baraglia, kindly contact me in order to be included in the mailing-list. 2014 ScheduleSpeaker: David Baraglia Title: Fundamentals of L^p and Sobolev Spaces Abstract: This will be a review of the fundamentals of L^p-spaces and Sobolev spaces following the textbook "Sobolev Spaces" by Adams. Speaker: Pedram Hekmati Title: Introduction to Sobolev Spaces Abstract: This is a continuation of last week's talk. We will first review distributions and weak derivatives. We then proceed to define Sobolev spaces, cover their basic properties and time permitting, begin discussing Sobolev embedding theorems. Speaker: Peter Hochs Title: Embeddings of Sobolev spaces Abstract: I will discuss which Sobolev spaces are contained in which other spaces of functions. We will also see that some of these inclusion maps are compact operators. This is important for showing that certain differential equations have finite-dimensional solution spaces. Speaker: David Baraglia Title: Sobolev spaces on compact manifolds Abstract: So far we have looked at Sobolev spaces for open subsets of Euclidean space. I will show how these results can be transported over to the setting of compact manifolds. In particular we will see that the Sobolev embedding and compactness theorems are simple consequences of the corresponding theorems in the Euclidean case. Speaker: Hang Wang Title: Dirac operators on 4-dimensional manifolds Abstract: In this talk, we shall introduce Spin^c structures, spin connections, Clifford algebra and Dirac operators on a 4-manifold. These are necessary elements for introducing Seiberg-Witten equations. My talk follows Moore’s “Lecture Note on Seiberg-Witten Invariants” Chapter 2. Speaker: Hang Wang Title: Spin^c structures and characteristic classes Abstract: From the perspective of algebra topology, we characterise spin and spin^c structures, using a mild introduction of characteristic classes. This will be used to show that an oriented 4-manifold admits a spin^c structure. We will also explain the topological terms in the Atiyah-Singer index formula for Dirac operators on a 4-manifold with coefficients in a line bundle. Speaker: Vincent Schlegel Title: The BV Formalism Abstract: In this talk, I will describe a method in quantum field theory called the BV formalism. The BV formalism - named for its pioneers Batalin and Vilkovisky - is designed to treat the problem of path integration via homological algebra and lies at the crossroads of mathematical physics, graded symplectic geometry and derived geometry. By way of a circuitous route through some of the common techniques of quantum field theory, I will first motivate and then describe the BV formalism, with particular emphasis on illuminating examples. Speaker: David Baraglia Title: Spin^c connections and their Dirac operators Abstract: I will continue with some background material for the Seiberg-Witten equations. This time we will look at Spin^c connections and their associated Dirac operators. I'll aim to cover the Weitzenbock formula and the index theorem. Speaker: David Baraglia Title: The Seiberg-Witten equations Abstract: I will introduce the Seiberg-Witten equations and say a few things about their importance in the study of smooth 4-manifolds. I'll prove some basic results about the equations in preparation for studying their moduli spaces in upcoming seminars. Speaker: Pedram Hekmati Title: The Seiberg-Witten Moduli Spaces Abstract: I will initiate the study of the moduli space of solutions to the Seiberg-Witten equations. The aim is to describe the Sobolev completion of the space of solutions modulo gauge transformations and show that it is a smooth Banach manifold away from singularities at so called reducible elements. Speaker: Pedram Hekmati Title: Compactness of the moduli space Abstract: I will complete the discussion on the Sobolev completion of the moduli space of solutions to the perturbed Seiberg-Witten equations and begin to prove compactness of the moduli space in the case when the 4-manifold is simply connected. Speaker: David Baraglia Title: Compactness of the Seiberg-Witten moduli space Abstract: After reviewing the Seiberg-Witten equations, I will give the proof of compactness of the moduli space. The proof is a nice application of the Sobolev compactness theorem and tools from the theory of elliptic operators. Speaker: David Baraglia Title: Compactness of the Seiberg-Witten moduli space part 2 Abstract: I will give the proof of compactness of the Seiberg-Witten moduli space. The proof is a nice application of the Sobolev compactness theorem and tools from the theory of elliptic operators. Speaker: Hang Wang Title: A dimension formula of Seiberg-Witten moduli spaces Abstract: We will prove the formula of the dimension of Seiberg-Witten moduli spaces. It is related to the index of the Dirac operator in the Seiberg-Witten equation. Speaker: David Baraglia Title: Donaldson's theorem via Seiberg-Witten theory Abstract: We have reached a point where we can give our first major application of Seiberg-Witten theory, namely a proof of Donaldson's diagonalisation theorem for compact smooth 4-manifolds with definite intersection form. Before getting to the proof, I will review some facts about the topology of 4-manifolds and their intersection forms. Speaker: David Baraglia Title: Donaldson's theorem via Seiberg-Witten theory part 2 Abstract: I will give the Seiberg-Witten theory proof of Donaldson's theorem on diagonalisation of intersection forms of definite 4-manifolds. Speaker: Hang Wang Title: Orientation of the moduli space Abstract: I will show that the smooth manifold formed by irreducible points of the moduli space of the Seiberg-Witten equation is oriented. Speaker: Peter Hochs Title: Seiberg-Witten invariants Abstract: I will discuss the definition of Seiberg-Witten invariants of Spin-c structures on compact 4-manifolds, and some applications and examples. Previous Years:Last updated: 21 October 2014 |