2014 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 5.57

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

2014 Seminars

• 18 March
  Speaker: 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.

• 25 March
  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.

• 1 April
  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.

• 8 April
  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.

• 15 April
  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.

• 29 April
  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.

• 9 May
  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.

• 13 May
  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.

• 20 May
  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.

• 3 June
  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.

• 10 June
  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.

• 26 August
  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.

• 9 September
  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.

• 16 September
  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.

• 23 September
  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.

• 30 September
  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.

• 14 October
  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.

• 21 October
  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.