Course manual 2018/2019

Course content

Mirror Symmetry investigates a strange connection between two types of geometry: symplectic geometry and algebraic geometry. It was originally discovered in theoretical physics as a duality between two models of string theory: the A- and the B-model. In the 1990's it also became important in mathematics because it could be used to calculate numbers in geometry that mathematicians had tried to find for many years. Since then it has become a main research topic in geometry, algebra and mathematical physics.

In this course we will explore the basic ideas behind mirror symmetry from the point of view of the homological mirror symmetry conjecture. This conjecture formulates an equivalence between two categories: the Fukaya category of a symplectic manifold and the derived category of coherent sheaves of an algebraic variety. We will introduce the mathematics needed to define these two categories such as homology, A-infinity algebras, Floer theory and derived categories. These concepts will be illustrated by some basic examples coming from surfaces. Finally we will work out the mirror correspondence in detail in the cases of the torus.

Study materials

Syllabus

A syllabus will be put on canvas

Objectives

Aan het eind van het vak kan de student:

Calculate simplicial homology and ext-groups
Work with A_infinity categories and their representations
Determine the fukaya category of certain surfaces and cotangent spaces.
Describe the module categories of path algebras and sheaf categories of certain spaces.
State the mirror symmetry conjectures and its generalizations
Illustrate the mirror symmetry conjecture in case of a torus

Teaching methods

Lecture

Theoretical course.

Learning activities

Activity	Hours
Hoorcollege	28
Tentamen	3
Self study	137
Total	168	(6 EC x 28 uur)

Attendance

The programme does not have requirements concerning attendance (OER-B).

Assessment

Item and weight	Details
Final grade
1 (100%) Tentamen

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Motivation from physics
- The road to superstrings
- The A model and the B model
- Open strings and Branes
- Suggested Reading:
  An introduction to topological strings by Collinucci and Wyder
  A mini-course on topological Strings by Vonk
  D-branes on CY-manifolds by Aspinwall
  Mirror Symmetry by Hori et al.
Homology and Cohomology
- The idea behind homology and cohomology
- Geometry: De Rham, Singular and Morse
- Algebra: Ext and Hochschild
- An important observation
- Suggested Reading: Singular and Simplicial, de Rham cohomology, Morse Homology,Resolutions and Extensions, Hochschild cohomology
The A-infinity Formalism
- The inadequacy of algebras
- A-infinity algebra
- The bar construction
- A-infinity categories and completion
- Suggested Reading: Keller, Kontsevich-Soibelman, Segal Chapter 2, Lu, Pamieri, Wu, Zhang
Fukaya Categories
- Intersection theory as a homology theory
- Floer homology
- Self intersections and Morse theory
- The Fukaya category
- Suggested Reading: Smith, Salamon, Oliver Faber's handwritten notes, Meknes
Matrix Factorizations and Derived categories of Coherent sheaves.
- Derived Categories of Coherent sheaves
- Example: The affine line and Mod-C[X]
- Example: The projective line and Tails-C[X,Y]
- The category of singularities and Orlov's theorem.
- Matrix factorizations
- Suggested Reading: Chen and Krause (2, 3, 5), Keller (2.10, 4.7 + figure 3 at the end), Ballard (3.1 and 3.3), Orlov, Brunning and Burban (Ch 1-3)
The first example: Elliptic curves.
- The A-model: lines on a torus.
- The B-model: sheaves on an elliptic curve.
- The correspondence
- Autoequivalences of the category.
- Suggested Reading: Polishchuk and Zaslow, Port, Brunning and Burban (B-side: Ch 4)
The SYZ-conjecture.
- Finding skyscrapers in the Fukaya category
- Moduli spaces of Special Lagrangians
- Stability conditions

Owner	Master Mathematics
Coordinator	dr. R.R.J. Bocklandt
Part of	Master Mathematics, year 1 Master Mathematical Physics, year 1

Mirror Symmetry