Mirror Symmetry
Course description
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 Bmodel. 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, Ainfinity 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.
Practicalities
The lectures take place on Tuesdays in the afternoon. More info can be found on datanose.The evaluation will consist of an exam (75%) and exercises (25%) More information about this will be made available later in the course.
Syllabus
The course will consist of seven chapters. Each of these chapters will take about 2 weeks. The students will get notes for each chapter with exercises they have to work out. The notes will appear gradually on this site but additional literature is already available for each chapter.Exam
Details about the final exam will be discussed during class.Structure of the course
The course will treat the following topics.
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 minicourse on topological Strings by Vonk
Dbranes on CYmanifolds 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 Ainfinity Formalism
 The inadequacy of algebras
 Ainfinity algebra
 The bar construction
 Ainfinity categories and completion
 Suggested Reading: Keller, KontsevichSoibelman, 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 ModC[X]
 Example: The projective line and TailsC[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 13)

The first example: Elliptic curves.
 The Amodel: lines on a torus.
 The Bmodel: sheaves on an elliptic curve.
 The correspondence
 Autoequivalences of the category.
 Suggested Reading: Polishchuk and Zaslow, Port, Brunning and Burban (Bside: Ch 4)

The SYZconjecture.
 Finding skyscrapers in the Fukaya category
 Moduli spaces of Special Lagrangians
 Stability conditions