Join the course

Introduction: 

Algebraic geometry is a branch of mathematics that studies the geometry of solutions of systems of polynomial equations, known as algebraic varieties. In this 4-lecture course, we will introduce undergraduates to the classical approach of algebraic geometry, with minimal prerequisites and a motivated problem-driven approach. 

The interest in algebraic geometry is not confined to pure mathematicians. In fact, algebraic geometry provides a powerful framework for studying a wide range of mathematical problems and has applications in many areas of science and computer science, including cryptography, computer vision, robotics, and physics.

We will start by discussing natural questions arising from other areas of mathematics such as integrals and basic number theory, which historically motivated the study of algebraic geometry. From there, we will develop algebraic geometry systematically using the classical approach. 

The topics we intend to cover include:

•  Lecture 1: motivations and background 

  • 1. Motivations and overview. 

    2. Review on some basic algebra. 

    3.  (weak) Nullstellensatz 

    4. Algebraic sets: Definition, examples, and properties

    5.  Zariski topology: Definition and properties

SLIDES OF LECTURE#1

VIDEO RECORDING

• Lecture 2: General Theory 

1.  Affine and projective varieties: Definition, examples, and properties – Regular functions and coordinate rings: Definition and examples

2.  Morphisms and rational maps: Definition and examples. 

3.  Dimension of varieties.

4.  Smoothness of varieties.

SLIDES OF LECTURE#2

     VIDEO RECORDING

•  Lecture 3: Curves 

1. Intersection theory on projective planar curves. 

2. Bezout theorem for curves.

3.  Applications. 

4. (optional) Divisors and Riemann-Roch for curves. 

•  Lecture 4: Surfaces, computations, and outlook

1. Cayley–Salmon theorem on Surfaces. 

2. An overview of computational techniques: Gr¨obner basis and Macaulay2.

3. Glimpse into schemes

Required Background:

For the background, we assume familiarity with linear algebra and basic analysis (e.g. partial derivatives), basic algebra (e.g. complex numbers and fundamental theorem of algebra), as well as basic definitions and facts about ideals and rings. We shall recall all the facts that we need from other areas of mathematics as we go along. 

Schedule: 

1. Lecture 1: July 08, 2023, 11AM CET (2PM Pakistan Time)

2. Lecture 2: July 15, 2023, 11AM CET (2PM Pakistan Time)

3. Lecture 3: July 22, 2023, 11AM CET (2PM Pakistan Time)

4. Lecture 4: July 29, 2023, 11AM CET (2PM Pakistan Time)

Please write an email on themathetf@gmail.com to us if you want to join the online lectures.