### Algebraic Geometry for Everyone

The aim of this course is to present the beauty and interest of algebraic geometry, rather than being pedantic about all the mathematical details. We will show some of the interesting (classical) results in the case of curves and surfaces, as well as working with concrete examples. We will briefly touch on the modern approach via schemes and recommend further reading for those interested.

Instructor: Fawzy Hegab, Berlin Mathematical School, Berlin

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

Motivations and overview.

Review on some basic algebra.

(weak) Nullstellensatz

Algebraic sets: Definition, examples, and properties

Zariski topology: Definition and properties

Lecture 2: General Theory

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

Morphisms and rational maps: Definition and examples.

Dimension of varieties.

Smoothness of varieties.

Lecture 3: Curves

Intersection theory on projective planar curves.

Bezout theorem for curves.

Applications.

(optional) Divisors and Riemann-Roch for curves.

Lecture 4: Surfaces, computations, and outlook

Cayley–Salmon theorem on Surfaces.

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

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:

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

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

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

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.