Undergraduate Course: Incidence Geometry (MATH11232)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Year 5 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  Classical incidence geometry is a branch of combinatorics which studies intersection patterns between finite families of points, lines, circles and other simple shapes. In this course we will study some core topics in incidence geometry from the perspective of a recently developed tool called the polynomial partitioning method. The first goal will to be to understand the SzemerediTrotter theorem, a fundamental result in the field. We then look at other applications of polynomial partitioning, such as counting circle tangencies, the famous Erdos unit distance problem. 
Course description 
Classical incidence geometry is a branch of combinatorics which studies intersection patterns between finite families of points, lines, circles and other simple shapes. For instance, given a set of lines L and points P in the plane, we wish to count the maximum number of times a point from P lies in a line from L.
This question turns out to be surprisingly deep, and relies on analytic properties of the plane. It was answered by SzemerediTrotter theorem, proved in 1983, which can be thought of as a `fundamental theorem of incidence geometry'.
In the last 1015 years there has been a revolution in the field of incidence geometry brought about by the introduction of the polynomial method. This has led to new and simpler proofs of the central results in the area (such as the SzemerediTrotter theorem) and spectacular breakthroughs on a number of longstanding conjectures. One remarkable feature of the polynomial method is that in many cases it provides very short proofs involving only elementary tools from linear algebra and polynomial arithmetic.
In this course we will study some core topics in incidence geometry from the perspective of the polynomial method and, in particular, polynomial partitioning. The first goal will to be to understand the SzemerediTrotter theorem. We then look at other applications of polynomial partitioning, such as counting circle tangencies, the famous Erdos unit distance problem and a generalisation of the SzemerediTrotter theorem due to SolymosiTao. We then move on to some more recent developments such as the finite field Kakeya theorem and the joints theorem, both of which have strikingly simple proofs. If time permits, in the last part of the course we will study the Kakeya problem in the Euclidean plane.

Information for Visiting Students
Prerequisites  Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling. 
High Demand Course? 
Yes 
Course Delivery Information
Not being delivered 
Learning Outcomes
On completion of this course, the student will be able to:
 Be able to work with the polynomial method in incidence geometry by proving unseen results using the methods of the course
 Correctly state the main definitions and theorems in the course
 Produce examples and counterexamples illustrating the mathematical concepts presented in the course
 Explain their reasoning about geometric counting problems clearly and precisely, using appropriate technical language.

Reading List
The course does not follow a textbook. Notes will be provided. A good reference book is:
Polynomial Methods in Combinatorics, Larry Guth (AMS University Lecture Series Volume: 64, 2016) 
Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  IGeo,Combinatorics,Incidence geometry,polynomial method 
Contacts
Course organiser  Dr Jonathan Hickman
Tel: (0131 6)50 5060
Email: Jonathan.Hickman@ed.ac.uk 
Course secretary  Mr Martin Delaney
Tel: (0131 6)50 6427
Email: Martin.Delaney@ed.ac.uk 

