Undergraduate Course: Incidence Geometry (MATH11232)
|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
|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 Szemeredi--Trotter 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.
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 Szemeredi--Trotter theorem, proved in 1983, which can be thought of as a `fundamental theorem of incidence geometry'.
In the last 10-15 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 Szemeredi--Trotter theorem) and spectacular breakthroughs on a number of long-standing 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 Szemeredi--Trotter 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 Szemeredi--Trotter theorem due to Solymosi--Tao. 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
|Pre-requisites||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?
Course Delivery Information
|Not being delivered|
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.
|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)
|Graduate Attributes and Skills
|Keywords||IGeo,Combinatorics,Incidence geometry,polynomial method
|Course organiser||Dr Jonathan Hickman
Tel: (0131 6)50 5060
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427