Undergraduate Course: Discrete Geometry (MATH10090)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||At the heart of many problems in continuous settings lies basic combinatorial issue involving incidence geometry; how lines in euclidean spaces interact.
Even the question of how many intersections can arise from an arbitrary configuration of lines in the plane is nontrivial and answered by the beautiful Szemeredi-Trotter theorem.
Traditionally, problems in incidence geometry are notoriously hard with clever ad hoc methods devised, resulting in incremental progress. Recently an old elementary method, the so-called polynomial method, has been applied to problems in discrete geometry with dramatic effect, often giving definitive resolutions to long-standing problems.
The polynomial method is surprisingly powerful, dating back to Thue and has been used to give an elementary proof of the famous Riemann Hypothesis for curves over finite fields.
We plan to introduce the method by proving the RH for superelliptic curves and then show how it can be used in a variety of problems in discrete geometry; to solve the joints problems (where one asks how many places can a configuration of lines in space meet in three independent directions), to prove the Kakeya and Nikodym problem over finite fields in all dimensions and to give an alternative proof of the Szemeredi-Trotter theorem.
- Overview of problems in discrete geometry, introduction to incidence geometry
- Introduction to polynomial method, the Schwartz-Zippel Lemma and the method of undetermined coefficients.
- Riemann hypothesis for curves in finite fields
- Finite field Kakeya and Nikodym problems
- The Joints Problem
- Cell decompositions in euclidean spaces.
- Applications of cell decompositions: Szemeredi-Trotter theorem and a projection theorem.
- Discussion of other problems: Erdos distance problem and Diophantine equations.
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
Honours Algebra (MATH10069)
||Other requirements|| Students might find it useful to have taken MATH10072 Combinatorics and Graph Theory.
Information for Visiting Students
Course Delivery Information
|Not being delivered|
| 1. Facility with the main ideas behind the polynomial method.
2. Ability to use the polynomial method in simple and concrete situations.
3. Capacity to identify the essential features of ideas and arguments introduced in the course and adapt them to other settings.
4. Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
5. Understand the statements and proofs of important theorems and be able to explain the key steps in proofs, sometimes with variation.
|Graduate Attributes and Skills
|Course organiser||Dr Martin Dindos
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045