MATH426-22S1 (C) Semester One 2022

# Geometry

15 points

Details:
 Start Date: Monday, 21 February 2022 End Date: Sunday, 26 June 2022
Withdrawal Dates
Last Day to withdraw from this course:
• Without financial penalty (full fee refund): Sunday, 6 March 2022
• Without academic penalty (including no fee refund): Sunday, 15 May 2022

## Description

The course deals with advanced topics in geometry

Geometry is the area of mathematics that studies the notions of shape, space and relative position. Attempts to prove Euclid’s parallel postulate led to the discovery of many different types of geometry.

In this course, we will focus on projective geometry as many other geometries such as affine, hyperbolic or Euclidean can be modelled using projective spaces. Projective geometry also forms the base for more advanced algebraic geometry.

We will investigate the role of the classical theorems of Desargues, and Pappus, study the cross ratio, investigate groups acting on projective spaces and study conic sections and quadrics. Finally, we will take an axiomatic approach to deal with non-Desarguesian projective planes.

Additional topics may include the link of projective spaces over a finite field with latin squares, coding theory and design theory.

## Learning Outcomes

• Competently use projective concepts such as homogeneous coordinates and cross-ratios over different fields.
• Prove basic but fundamental theorems about collineations, polarities and conics in projective planes and spaces.
• Relate the study of group theory to projective geometry.
• Explore projective planes from an axiomatic point of view.
• Develop a good understanding of what a mathematical proof entails.
• Develop written and oral communication skills, emphasising the ability to explain what the mathematics means.

This course will provide students with an opportunity to develop the Graduate Attributes specified below:

 Critically competent in a core academic discipline of their award Students know and can critically evaluate and, where applicable, apply this knowledge to topics/issues within their majoring subject. Employable, innovative and enterprising Students will develop key skills and attributes sought by employers that can be used in a range of applications.

## Prerequisites

Subject to approval of the Head of School.

## Timetable 2022

Students must attend one activity from each section.

Activity Day Time Location Weeks Lecture A 01 Tuesday 13:00 - 14:00 Jack Erskine 505 21 Feb - 10 Apr 2 May - 5 Jun Lecture B 01 Wednesday 14:00 - 15:00 Jack Erskine 505 21 Feb - 10 Apr 2 May - 5 Jun Lecture C 01 Friday 13:00 - 14:00 Jack Erskine 505 21 Feb - 10 Apr 2 May - 5 Jun

Assignments 20%
Test 25%
Presentation 25%
Exam 30%