## PURE MTH 3024 - Finite Geometry III

### North Terrace Campus - Semester 2 - 2018

Projective geometry is one of the important modern geometries introduced in the 19th century. Projective geometry is more general than our usual Euclidean geometry, and it has useful applications in Information Security, Statistics, Computer Graphics and Computer Vision. The majority of this course will be on projective planes. Topics covered are: projective planes, homogeneous coordinates, field planes, collineations of projective planes, conics in field planes, k-arcs in projective planes, projective geometry of general dimension, quadrics and ovoids in 3-dimensional projective space.

• General Course Information
##### Course Details
Course Code PURE MTH 3024 Finite Geometry III School of Mathematical Sciences Semester 2 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y MATHS 1012 Projective geometry is one of the important modern geometries introduced in the 19th century. Projective geometry is more general than our usual Euclidean geometry, and it has useful applications in Information Security, Statistics, Computer Graphics and Computer Vision. The majority of this course will be on projective planes. Topics covered are: projective planes, homogeneous coordinates, field planes, collineations of projective planes, conics in field planes, k-arcs in projective planes, projective geometry of general dimension, quadrics and ovoids in 3-dimensional projective space.
##### Course Staff

Course Coordinator: Dr Susan Barwick

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
1. Demonstrate a deep understanding of the axiomatic approach to projective spaces.
2. Be able to perform calculations in Desarguesian planes and projective 3-spaces.
3. Classify the structure of collineations of projective planes.
4. Demonstrate an understanding of the theory of conics in field planes.
5. Apply the theory to solve problems of varying levels of difficulty.
6. Demonstrate skills in communicating mathematics orally and in writing.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1,2,3,4,5
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
1,2,3,4,5
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
6
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
5,6
Intercultural and ethical competency
• adept at operating in other cultures
• comfortable with different nationalities and social contexts
• able to determine and contribute to desirable social outcomes
• demonstrated by study abroad or with an understanding of indigenous knowledges
6
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
6
• Learning Resources
None.
##### Recommended Resources
The material in the course is closely related to the textbook:
L.R.A. Casse, Projective Geometry, An Introduction.

##### Online Learning
This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, etc.
• Learning & Teaching Activities
##### Learning & Teaching Modes
The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload Hours Lectures 30 90 Tutorials 5 25 Online Quizzes 6 Assignments 5 35 Total 156
##### Learning Activities Summary
Lecture Outline
1. Projective planes, 3 lectures (extended Euclidean plane, finite projective planes)
2. Projective space, 2 lectures (extended Euclidean 3-space, r-dimensional projective space)
3. Field planes, 3 lectures (fields, homogeneous coordinates, subplanes)
4. Collineations, 7 lectures (homographies, automorphic collineations, fundamental theorem of field planes, central collineations, elations, homologies)
5. PG(r,F), 1 lecture
6. Conics in PG(2,F), 5 lectures (J's equation, polarity, conics when charF=2, conics in the real projective plane)
7. Combinatorial structures (latin squares, incidence matrices, difference set, 8 lectures
##### Small Group Discovery Experience
The final topic on Combinatorial Structures will be conducted in a flipped classroom mode, and students will work in small groups of 2-3 students during the lecture time slots.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment Task Weighting Learning Outcomes exam 70% all assignments 15% all tutorials and quizzes 15% all
##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail
Assignments are due in weeks 3, 5, 7, 9, 11, each is worth 3%, contributing a total of 15%.

Tutorials and online quizzes will be regularly held to enable active learning. These contribute a total of 15%.
##### Submission
Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments accepted at the discretion of the lecturer. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons.  Documentation is required and the lecturer must be notified as soon as possible.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

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