This course introduces students to key concepts and techniques in advanced algebra. Topics will be taken from contemporary research areas in Algebra.

Possible topics include:

Algebraic number theory, Analytic number theory, Algebraic geometry and scheme theory, Sheaf theory, Derived and Triangulated categories, Algebraic curves and Riemann surfaces

Note: This is an HPC. It emphasizes mathematical rigour and proof and continues the development of modern algebra from an abstract viewpoint.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of advanced algebra and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of advanced algebraic techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanced algebra
4. Apply problem-solving using advanced algebraic techniques applied to diverse situations in physics, engineering and other mathematical contexts

Required Resources

Required textbook: Basic Algebraic Geometry, Part I by Igor Shafarevich

Recommended Resources

Recommended student system requirements 

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

For more information please see https://www.anu.edu.au/students/systems/recommended-student-system-requirements

Staff Feedback

Students will be given feedback in the following forms in this course:

• written comments
• verbal comments
• feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Other Information

The lectures and workshops will be live-streamed and recorded for remote participants. Remote participants will also be able to engage in in-person activities via Zoom and other online tools.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	What is algebraic geometry? The first examples: algebraic subsets of affine space	Submission of assignment 0
2	Regular functions and the Nullstellensatz	Submission of assignment 1
3	Regular maps	Submission of assignment 2
4	General algebraic varieties	Submission of assignment 3
5	Functions and maps on projective varieties	Submission of assignment 4
6	Products and the Segre embedding	Mid-semester exam
7	Completeness of projective varieties	Submission of assignment 5
8	Irreducibility	Submission of assignment 6
9	Rational maps	Submission of assignment 7
10	Dimension	Submission of assignment 8
11	Grassmannians	Submission of assignment 9
12	Tangent spaces and singularities	Submission of assignment 10

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Assignments x 10	50 %	*	*	1,2,3,4
Mid-semester Exam	25 %	30/08/2021	24/09/2021	1,2,3,4
Final Exam	25 %	04/11/2021	02/12/2021	1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information 
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Integrity . In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Examination(s)

There will be a mid-semester examination and a final examination. A passing mark in both examinations is a hurdle in successful completion of the course.

Assessment Task 1

Assignments x 10

There will be weekly assignments (11 in total) each worth 5% of the final mark. The assignment with the lowest mark will be dropped from the calculation.

Assessment Task 2

Mid-semester Exam

The mid-semester exam may be

(a) an in-person exam, administered by the lecturer in a covid safe manner. In this case, remote participants will be able to take an exam invigilated through Zoom.

(b) a take-home exam.

The convener will clarify the format after taking into account pandemic requirements and other considerations and will notify all enrolled students at least two weeks prior to the start of the mid-semester exam period.

The exam will count towards 25% of the final mark.

Assessment Task 3

Final Exam

The final exam may be

(a) an in-person exam, administered by the lecturer in a covid safe manner. In this case, remote participants will be able to take an exam invigilated through Zoom.

(b) a take-home exam.

The convener will clarify the format after taking into account pandemic requirements and other considerations, and will notify all enrolled students at least two weeks prior to the start of the end of semester exam period

The exam will count towards 25% of the final mark.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. At its heart, academic integrity is about behaving ethically, committing to honest and responsible scholarly practice and upholding these values with respect and fairness.

The ANU commits to assisting all members of our community to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to be familiar with the academic integrity principle and Academic Misconduct Rule, uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with.

The Academic Misconduct Rule is in place to promote academic integrity and manage academic misconduct. Very minor breaches of the academic integrity principle may result in a reduction of marks of up to 10% of the total marks available for the assessment. The ANU offers a number of online and in person services to assist students with their assignments, examinations, and other learning activities. Visit the Academic Skills website for more information about academic integrity, your responsibilities and for assistance with your assignments, writing skills and study.

Online Submission

You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. This course does not use Turnitin.

Hardcopy Submission

All assignment submission is electronic (see Wattle for more details).

Late Submission

Policy regarding late submission is detailed below:

• Late submission for assignments not permitted. To compensate for the strict late assignment policy, the lowest assignment score will be dropped.
• Late submission for take-home examination not permitted. Late submission is not accepted for take-home examinations. A mark of 0 will be awarded.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Assignments will be returned electronically.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Not permitted

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students