# Part A Mathematics 2019-20

The confirmed synopses for Part B 2019-20 will be available on the course management portal https://courses.maths.ox.ac.uk/ before the start of Michaelmas Term 2019.

**Three or Four Year Course**

All students who complete Parts A and B will be classified. Those who have achieved honours and who wish to graduate at this point may supplicate for a BA.

Students wishing to take the four-year course should register to do so at the beginning of their third year, and will be permitted to do so on the basis of an upper second class performance, or better, in the third year classification. They will take Part C in their fourth year, be awarded a separate classification and, if successful, may supplicate for an MMath.

**Masters in Theoretical and Mathematical Physics**

There is a Mathematical Physics stream as an alternative to Part C (the fourth year). Students who move on to this stream and successfully complete the year will be awarded an MMathPhys.

Mathematics students interested in transferring to the MMathPhys will need to make an application during their third year. This option will not be available to students on the joint degrees.

Interested students should bring this up with their tutors. Full details relating to this masters will be in the MMathPhys handbook, including details of those second and third year options which are suggested background or recommendations for the masters and a description of the application process and deadlines. Further details available online: http://mmathphys.physics.ox.ac.uk/.

**Pathways to Part B**

Most, but not all, third year options (Part B) have certain pre-requisites from Part A. Whilst the courses that will be offered in Part B in a year's time (to follow on from the Part A courses detailed in this supplement) have not been wholly decided there is not substantial change year-on-year in the list of Part B options offered.

What follows is a list envisaging how the Part A options for 2019-20 would be pre-requisites or useful knowledge for the Part B options for 2019-20. You will note that there are also a good number of courses that have no prerequisites.

**A3 Rings and Modules****A4 Integration****A5 Topology****A6 Differential Equations 2****A7 Numerical Analysis**- no Part B courses explicitly require this.**A8 Probability****A9 Statistics****A10 Fluids and Waves****A11 Quantum Theory****ASO: Number Theory****ASO: Group Theory****ASO: Projective Geometry****ASO: Introduction to Manifolds****ASO: Integral Transforms****ASO: Calculus of Variations****ASO: Graph Theory****ASO: Special Relativity**- no Part B courses explicitly require this.**ASO: Mathematical Modelling in Biology**

Essential for B2.1: Introduction to Representation Theory

Essential for B2.2: Commutative Algebra

Essential for B3.1: Galois Theory

Essential for B3.4: Algebraic Number Theory

Recommended for B4.1: Banach Spaces

Recommended for B4.2: Hilbert Spaces

Essential for B4.3: Distribution Theory and Fourier Analysis: An Introduction

Essential for B8.1: Probability, Measure and Martingales

Essential for B8.2: Continuous Martingales and Stochastic Calculus

Recommended for B8.3: Mathematical Models of Financial Derivatives

Recommended for B3.2: Geometry of Surfaces

Useful for B3.3: Algebraic Curves

Essential for B3.5: Topology and Groups

Essential for B5.2: Applied Partial Differential Equations

Recommended for B5.3: Viscous Flow

Useful for B5.5: Further Mathematical Biology

Useful for B5.6: Nonlinear Systems

Useful for B5.1: Stochastic Modelling of Biological Processes

Essential for B8.1: Probability, Measure and Martingales

Essential for B8.2: Continuous Martingales and Stochastic Calculus

Essential for B8.3: Mathematical Models of Financial Derivatives

Useful for B8.4: Information Theory

Essential for SB3a: Applied Probability

Essential for SB1 Applied and Computational Statistics

Useful for SB2a Foundations of Statistical Inference

Useful for SB4 Actuarial Science

Essential for SB1 Applied and Computational Statistics

Essential for SB2a Foundations of Statistical Inference

Essential for SB3b Statistical Life-time Models

Recommended for B5.2: Applied Partial Differential Equations

Essential for B5.3: Viscous Flow

Essential for B5.4: Waves and Compressible Flow

Essential for B7.3: Further Quantum Theory

Essential C7.4: Introduction to Quantum Information

Useful for B3.1: Galois Theory

Recommended for B3.4: Algebraic Number Theory

Recommended for B2.1: Introduction to Representation Theory

Recommended for B3.1: Galois Theory

Recommended for B3.4: Algebraic Number Theory

Recommended for B3.5: Topology and Groups

Recommended for B3.3: Algebraic Curves

Useful for B3.2: Geometry of Surfaces

Useful for B3.3: Algebraic Curves

Recommended for B4.3: Distribution Theory and Fourier Analysis: An Introduction

Useful for B4.2: Hilbert Spaces

Recommended for B4.3: Distribution Theory and Fourier Analysis: An Introduction

Useful for B5.2: Applied Partial Differential Equations

Recommended for B5.4: Waves and Compressible Flow

Recommended for B5.2: Applied Partial Differential Equations

Essential for B7.1: Classical Mechanics

Recommended for B8.5: Graph Theory

Useful for B5.1: Stochastic Modelling of Biological Processes

Useful for B5.2: Applied Partial Differential Equations

Useful for B5.3: Viscous Flow

Useful for B5.4: Waves and Compressible Flow

Essential for B5.5: Further Mathematical Biology

Recommended for B5.6: Nonlinear Systems

*The following Part B courses in 2019-20 have no prerequisites from Part A. *

- B1.1: Logic
- B1.2: Set Theory
- B6.1 Numerical Solution of Differential Equations I
- B6.2 Numerical Solution of Differential Equations II
- B6.3 Integer Programming
- B7.2: Electromagnetism
- BEE/BOE Mathematical/Other Mathematical Extended Essay
- BN1.1 Mathematics Education
- BN1.2 Undergraduate Ambassadors' Scheme
- BSP Structured projects
- BO1.1: History of Mathematics
- OCS1 Lambda Calculus and Types
- OCS2 Computational Complexity
- N101 Early Modern Philosophy
- N102 Knowledge and Reality
- N122 Philosophy of Mathematics
- N127 Philosophical Logic

The examination syllabus, as referred to in the *Examination Regulations*, and synopses have been approved by the Mathematics Teaching Committee for examination in 2020. Please see the current edition of the *Examination Regulations* (https://www.admin.ox.ac.uk/examregs/) for the full regulations governing these examinations. Examination Conventions can be found at: https://www.maths.ox.ac.uk/members/students/undergraduate-courses/examin....

The Part A examination **syllabus** is the mathematical material of the synopses, as separately detailed by paper below. The course synopses on the course webpages also give additional detail to the syllabus (for example, showing how the material is split by lectures) and are also accompanied by lists of recommended reading.

**Honour School of Mathematics - Part A**

For Part A, each candidate shall be required to offer nine or ten written papers. These papers must include:

- A0 - Linear Algebra (1.5 hours)
- A1 - Differential Equations 1 (1.5 hours)
- A2 - Metric Spaces and Complex Analysis (3 hours)
- ASO - Short Options (1.5 hours)

and five or six papers from the Long Options (each 1.5 hours long).

- A3 Rings and Modules
- A4 Integration
- A5 Topology
- A6 Differential Equations 2
- A7 Numerical Analysis
- A8 Probability
- A9 Statistics
- A10 Fluids and Waves
- A11 Quantum Theory

Paper ASO will examine the nine Short Options (Number Theory, Group Theory, Projective Geometry, Introduction to Manifolds, Integral Transforms, Calculus of Variations, Graph Theory, Special Relativity, and Mathematical Modelling in Biology). Students are recommended to take three of these Short Options.

Part A shall be taken on one occasion only (there will be no resits). At the end of the Part A examinations, a candidate will be awarded nine or ten `University Standardised Marks' (USMs). The USM from Paper A2 will have twice the weight of the USMs awarded for the other papers. A weighted average of these USMs will be carried forward for the classification awarded at the end of the third year, with this average from the second year papers counting for 40%.

**Additional Long Option**

Students are permitted to take an additional long option at Part A. A student taking 5 long options will still have each of them counting as a unit's weight towards their overall second year mark.

For a student taking 6 long options, their best 4 papers (following the exams) will count one unit each and their worst 2 papers will count half a unit each. Thus these 6 papers will overall still have a weight of 5 units.

The results from all 6 papers will appear on the student's exam transcript.

The aim of the above scoring system is to ensure anyone taking on an extra option will not do so lightly (all marks will be reported and all count to some extent) but also that a student will not get a lower overall mark for having taken on the extra workload (the given scoring system was a fair compromise looking at several years' data sets).

We are anticipating that most students will not wish to take on the extra workload, and that in most years it would be some subset of the first-class students wishing to take 6 long options.

**Core Material**

The examination syllabi of the three core papers A0, A1 and A2 shall be the mathematical content of the synopses for the courses

- Linear Algebra
- Differential Equations 1
- Metric Spaces and Complex Analysis

**Options**

The examination syllabi of the options paper, A3-A11, shall be the mathematical content of the synopses for the courses

- Rings and Modules
- Integration
- Topology
- Differential Equations 2
- Numerical Analysis
- Probability
- Statistics
- Fluids and Waves
- Quantum Theory

**Short Options**

The examination syllabi of the short options paper ASO shall be the mathematical content of the synopses for the courses

- Number Theory
- Group Theory
- Projective Geometry
- Introduction to Manifolds
- Integral Transforms
- Calculus of Variations
- Graph Theory
- Special Relativity
- Mathematical Modelling in Biology