This page contains a collection of syllabuses for undergraduate
mathematics degrees. Syllabuses are incredibly valuable in helping one
to undestand what collection of material is reasonably comprehensive for
a given course. By compiling a few (especially ones of high quality),
it’s easy to see the trends that emerge and what is collectively deemed
a good basis of a student’s mathematical education.
In addition to Cambridge, I’m intending to include information from
Standford and MIT for a window into some top American schools.
Cambridge
This is a formatted summary of pieces of the full schedule found here.
An undergraduate mathematics degree at Cambridge consists of three
parts: IA, IB, and II. Each part seems to be assessed separately. Part
IA is a small set of courses that all students must take. Part IB begins
to allow for individual interests in course selection. Part II has the
widest course selection and gives even more opportunity for
specificity.
What is additionally interesting is the inclusion of computational
projects which use tools like MATLAB to solve problems. This seems like
an especially practical skillset for the modern mathematician.
For the Americans like me, Michaelmas Term is the term spanning
October through December, and Lent term runs from January to March. This
means that in Part IA, courses on Groups, Vectors and Matrices, Numbers
and Sets, and Differential Equations are taught first, followed by a
block containing Analysis I, Probability, Vector Calculus, and Dynamics
and Relativity.
I still need to transfer and format the information about Part IB and
Part II to this page.
Part IA
Groups
24 lectures, Michaelmas Term
Examples of groups.
Axioms for groups.
Examples from geometry: symmetry groups of regular polygons, cube,
tetrahedron.
Permutations on a set; the symmetric group.
Subgroups and homomorphisms.
Symmetry groups as subgroups of general permutation groups.
The Mobius group; cross-ratios, preservation of circles, the point
at infinity.
Conjugation.
Fixed points of Mobius maps and iteration.
Lagrange’s theorem
Cosets.
Lagrange’s theorem.
Groups of small order (up to order 8).
Quaternions.
Fermat-Euler theorem from the group-theoretic point of view.
Group actions
Group actions; orbits and stabilizers.
Orbit-stabilizer theorem.
Cayley’s theorem (every group is isomorphic to a subgroup of a
permutation group).
Conjugacy classes.
Cauchy’s theorem.
Quotient groups
Normal subgroups, quotient groups and the isomorphism theorem.
Matrix groups
The general and special linear groups; relation with the Mobius
group.
The orthogonal and special orthogonal groups.
Proof (in \mathbb{R}^3) that every
element of the orthogonal group is the product of reflections and every
rotation in \mathbb{R}^3 has an
axis.
Basis change as an example of conjugation.
Permutations
Permutations, cycles and transpositions.
The sign of a permutation.
Conjugacy in S_n and in A_n.
Simple groups; simplicity of A_5.
[4]
Appropriate Books
M.A. Armstrong Groups and Symmetry. Springer–Verlag 1988
† Alan F Beardon Algebra and Geometry. CUP 2005
R.P. Burn Groups, a Path to Geometry. Cambridge University
Press 1987
J.A. Green Sets and Groups: a first course in Algebra.
Chapman and Hall/CRC 1988
W. Lederman Introduction to Group Theory. Longman 1976
Nathan Carter Visual Group Theory. Mathematical Association
of America Textbooks
Vectors and Matrices
24 lectures, Michaelmas Term
Complex numbers
Review of complex numbers, including complex conjugate, inverse,
modulus, argument and Argand diagram.
Informal treatment of complex logarithm, nth roots and complex powers.
de Moivre’s theorem.
Vectors
Review of elementary algebra of vectors in \mathbb{R}^3, including scalar product.
Brief discussion of vectors in \mathbb{R}_n and Cn; scalar product and the
Cauchy–Schwarz inequality.
Concepts of linear span, linear independence, subspaces, basis and
dimension.
Suffix notation: including summation convention, \delta_{ij}, and \epsilon_{ijk}.
Vector product and triple product: definition and geometrical
interpretation.
Solution of linear vector equations.
Applications of vectors to geometry, including equations of lines,
planes and spheres.
Matrices
Elementary algebra of 3 \times 3
matrices, including determinants.
Extension to n \times n complex
matrices.
Trace, determinant, non-singular matrices and inverses.
Matrices as linear transformations; examples of geometrical actions
including rotations, reflections, dilations, shears; kernel and image,
rank–nullity theorem (statement only).
Simultaneous linear equations: matrix formulation; existence and
uniqueness of solutions, geometric interpretation; Gaussian
elimination.
Symmetric, anti-symmetric, orthogonal, hermitian and unitary
matrices.
Decomposition of a general matrix into isotropic, symmetric
trace-free and antisymmetric parts.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors; geometric significance.
Proof that eigenvalues of hermitian matrix are real, and that
distinct eigenvalues give an orthogonal basis of eigenvectors.
The effect of a general change of basis (similarity
transformations).
Diagonalization of general matrices: sufficient conditions; examples
of matrices that cannot be diagonalized.
Canonical forms for 2 \times 2
matrices.
Discussion of quadratic forms, including change of basis.
Classification of conics, cartesian and polar forms.
Rotation matrices and Lorentz transformations as transformation
groups.
Appropriate Books
Alan F Beardon Algebra and Geometry. CUP 2005
Gilbert Strang Linear Algebra and Its Applications. Thomson
Brooks/Cole, 2006
Richard Kaye and Robert Wilson Linear Algebra. Oxford
science publications, 1998
D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian
Tensors. Nelson Thornes 1992
E. Sernesi Linear Algebra: A Geometric Approach. CRC Press
1993
James J. Callahan The Geometry of Spacetime: An Introduction to
Special and General Relativity. Springer 2000
Numbers and Sets
24 lectures, Michaelmas Term
Introduction to number
systems and logic
Overview of the natural numbers, integers, real numbers, rational
and irrational numbers, algebraic and transcendental numbers.
Brief discussion of complex numbers; statement of the Fundamental
Theorem of Algebra.
Ideas of axiomatic systems and proof within mathematics; the need
for proof; the role of counter-examples in mathematics.
elementary logic; implication and negation; examples of negation of
compound statements.
Proof by contradiction.
Sets, relations and functions
Union, intersection and equality of sets.
Indicator (characteristic) functions; their use in establishing set
identities.
Functions; injections, surjections and bijections.
Relations, and equivalence relations.
Counting the combinations or permutations of a set.
The Inclusion-Exclusion Principle.
The integers
The natural numbers: mathematical induction and the well-ordering
principle.
Examples, including the Binomial Theorem.
Elementary number theory
Prime numbers: existence and uniqueness of prime factorisation into
primes; highest common factors and least common multiples.
Euclid’s proof of the infinity of primes.
Euclid’s algorithm.
Solution in integers of ax + by =
c.
Modular arithmetic (congruences).
Units modulo n.
Chinese Remainder Theorem.
Wilson’s Theorem; the Fermat-Euler Theorem.
Public key cryptography and the RSA algorithm.
The real numbers
Least upper bounds; simple examples.
Least upper bound axiom.
Sequences and series; convergence of bounded monotonic
sequences.
Irrationality of \sqrt{2} and e.
Decimal expansions.
Construction of a transcendental number.
Countability and
uncountability
Definitions of finite, infinite, countable and uncountable
sets.
A countable union of countable sets is countable.
Uncountability of \mathbb{R}.
Non-existence of a bijection from a set to its power set.
Indirect proof of existence of transcendental numbers.
Appropriate Books
R.B.J.T. Allenby Numbers and Proofs. Butterworth-Heinemann
1997
R.P. Burn Numbers and Functions: steps into analysis.
Cambridge University Press 2000
H. Davenport The Higher Arithmetic. Cambridge University
Press 1999
A.G. Hamilton Numbers, sets and axioms: the apparatus of
mathematics. Cambridge University Press 1983
C. Schumacher Chapter Zero: Fundamental Notions of Abstract
Mathematics. Addison-Wesley 2001
I. Stewart and D. Tall The Foundations of Mathematics.
Oxford University Press 1977
Differential Equations
24 lectures, Michaelmas Term
Basic calculus
Informal treatment of differentiation as a limit, the chain rule,
Leibnitz’s rule, Taylor series, informal treatment of O and o
notation and l’Hopital’s rule; integration as an area, fundamental
theorem of calculus, integration by substitution and parts.
Informal treatment of partial derivatives, geometrical
interpretation, statement (only) of symmetry of mixed partial
derivatives, chain rule, implicit differentiation.
Informal treatment of differentials, including exact
differentials.
Differentiation of an integral with respect to a parameter.
First-order linear
differential equations
Equations with constant coefficients: exponential growth, comparison
with discrete equations, series solution; modelling examples including
radioactive decay.
Equations with non-constant coefficients: solution by integrating
factor.
Nonlinear first-order
equations
Separable equations.
Exact equations.
Sketching solution trajectories.
Equilibrium solutions, stability by perturbation; examples,
including logistic equation and chemical kinetics.
Discrete equations: equilibrium solutions, stability; examples
including the logistic map.
Higher-order linear
differential equations
Complementary function and particular integral, linear independence,
Wronskian (for second-order equations), Abel’s theorem.
Equations with constant coefficients and examples including
radioactive sequences, comparison in simple cases with difference
equations, reduction of order, resonance, transients, damping.
Homogeneous equations.
Response to step and impulse function inputs; introduction to the
notions of the Heaviside step-function and the Dirac
delta-function.
Series solutions including statement only of the need for the
logarithmic solution.
Multivariate functions:
applications
Directional derivatives and the gradient vector.
Statement of Taylor series for functions on \mathbb{R}_n.
Local extrema of real functions, classification using the Hessian
matrix.
Coupled first order systems: equivalence to single higher order
equations; solution by matrix methods.
Non-degenerate phase portraits local to equilibrium points;
stability.
Simple examples of first- and second-order partial differential
equations, solution of the wave equation in the form f(x + ct) + g(x − ct).
Appropriate Books
J. Robinson An introduction to Differential Equations.
Cambridge University Press, 2004
W. E. Boyce and R. C. DiPrima Elementary Differential Equations
and Boundary-Value Problems (and associated web site: google Boyce
DiPrima). Wiley, 2004
G. F. Simmons Differential Equations (with applications and
historical notes). McGraw-Hill 1991
D. G. Zill and M.R. Cullen Differential Equations with Boundary
Value Problems. Brooks/Cole 2001
Analysis I
24 lectures, Lent Term
Limits and convergence
Sequences and series in \mathbb{R}
and \mathbb{C}.
Correlation coefficient, bivariate normal random variables.
Inequalities and limits
Markov’s inequality, Chebyshev’s inequality.
Weak law of large numbers.
Convexity: Jensen’s inequality for general random variables, AM/GM
inequality.
Moment generating functions and statement (no proof) of continuity
theorem.
Statement of central limit theorem and sketch of proof.
Examples, including sampling.
Appropriate Books
W. Feller An Introduction to Probability Theory and its
Applications, Vol. I. Wiley 1968
G. Grimmett and D. Welsh Probability: An Introduction.
Oxford University Press 2nd Edition 2014
S. Ross A First Course in Probability. Prentice Hall
2009
D.R. Stirzaker Elementary Probability. Cambridge University
Press 1994/2003
Vector Calculus
24 lectures, Lent Term
Curves in \mathbb{R}^3
Parameterised curves and arc length, tangents and normals to curves
in \mathbb{R}^3; curvature and
torsion.
Integration in \mathbb{R}^2 and \mathbb{R}^3
Line integrals.
Surface and volume integrals: definitions, examples using Cartesian,
cylindrical and spherical coordinates; change of variables.
Vector operators
Directional derivatives.
The gradient of a real-valued function: definition; interpretation
as normal to level surfaces; examples including the use of cylindrical,
spherical and general orthogonal curvilinear coordinates.
Divergence, curl, and \nabla^2 in
Cartesian coordinates, examples; formulae for these operators (statement
only) in cylindrical, spherical and general orthogonal curvilinear
coordinates.
Solenoidal fields, irrotational fields and conservative fields;
scalar potentials.
Vector derivative identities.
Integration theorems
Divergence theorem, Green’s theorem, Stokes’s theorem, Green’s
second theorem: statements; informal proofs; examples; application to
fluid dynamics, and to electromagnetism including statement of Maxwell’s
equations.
Laplace’s equation
Laplace’s equation in \mathbb{R}^2
and \mathbb{R}^3: uniqueness theorem
and maximum principle.
Solution of Poisson’s equation by Gauss’s method (for spherical and
cylindrical symmetry) and as an integral.
Cartesian tensors in \mathbb{R}^3
Tensor transformation laws, addition, multiplication, contraction,
with emphasis on tensors of second rank.
Isotropic second and third rank tensors.
Symmetric and antisymmetric tensors.
Revision of principal axes and diagonalization.
Quotient theorem.
Examples including inertia and conductivity.
Appropriate Books
H. Anton Calculus. Wiley Student Edition 2000
T. M. Apostol Calculus. Wiley Student Edition 1975
M. L. Boas Mathematical Methods in the Physical Sciences.
Wiley 1983
D. E. Bourne and P. C. Kendall Vector Analysis and Cartesian
Tensors. 3rd edition, Nelson Thornes 1999
E. Kreyszig Advanced Engineering Mathematics. Wiley
International Edition 1999
J. E. Marsden and A. J. Tromba Vector Calculus. Freeman
1996
P. C. Matthews Vector Calculus. SUMS (Springer Undergraduate
Mathematics Series) 1998
K. F. Riley, M. P. Hobson, and S. J. Bence Mathematical Methods
for Physics and Engineering. Cambridge University Press 2002
H. M. Schey Div, grad, curl and all that: an informal text on
vector calculus. Norton 1996
M. R. Spiegel Schaum’s outline of Vector Analysis. McGraw
Hill 1974
Dynamics and Relativity
24 lectures, Lent Term
Basic concepts
Space and time, frames of reference, Galilean transformations.
Newton’s laws.
Dimensional analysis.
Examples of forces, including gravity, friction and Lorentz.
Newtonian dynamics of a
single particle
Equation of motion in Cartesian and plane polar coordinates.
Work, conservative forces and potential energy, motion and the shape
of the potential energy function; stable equilibria and small
oscillations; effect of damping.
Angular velocity, angular momentum, torque.
Orbits: the u(\theta) equation;
escape velocity; Kepler’s laws; stability of orbits; motion in a
repulsive potential (Rutherford scattering).
Rotating frames: centrifugal and Coriolis forces.
Brief discussion of Foucault pendulum.
Newtonian dynamics
of systems of particles
Momentum, angular momentum, energy.
Motion relative to the centre of mass; the two body problem.
Variable mass problems; the rocket equation.
Rigid bodies
Moments of inertia, angular momentum and energy of a rigid
body.
Parallel axis theorem.
Simple examples of motion involving both rotation and translation
(e.g. rolling).
Special relativity
The principle of relativity.
Relativity and simultaneity.
The invariant interval.
Lorentz transformations in (1 +
1)-dimensional spacetime.
Time dilation and length contraction.
The Minkowski metric for (1 +
1)-dimensional spacetime.
Lorentz transformations in (3 + 1)
dimensions.
4–vectors and Lorentz invariants.
Proper time.
4–velocity and 4–momentum.
Conservation of 4–momentum in particle decay.
Collisions.
The Newtonian limit.
Appropriate Books
D. Gregory Classical Mechanics. Cambridge University Press
2006
G. F. R. Ellis and R. M. Williams Flat and Curved
Space-times. Oxford University Press 2000
A. P. French and M. G. Ebison Introduction to Classical
Mechanics. Kluwer 1986
T. W. B. Kibble and F. H. Berkshire Introduction to Classical
Mechanics. Kluwer 1986
M. A. Lunn A First Course in Mechanics. Oxford University
Press 1991
P. J. O’Donnell Essential Dynamics and Relativity. CRC Press
2015
W. Rindler Introduction to Special Relativity. Oxford
University Press 1991
E. F. Taylor and J. A. Wheeler Spacetime Physics: introduction to
special relativity. Freeman 1992
Part IB
Not all of the following courses are required. Rather, it is
recommended that students take a subset that corresponds to their
interests.
It’s also worth noting that this material is accompanied by some
projects which train the use of e.g. matlab to solve problems (or at
least build intuition around them). A manual for these problems can be
found here.
This section is not complete.
Linear Algebra
24 lectures, Michaelmas Term
Groups, Rings, and Modules
24 lectures, Lent Term
Analysis and Topology
24 lectures, Michaelmas Term
Geometry
24 lectures, Lent Term
Part IB Analysis and Topology is an essential prerequisite.