Courses

Vector fields and groups of linear transformations. The exponential map. Linear groups and their Lie algebras. Connectedness. Closed subgroups. The classical groups. Manifolds, homogeneous spaces and Lie groups. Integration and representation.

Classification of second order partial differential equations. Legendre’s equation/polynomial. Gamma function and Bessel equation. Laplace transforms/equations. Fourier series. Boundary value problems. Application to heat conduction, vibrating strings...

Galilean relativity, postulates of special relativity; Lorentz transformations. Lorentz-Fitzgerald contraction, time dilation. 4-vectors, relativistic mechanics, kinematics and force, conservation laws; decay of particles; collision problems, covariant...

Elementary topology of the complex plane. Complex functions and mappings. The derivative. Differentiability and analyticity. Harmonic functions. Integrals. Maximum modulus, Cauchy-Gorsat, Cauchy theorems. Applications. Taylor and Laurent series, zeros and...

Modules, submodules, homomorphism of modules. Quotient modules, free (finitely generated) modules. Exact sequences of modules. Direct sum and product of modules. Chain conditions, Noetherian and Artinian modules. Projective and injective modules. Tensor...

Finite dimensional normed vector spaces.   Equivalent norms. Banach spaces.Infinite-dimensional normed vector spaces--Hamel and Schauder bases; separability. Compact linear operators on a Banach space. Complementary subspaces and the open-mapping...

Manifold, submanifold, differentiability of maps between manifolds, the tangent space, the tangent bundle and the tangent functor. Vector bundle.  The exterior algebra, the notion of a differentiable form on a manifold, singular n-chains and integration...

Arclength, curvature and torsion of a curve. Geometry of surfaces. Curvature, first and second fundamental form, Christofel symbols. Geodesics. Parallel vector fields. Surfaces of constant Gaussian curvature. Introduction to manifolds, tangent spaces and...

Generalisation of the Riemann (R) integral (eg Kurzweil-Henstock (KH) integral). Lebesgue (L) integral. Relationship between the KH-integrable, L-integrable and R-integrable functions. Convergence theorems. Measurability. Measure.

Linear and affine maps between normed vector spaces. Limits, continuity, tangency of maps and the derivative as a linear map. Component-wise differentiation, partial derivatives, the Jacobian as the matrix of the linear map. Generalized mean value theorem....