**This workshop will aim to touch upon some of these topics **

**Algebra**

- Fields, subrings. homomorphism of rings. Kernel and image of a homomorphism. Characteristic of a ring.
- Quotient rings. Prime ideals, maximal ideals and their characterization.
- Polynomial rings. Divisibility. units. Factorization in a ring. Irreducible and prime elements in a ring.
- Unique factorization domain, principal ideal domain and euclidean domains.

**Analysis**

- Riemann-Darboux integral
- Pointwise and uniform convergence:
- Complex Analysis: Continuity, differentiation, analytic/holomorphic functions, Cauchy-Riemann equations,
- Introduction to metric spaces

**Linear Algebra
**

- Structure of a single linear transformation: Invariant subspaces, semi-simple and nilpotent maps;
- Characteristic polynomial and minimal polynomial;
- Primary decomposition, cyclic subspaces and decomposition into cyclic subspaces, Jordan and rational canonical forms.
- Spectral theorem for orthogonal and unitary linear maps. Singular value decomposition.
- Modules: Definition and examples, submodules, direct sums, module homomorphisms, quotient modules, torsion elements, free modules

**Differential Equations
**

- BVPs and greens functions, Lipchitz continuity, Lyapunov stability
- Existence and uniqueness theorem on ODE namely Picards iteration and Pieno theorem
- Solution of stiff equations using asymptotic expansion
- Introduction to numerics, Fundamental matrix solutions and exponential of matrices