The exhaustive list of topics in Random Matrix Theory and Applications in which we provide **Help with Homework Assignment** and **Help with Project** is as follows:

**Maximum entropy approach of complex systems:**- Probability and information entropy: the role of the relevant physical parameters as constraints.
- The role of symmetries in motivating a natural probability measure e.g Gaussian or uniform etc.
- The maximum entropy criterion in the context of statistical inferences.

**Random matrix ensembles:**- Nature of ensemble: Role of symmetry, interactions and other system conditions.
- Basis invariance vs Basis dependence of the ensemble and their transformation properties.
- Stationary vs non-stationary ensembles.
- Conservative systems and Gaussian ensembles of Hermitian matrices: ten standard types.
- Non-conservative systems and ensembles of non-Hermitian matrices: Ginibre ensembles.
- Time-periodic systems and circular ensembles of unitary matrices.
- Laguerre ensembles.
- Multi-cut ensembles.

**Correlations and fluctuation measures:**- Fluctuation measures of eigenvalues e.g. number variance, spacing distribution, spectral rigidity, gap probabilities etc.
- Fluctuations measures of eigenfunctions e.g local intensity distribution, inverse participation ratio, local density of states etc.
- Level density and level repulsion: role of global symmetries.
- Eigenfunction localization: role of interactions and disorder.
- Behavior at the edge of the spectrum.
- Critical level statistics and multifractality of eiegnfunctions.
- Universality of fluctuations measures.

**System dependent random matrix ensembles:**- Varying system conditions and transition between stationary ensembles.
- Common mathematical formulation of fluctuation measures for multi-parametric Gaussian ensembles.
- Connection to one and two dimensional Calogero-Sutherland Hamiltonian of interacting particles.
- Phase transition and critical ensembles.
- Correlated random matrix ensembles.

**Random matrices to quantum systems:**- Random matrix theory of quantum transport.
- Random matrix theory of quantum chaotic systems.
- Disordered systems.
- Quantum gravity.
- Nuclear resonances, Atoms, molecules etc..

**Random matrices to classical systems:**- Financial systems e.g stock market fluctuations.
- Biological systems e.g signals received by brain.
- Atmospheric correlations.
- Complex networks e.g traffic systems.
- Light propagation through random media.
- Elastomechanics.
- Number theoretic systems e.g. Reimann-zeta function.