Hello! Hope you are doing good

The syllabus of NISER is quite vast. Howevwr, you can refer the following:

Vector spaces, Subspaces, Linear independant/ dependent, Basis and dimension, Linear transformations and Matrix representations. Properties of matrices, Rank, System of linear equations/ solutions, Eigen values and Eigen vectors, Diagonalization of matrices. (4 lectures)
Groups, Subgroups, Homomorphism Theorems, Automorphisms, Cayley’s Theorem, Symmetric groups, Group actions, Conjugacy classes, Sylow’s Theorem

Analysis:
Several variable calculus : Review of Limit and continuity, Partial derivatives, Total derivatives, Derivative as linear transformation, algebra of derivatives, Chain rule and its applications, Inverse function theorem and Implicit function theorem
Complex analysis: Recall complex numbers and the complex plane, functions on complex plane, holomorphic functions, C-R equation, power series, Integration along curves. The Cauchy-Goursat's theorem, Morera's theorem. Holomorphic functions and its power series expansion

Topology:
Definition of topology and playing with some examples, Basis and Subbasis for a topology, Product topology and Subspace topology, Closed sets and limit points (notions of closure, interior, neighbourhood, limit point, T1 and T2 spaces), Continuous functions, Quotient topology, Topological groups.

Hope it helps

Hello Aspirant,

In the given below link you can find the the syllabus of NEST which is is conducted, jointly, by the National Institute of Science Education and Research (NISER) and University of Mumbai - Department of Atomic Energy Centre for Excellence in Basic Sciences (UM-DAE CEBS).

Link: https://university.careers360.com/articles/nest

Good Luck