"Partition function for Chern-Simons(CS) theory on any Seifert manifold M with a gauge group G and level K can be written as a sum over integrable representations of the corresponding affine Lie algebra of the boundary Wess-Zumino-Witten Model. We consider the partition function for U(N) Chern-Simons theory with level K written as a sum over Integrable representation(for odd K) at large N, and show a natural manifestation of matrix integrals for CS theory studied earlier. For small coupling the dominant representations are characterised by Young diagrams with number of boxes on the top-most row being less than the level K. On the other hand at some critical value of the coupling, the dominant representations are always characterised by Young diagrams with exact K number of boxes on the topmost row. The restriction over representations dictate some constraints on the eigenvalue distribution of the matrix model as well. For CS theory on S2 × S1 this approach naturally describes the discreteness of the eigenvalues of the corresponding holonomy matrix which in turn translates into emergence of new phases. Our ongoing work deals with the effect of discreteness of the eigenvalues for the CS theory on S3."

"One of the major aims of theoretical computer science is to understand what is the most efficient way to perform a given task with limited computational resources. In this thesis, some absolutely tight lower bounds are shown for certain restricted models of computation. More specifically, this thesis studies the questions of proving tight lower bounds for sums of read-once formulas, and of proving tight lower bounds for tropical formulas."

"In this thesis we present improved approximation algorithms for the stochastic matching problem where the input is a stochastic subgraph of an edge-weighted graph probed under patience constraints on vertices. We also present improved results on efficiently approximating the maximum independent sets in geometric intersection graphs of special types called $B_1, B_2$-VPG graphs."

"Chemotaxis is the directed motion of organisms in response to a chemical gradient. E.coli chemotaxis is one of the most well-studied systems in biology. The motion of an E. coli consists of run-and-tumble modes. In presence of nutrient bacteria accumulate in the region of higher concentrations by modulating its run durations. We are interested in two broad questions. The first one is related to chemotactic performance of an E. coli cell and the second one is related to its run- and-tumble motion.We identify a set of well-defined response functions which characterize different aspects of chemotactic performance and investigate how these different response functions depend on the external environment and the internal biochemical pathway of the E. coli cell and at what conditions the chemotactic performance becomes most efficient. The second question deals with a simple run-and-tumble random walk whose switching frequencies between run and tumble mode depend on a stochastic signal. We are interested in characterizing the effect of signaling noise on the long- time behavior of the random walker and investigate how the fluctuations present in the input signal affect the motion."