This thesis consists of parameterized complexity analysis of two classes of problems: (i) graph partitioning problems and (ii) crossing minimisation problems. The first of these, graph partitioning problems, involve testing whether it is possible to partition the vertex set of a graph subject to a set of constraints, or testing whether it is possible to modify a graph slightly by way of vertex deletions, edge deletions etc. so that the vertex set of the resulting graph can be partitioned subject to a set of constraints. The second part of the thesis deals with crossing minimisation problems. Here, the goal is to test if a given graph embedded in the plane contains a certain subgraph with at most a given number of pairwise edge crossings. The thesis contains several new results in these two directions in the realm of Parameterized Complexity. Topic: Thesis Defense Time: Sep 24, 2021 02:00 PM India Join Zoom Meeting https://zoom.us/j/94542723307?pwd=bUg5cnVYMEMwUThoZklQbDlSOC9lZz09 Meeting ID: 945 4272 3307 Passcode: 543634

In recent years, periodic driving has emerged as a powerful tool for the coherent control of quantum many-body systems. Periodically driven (Floquet) systems provide a versatile platform for realizing non-equilibrium phases of matter, which may not have any equilibrium analog. However, the driving can cause these systems to heat up, presenting a major obstacle to these endeavors. In this talk, I will present some of our investigations on this issue. In the first half of the talk, I will discuss a scheme to stabilize a Floquet Bose-Einstein condensate loaded in a one-dimensional optical lattice. I will demonstrate that the heating rate of this system can be controlled by suitably designing the transverse confinement. In the second half of the talk, I will propose a route to realize a discrete time crystal. A time crystal is a fascinating non-equilibrium phase of matter that exhibits spontaneous time-translation-symmetry breaking. I will demonstrate that a many-body echo protocol can be employed to evade thermalization in a periodically driven Ising spin chain, leading to the emergence of an eternal time crystal. I will conclude with a brief overview of future research directions. Google meet link: meet.google.com/kyv-mrdd-ybp