String theory, knot theory, thermal QCD and string cosmology

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My research proposal may be divided into four broad categories: string cosmology, string theory, thermal QCD and knot theory. Let me start by string cosmology. A recent challenge is to construct de Sitter vacua, i.e a vacua with positive cosmological constant, from string theory. This has turned out to be very challenging due to certain no-go conditions, also called the swampland criteria. One of my aim is to find a concrete way to overcome the no-go conditions and show how de Sitter vacua may come about or justify that there could be no de Sitter solution possible in string theory. My second research proposal is in string theory. One of the issue that is important in string theory is the choice of the stable vacuum. Stabilization of moduli require switching on background fluxes, and also non-perturbative effects, leading to non-Calabi-Yau manifolds, or more generically, non-Kahler manifolds. Despite ubiquitous, concrete constructions of these manifolds are harder because they require sophisticated techniques in differential geometry. Thus one of my proposal here is to have generic constructions of these vacua. This is important otherwise string theory will be unable to reproduce the standard model. My third proposal is related to thermal QCD. We have been able to construct the dual gravitational description that can explain the dynamics of QCD like theories at all energy scales. However many things remain to be shown. For example it is as yet unknown how bulk viscosity in such a theory works at intermediate couplings, i.e couplings between weak and strong 't Hooft couplings. Knowing this will be an immense progress in the literature because the present techniques used to study dynamics at intermediate couplings are pretty much inconclusive. Another related thing is color superconductivity which may be studied using the same dual gravitational model. Interestingly, the gravity dual constructed by us is not just to study QCD, but also to study other holographic questions like entanglement entropies etc. Since our model gives rise to the gravity dual of a theory that is non-conformal, answering questions related to entanglement entropies (EE) etc. will shed light on these issues when non-conformality is switched on. My last proposal is on knot theory. This is a new direction that I have started recently, and is based on our findings that many of the results of topological field theory my be derived from certain gravitational background in M-theory. This surprising construction not only explains the concept of topological twisting, but also directly reproduces the boundary Chern-Simons theory as well as the knot invariants. However the puzzling thing is that the coefficients of a given knot polynomial have one-to-one correspondence to the number of solutions of a certain differential equation. One of my proposal is to prove this conjecture. Related topics like the construction of  the colored Jones and Kaufmann polynomials require further research.