In this thesis, the main objective is to understand the electronic transport in strongly correlated electron systems. Our study is inspired by the anomalous linear in temperature resistivity (strange metallicity) observed in High temperature superconductors, twisted bilayer graphene and heavy fermion compounds. We have considered Hubbard model as a paradigmatic model to understand the interplay of kinetic energy and coulomb repulsion in these systems. We study transport in Hubbard model with an emphasis on the role of two particle correlations (charge and spin). We developed a self consistent theory for U infinity Hubbard model in infinite dimensions. In doing so, we derived a Dyson equation for the fermionic Green’s function where self energy is given in terms of local fermionic Green’s function and charge + spin correlation functions. Charge and spin correlation functions are expressed in terms of current-current correlation function which we calculate using a bubble diagram. We find a Fermi Liquid (FL) at extremely low temperatures marked by resistivity qudratic in temperature followed by a linear in temperature resistivity at intermediate and high temperatures. By comparing the energy scales of spin and charge fluctuations with respect to the thermal fluctuations, we differentiate between classical and quantum regions in the resistivity vs temperature phase digram for various dopings. We also find that the transport in this model is governed by the coupling of local, massless bosonic fluctuations with fermions and the intermediate temperature linear in temperature resistivity (where bosonic fluctuations are quantum in nature) might correspond to strange metal regime. Since the previous approach doesn’t differentiate between the spin and charge fluctuations qualitatively, we also study finite U Hubbard model using Dynamical Mean Field Theory (DMFT) + Numerical Renormalization Group (NRG). By studying the systematic evolution of charge and spin fluctuations for various dopings and temperatures, we found that the spin fluctuations become incoherent earlier, while charge dynamics remains quantum over a broader temperature range. This provides a microscopic insight into the complex, two-stage, Fermi to non Fermi liquid to bad metal crossovers seen in transport data, in particular in the dc resistivity.

Algorithms that use sample access to an unknown probability distribution to learn and estimate properties of the probability distribution have wide-ranging applications. In the setting of property testing, the goal is to design algorithms that minimize the sample complexity. In this talk, we will look at algorithms that obtain a trade-off between the sample complexity and space complexity. We will study the identity testing problem in the conditional access model, and the tolerant testing problem in the sample access model. In both models, we will obtain algorithms that achieve almost tight trade-offs between the sample complexity and space complexity. We will then see algorithms for testing properties of structured distributions that obtain optimal trade-offs.