"Apart from my graduate years spent on S-matrix theory, my interests have focused on quantum field theory in one form or another, with a weakness for exact solutions and low dimensions. In the earlier period, I applied it to particle physics, to find the quark-gluon coupling constant using finite-energy sum rules, and in collaboration with Witten, exact S-matrices for certain field theories. I worked extensively on the Gross-Neveu model, discovering there the only known example of self-triality, exemplified by a theory which has three equivalent, mutually dual, representation, all with the same Lagrangian. The transformation I discovered there was used by Witten to establish, in a transparent way, world-sheet supersymmetry in string theory. It has appeared more recently in condensed matter systems with SO(8) symmetry. I then switched to statistical mechanics. Here my best efforts were in systems with random impurities. G. Murthy and I worked out several exact solutions for such systems. These solutions are particularly useful since intuition is not a very reliable guide in systems with competing interactions, e.g. a random magnet in which each spin receives conflicting signals from its neighbors on which way to align. I also worked out the exact long-distance limit of an Ising system with random bonds spread around the critical value using bosonization and also resolved a controversy. I then turned to systems where quantum mechanics played a role and worked out an exact solution to a metal insulator transition in one dimension, and the physics of antiferromagnets, sometimes in collaboration with Read or Sachdev, who have also been my sounding boards on countless occasions. Subsequently I applied the renormalization group to nonrelativistic Fermi systems. The presence of a Fermi-surface necessitates a modification of the standard RG procedure which integrates out all high energies. I was able, using this new approach, to recover the results of Landau's fermi liquid theory, a notoriously mysterious and difficult subject. The RG approach is simpler, more straightforward, and also contains information about the instabilities which invalidate the Fermi-liquid approximation. This idea has also found applications in searches for non-Fermi liquids in high T_c materials and in physics at finite matter density. Of late, I have been working on developing a Hamiltonian description of the Fractional Quantum Hall Effect, often in collaboration with G. Murthy. The theory has now reached a point where one can compute gaps for all observed fractions, finite temperature properties like polarization and relaxation rate. The theory manages to map the electronic variables in terms of which the problem is originally posed to the Composite Fermion variables that describe the ultimate quasiparticles. Murthy and I have extended this approach to fractional Chern insulators."
More info: http://pantheon.yale.edu/~rshankar/