Our key past accomplishments are described here.

1. Discovery of the role of anisotropy and the steady state selection mechanism in diffusion-controlled growth, stability analysis of dendrites, in collaboration with Eshel Ben-Jacob, Jim Langer.
   
   
2. First experimental exploration of the pattern phase diagram for diffusion-controlled growth, from our identification of the analogy with Hele-Shaw flow, in collaboration with Eshel Ben-Jacob.
   
   
3. First large-scale, systematic simulations of phase ordering in 2 and 3 dimensions for quenched systems with continuous symmetries, including the calculation of universal scaling functions for topological defect correlation functions. Systems studied include XY models (superfluids, magnets), liquid crystals (isotropic-nematic), superconductors, and miscut crystal surfaces. This program was influenced by collaboration with Yoshi Oono at Illinois.
   
   
4. Developed the first order parameter theory for the vulcanization transition in cross-linked polymers, which proved to be capable of predicting architectural, thermodynamic and dynamic scaling properties of cross-linked systems. In collaboration with Paul Goldbart at Illinois.
   
   
5. First to suggest a d-wave pairing state for the cuprate superconductor YBCO, and to review experimental evidence to test this hypothesis. Discovery of the power law nature of the penetration depth at low temperatures in data previously thought to be iron-clad evidence for s-wave pairing. Calculation of the influence of impurity scattering on the low temperature penetration depth and successful comparison with experimental data ruled out s-wave pairing, and was quantitatively consistent with d-wave pairing.
   
   
6. First to observe a convincing critical fluctuation regime in YBCO (now known as 3D XY scaling) in data analysis of experimental work with the University of British Columbia superconductivity group. In the best samples, an unprecedented 3 decades of power law scaling near the superconducting transition can be observed, a record (I believe) for any solid state system.
   
   
7. First comprehensive treatment of the dynamic critical phenomena at the superconducting transition, resolving puzzles in data from computer simulation and experiment; nonlinear scaling theory for the dynamic critical behavior of the superconducting transition, verified in collaborative work with Dale van Harlingen's experimental group at Illinois.
   
   
8. First multiscale phase field model code, using adaptive mesh refinement and finite elements. Capable of simulating complex, realistic solidification microstructures, even in 3 dimensions, with fluid flow effects. In collaboration with Jon Dantzig.
   
   
9. Development of the renormalization group for deterministic partial differential equations (PDEs) describing non-equilibrium phenomena. First analytic calculation of anomalous dimensions in a nonlinear diffusion equation. Development of numerical RG methods for partial differential equations. In collaboration with Yoshi Oono.
   
   
10. Development of the renormalization group for singular perturbation problems, unifying boundary layer, matched asymptotics, multiple scales analysis and reductive perturbation theory. Our theory enables coarse-grained amplitude and phase equations to be derived from underlying PDEs. In collaboration with Yoshi Oono.

Note: I like to collaborate with my students, postdocs and faculty colleagues, and the full credits for these works can be found from the publications themselves, where typically, students are listed first. Principal collaborators are mentioned here.