Phase transitions, Domain walls and minimal surfaces

I shall describe how a 25 year-old conjecture of DeGiorgi, which originated from problems in phase transitions, is closely related to Gibbons' conjectures on scalar field theories and to the Bernstein problem for minimal graphs. So far, the conjecture has only been solved in low dimensions and the progress (one dimension at a time) seems to be slow in spite of a fierce international competition for a complete resolution. I will also describe how UBC is keeping in the lead, thanks to major contributions by Martin Barlow, Changfeng Gui and their collaborators. The mathematics involved are basic, varied and great!