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A classical theorem of Bernstein says that entire minimal graphs over R^2 are planes. This result was extended to R^n for n < 8 in works of Fleming, De Giorgi, Almgren, and Simons. Finally, Bombieri - De Giorgi - Giusti constructed a non-flat entire minimal graph over R^8, settling the…

We discuss our result on a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kahn and Kalai. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-…