Joint intersectivity and polynomial corners

The polynomial Szemer\'edi theorem of Bergelson and Leibman gives broad conditions under which polynomial patterns must appear in every positive-density subset of Z^d. When the polynomials do not vanish at zero, the correct replacement condition is conjecturally joint intersectivity, a longstanding open problem. In this talk I will discuss a proof of this conjecture for pairs of polynomials, corresponding to length-three patterns. The main new difficulty is the diagonal case, where the two polynomials coincide, a situation that falls outside the scope of previous methods. I will explain the ergodic mechanism behind the argument and how methods from additive combinatorics, especially work of Peluse, Prendiville, and Leng, have fruitfully enriched the ergodic toolbox in recent years. This is joint work with Borys Kuca.