closed hypersurface of Euclidean space with constant mean curvature. In 1971, James Serrin utilized Alexandrov's technique to prove that highly symmetric solutions of certain second-order elliptic partial differential equations must be supported on symmetric domains. Nirenberg realized that Serrin's work could be reformulated so as to prove that solutions of second-order elliptic partial differential equations inherit symmetries of their domain and of the equation itself. Such results do not hold automatically, and it is nontrivial to identify which special features of a given problem are relevant. For example, there are many harmonic functions on Euclidean space which fail to be rotationally symmetric, despite the rotational symmetry of the Laplacian and of Euclidean space. Nirenberg's first results on this problem were obtained in collaboration with Basilis Gidas and Wei-Ming Ni. They developed a precise form of Alexandrov and Serrin's technique, applicable even to fully nonlinear elliptic and parabolic equations. In a later work, they developed a version of the Hopf lemma applicable on unbounded domains, thereby improving their work in the case of equations on such domains. Their main applications deal with rotational symmetry. Due to such results, in many cases of geometric or physical interest, it is sufficient to study ordinary differential equations rather than partial differential equations. Later, with Henri Berestycki, Nirenberg used the Alexandrov−Bakelman−Pucci estimate to improve and modify the methods of Gidas−Ni−Nirenberg, significantly reducing the need to assume regularity of the domain. In an important result with Srinivasa Varadhan, Berestycki and Nirenberg continued the study of domains with no assumed regularity. For linear operators, they related the validity of the maximum principle to positivity of a first eigenvalue and existence of a first eigenfunction. With Luis Caffarelli, Berestycki and Nirenberg applied their results to symmetry of functions on cylindrical domains. They obtained in particular a partial resolution of a well-known conjecture of Ennio De Giorgi on translational symmetry, which was later fully resolved in Ovidiu Savin's Ph.D. thesis. They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of Maria Esteban and Pierre-Louis Lions. Functional inequalities Nirenberg and Emilio Gagliardo independently proved fundamental inequalities for Sobolev spaces, now known as the Gagliardo–Nirenberg–Sobolev inequality and the Gagliardo–Nirenberg interpolation inequalities. They are used ubiquitously throughout the literature on partial differential equations; as such, it has been of great interest to extend and adapt them to various situations. Nirenberg himself would later clarify the