turkmath.org

De Giorgi's conjecture (1978) brings together three groups of mathematicians: one specializing in nonlinear partial differential equations, another in differential geometry, more specially on minimal surfaces and constant mean curvature surfaces, and in mathematical physics on phase transitions. Classifying solutions of PDEs has been a very interesting topic. We begin by various celebrated classification results for solutions of elliptic PDEs such as Lane-Emden conjecture and De Giorgi's conjecture. These conjectures have attracted many experts in the field for a few decades. Later in this talk, we state counterparts of these conjectures to systems of equations and we provide idea of proofs in lower dimensions. To provide such counterparts we need to introduce a few novel concepts. Part of this talk is based on joint works with Nassif Ghoussoub..