Abstract: Four-manifolds are fascinating objects. Their uncanny and challenging nature, how several areas (geometry, mathematical physics, PDE's, knot theory, analysis, to name a few) come together in their study, the fact that the more we know the more we realize how much it still lacks to be understood, are several reasons (among many others) that make four-manifold theory a lovely area of study. Moreover, having in mind its physical relevance in our understanding of nature, it is highly intriguing to see how dimension 4 hosts a myriad of wild and disparate mathematical phenomena, which no other dimension does. The purpose of this talk is to sample the traits of four-manifolds that set them apart. In order to be able to appreciate the loveliness of their behavior better, we will compare them to their counterparts in other dimensions. Along the way we will observe how several areas of mathematics come together in efforts to understand these manifolds, and a state-of-the-art technique in their study will be discussed.
