We present a perspective on several current research directions relevant to the mathematical design of new materials. We discuss: (i) design problems for phase-transforming and shape-morphing materials, (ii) epitaxy as an approach of central importance in the design of advanced semiconductor materials, (iii) selected design problems in soft matter, (iv) mathematical problems in magnetic materials, (v) some open problems in liquid crystals and soft materials and (vi) mathematical problems on liquid crystal colloids. The presentation combines topics from soft and hard condensed matter, with specific focus on those design themes where mathematical approaches could possibly lead to exciting progress.

This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

我们提出了一些与新材料的数学设计相关的当前研究方向的观点。我们讨论：（i）相变和形状变形材料的设计问题；（ii）外延作为先进半导体材料设计中的重要方法；（iii）选定的软性设计问题；（iv）数学磁性材料中的问题，（v）液晶和软材料中的一些未解决问题，以及（vi）液晶胶体中的数学问题。该演讲结合了软和硬凝聚物的主题，并特别关注那些数学方法可能会导致令人兴奋的进展的设计主题。

本文是主题“复杂材料的数学设计主题”的一部分。