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The concept of ‘holonomy’ originally arose in the late 19th century in mechanics as a way to describe motion subject to constraints, such as a ball rolling on a table without slipping or twisting.  However, its profound connection with the notion of curvature in differential geometry and generalizations of ‘conserved quantities’ under parallel translation was realized quite early in the 20th century, with fundamental contributions by É. Cartan, J. Schouten, E. Kähler and others in the decades 1910-30.  A more systematic exploration in the 1950s by M. Berger and S.-s. Chern clarified its foundations and inspired conjectures such as Calabi’s conjecture about compact Kähler manifolds with trivial canonical bundle, famously solved by S.-T. Yau in the 1970s.
        In the last 40 years, there have been many important advances in the so-called ‘exceptional’ cases, inspired by the desire to extend the ideas and results in Calabi–Yau geometry to these cases, with the ultimate goal of applying such results to physical theories going beyond string theory.
        The current state of the subject still presents enormous challenges in geometric analysis, topology, and theoretical physics, and I hope to give the audience a sense of this rapidly developing frontier.