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Mirror symmetry is a duality in string theory: one physical theory has two very different mathematical incarnations. Roughly, from a mathematical point of view: there are pairs X and Y of Calabi--Yau manifolds such that the symplectic geometry of X is equivalent to the complex geometry of Y and vice versa. This is made precise via the homological mirror symmetry conjecture of Kontsevich (1994). The Strominger--Yau--Zaslow conjecture (1996) asserts that X and Y admit dual Lagrangian torus fibrations over a common base; then the physical duality reduces to a simpler duality called T-duality on the torus fibers. I will give a gentle introduction to the mathematics of mirror symmetry, including some brief remarks on the relation to physics (which I hope can be expanded upon on another occasion by someone more qualified than myself). This is a wide-ranging and challenging topic, spanning complex geometry, symplectic geometry, differential geometry, and homological algebra, but I will attempt to convey the essence of the mirror correspondence as mathematicians currently understand it, with a minimum of prerequisites and technical details. I will teach a graduate class at UMass in Fall 2025 giving a more extensive introduction to the mathematics of mirror symmetry. 

References: Kontsevich: Homological algebra of mirror symmetry; Strominger--Yau--Zaslow: Mirror symmetry is T-duality

It is well-know that an inspiration for the development Cartan's concept of generalized spaces was Einstein's theory of General Relativity describing gravity as the curvature of spacetime. Less well-known is the second context Cartan had in mind: the generalized theory of elasticity developed by the Cosserat brothers. Cartan's study of the geometrical structures forming the foundations for both theories led to a generalization of the notion of curvature given by two parts; a rotational component which corresponded to the curvature known from Riemannian geometry, and a translational component which seemed to naturally be described by a rotational quantity. The translational curvature was named torsion by Cartan. This talk will aim to give an overview of Cartan's picture of torsion inspired by both General Relativity and generalized elasticity, and to (hopefully) answer the question of why Cartan sought to describe the translational curvature by a rotational quantity.