Asvin Gothandaraman, speaking at the Institute for Advanced Study workshop on Hodge Theory and O-minimality, presents a non-commutative counterpart to the symmetric product of a space, built from n-by-n matrix valued points on a variety X. He gives a precise formula for the cohomology of this new space in broad generality, framing the result as a first step into a wider non-commutative extension of classical constructions. Along the way he draws an analogy with Weil restriction, using it to motivate where the theory might go next. The talk is joint work with Yifeng Huang, Ruofan Jiang, and Yifan Wei, and Gothandaraman credits Wei, currently on the academic job market, with driving much of the underlying research. The seminar runs an hour and stays close to the technical formula and its proof strategy, aimed at an audience already fluent in algebraic geometry and cohomology.