The theory of support varieties, studied by Avramov, Buchweitz, Pollitz and many others has been used to reveal asymptotic properties of complete intersection maps. One such result says that, for a finitely generated module over a local complete intersection ring, the degree of the polynomial modeling its Betti numbers coincides with that of its (derived) dual. In this talk, we introduce a higher-order refinement of this support theory, called the cohomological jump loci. After establishing foundational properties of this theory, we use it to detect information not encoded by the support varieties alone. In particular, we show that the leading coefficients of the aforementioned polynomials coincide. Speaker(s): Daniel McCormick (University of Utah)