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Hilbert’s seventeenth problem famously showed that not every nonnegative polynomial admits a sum-of-squares (SoS) representation. Hilbert also identified several special classes for which nonnegativity and SoS are equivalent, including univariate polynomials, quadratic polynomials, and bivariate quartics. In this talk, we will extend this connection to new classes of multivariate quartically regularized polynomials arising as subproblems …

Neural networks exhibit rich geometric and topological structure across their weights, activations, and representation spaces. This talk surveys a number of methods for probing these structures, providing complementary perspectives on how representations evolve across depth, encode structure, and respond to perturbations. I will highlight a recent piece of work studying the effect of adversarial influence …