Recorded 08 March 2025. J Maurice Rojas of Texas A&M University presents "Breaking Complexity Barriers in Real Algebraic Geometry" at IPAM's LatMath 2025 Workshop. Abstract: Real solutions to large (nonlinear) systems of equations have long been central in many engineering applications, and are now known to be fundamentally important in complexity theory. For instance, recent work has shown that knowing enough about counting solutions implies new separations of complexity classes related to the P vs. NP problem. The key is to pay close attention to the underlying structure of the equations. We’ll review the connections between solving polynomial equations, algorithmic complexity, fewnomial theory, and combinatorics. We’ll also highlight a recent connection between the abc-Conjecture and speeding up equation solving over the real numbers. We assume no background in algebraic geometry. The main results are joint with Weixun Deng, Alperen Ergur, and Grigoris Paouris. Learn more online at: https://www.ipam.ucla.edu/programs/special-events-and-conferences/latmath-2025/?tab=overview