Explore the fascinating world of face numbers in centrally symmetric polytopes and spheres in this 55-minute theory seminar lecture by Isabella Novik from the University of Washington. Delve into the upper bound problem, examining the maximum number of i-dimensional faces possible in fixed-dimension complexes with a set number of vertices. Compare and contrast the answers for centrally symmetric polytopes and centrally symmetric triangulations of spheres. Survey the surprising developments in this field over the past five decades and learn about the current state of research. Gain insights into simplicial complexes, the Upper Bound Conjecture, properties of cyclic polytopes, and the Upper Bound Theorem. Discover how neighborly a centrally symmetric polytope can be and explore a theorem with a sketch of its proof. Conclude with a summary of key points and open problems in this intriguing area of mathematical research.