Undergraduate Mathematics Seminar

Abstract: Second-degree polynomials determine well-known curves on the plane (ellipses, parabolas, hyperbolas) and surfaces in the 3-space (paraboloids and hyperboloids of various kinds, etc.). Plane curves that can be determined by a third-degree polynomial, known as elliptic curves, are the topic of much of current research. One of the millennium million-dollars problems deals with them. In the talk, I'll discuss some applications of third-degree curves to plane geometry and number theory. I'll also sketch the proof of the "27 lines theorem": for every smooth third-degree surface in the complex projective 3-space there are exactly 27 lines lying on it.