AS/SC/AK/ MATH 4150X 3.0 W

Topics in Geometry: Algebraic Curves

      A self-contained introduction to algebraic geometry.
      Algebraic geometry is, roughly speaking, the study of geometric objects that can be defined "algebraically", i.e., by means of polynomials. (A one-dimensional object is a "curve"; a two-dimensional one is a "surface", etc.) Thus, the curves y = cos x and y=e^x from first-year calculus are not algebraic, but the curves y=x², y = x³, 4x² + 9y² = 36 , etc. are algebraic. Since "curves" are the simplest interesting geometric objects, perhaps the natural way to get an exposure to algebraic geometry is to begin with a study of algebraic curves, the topic of this course.
      Algebraic geometry has both differential geometry and algebraic number theory as mathematical neighbours. Apart from its own intrinsic beauty, algebraic geometry has found applications both within mathematics (algebraic curves figure fundamentally in the famous recent proof by Wiles of Fermat's Last Theorem) and outside it (modern theoretical physics, coding theory).
      Both the text and the grading scheme will be announced later.
      Students interested in 4150X should speak with the instructor before the course begins (in part, to determine whether they have adequate preparation for it). It is recommended that students have already taken MATH 3020 6.0 before taking this course, in addition to the official prerequisites below.