Algebraic geometry is the study of the zeros of polynomials. It has many applications to the sciences.
It is a very old subject that goes back to the ancient Greeks who considered conic sections, circles, ellipses, parabolae, hyperbolae, pairs of lines and double lines.
The modern era of algebraic geometry begins with the introduction of Cartesian coordinates. The animation depicts a smooth cubic surface. Cubic surfaces have received a lot of attention and an early success of the subject was the discovery that every smooth cubic surface contains exactly 27 lines.
Algebraic geometry continues to be a very active area of research, with connections to many other areas of mathematics including algebra, combinatorics, complex analysis, differential geometry, logic, mathematical physics, number theory, representation theory, symplectic geometry and topology.
The algebraic geometry group at UCSD has broad interests covering many different areas of research in algebraic geometry including classical algebraic geometry, birational geometry, Hodge theory, enumerative geometry and moduli theory.
Image Credit: 27 Lines on a Cubic Surface by Greg Egan