Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. Quotes While the move from dimension 2 to dimension 3 appears to be the obvious step there is a sense in which one should move from 2 to 4. This comes from the consideration of complex algebraic geometry. For complex dimension 1 this theory was started by Abel and continued by Riemann. For algebraic varieties of complex dimension n the real dimension is 2n, so the case n = 2 leads to 4-dimensional real manifolds. The key figures in the topology of higher-dimensional algebraic varieties were Lefschetz, Hodge, Cartan and Serre. While general algebraic geometry was one of the major developments of the second half of the 20th century, the topology of real 4-manifolds had a great surprise in store when Simon Donaldson made spectacular discoveries opening up an entirely new area. James Carlson The Poincare Conjecture https://books.google.com/books?id=i0slBQAAQBAJ&pg=PA3 16 October 2014 American Mathematical Society 978-0-8218-9865-9 3–4 Should you just be an algebraist or a geometer? is like saying Would you rather be deaf or blind? If you are blind, you do not see space: if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties. Michael Atiyah (2000). Mathematics in the 20th Century: geometry versus algebra, Mathematics Today, 37(2), 46- 53. In the various forms of geometry (differential, metric, affine, algebraic), the central object is the variety, considered as a set of points. Pierre Cartier A country of which nothing is known but the name Grothendieck and “motives” External links Category:Algebra Category:Geometry