Recommended articles

• On Proof and Progress in Mathematics
  Interesting and well-written thought about different aspects of how it is to be a mathematican, what is a proof, how mathematicians think, et cetera. Really worth a read.
• "The structure of algebraic threefolds: an introduction to Mori's program", János Kollár
  Good introductory text about the classification of threefolds. Its focus is on ideas and not on proofs. I skipped last part.
• "A mad day's work: From Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry", Pierre Cartier
  A short biographical sketch of Grothendieck together with connections between functional analysis, algebraic geometry and theoretical physics. Very interesting.
• "Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory", B. Mazur
  Very fun read about how the concept of "deformation" and "stability" shows up in different areas of mathematics.
• "Tropical Mathematics", D. Spayer, B. Sturmfels
  A short and very readable introduction to tropical mathematics and applications, with lots of references.
• "The octonions", J. Baez
  A survey article on the octonions with lots of connections to other parts of mathematics, including physics. Very interesting read.
• Stacks for everybody, Barbara Fantechi
  Stacks (or "stabler" in Norwegian) are extremely abstract, and this article does an excellent job of explaining them without being too technical.
• Today's menu: geometry and resolution of singular algebraic surfaces
  An illustrated guide to resolution of algebraic surfaces with lots of explanations and references. Fantastic reading.