- How did Jean-Pierre Serre use sheaf theory to give topologists a rigorous language for local-to-global questions?
Topology
The study of properties preserved under continuous deformation.
- What is cobordism theory, and how did René Thom use it to classify smooth manifolds?
- How did René Thom's cobordism work later lead him to develop catastrophe theory?
- What does it mean for the seven-dimensional sphere to admit 'exotic' smooth structures?
- Why did some mathematicians initially disbelieve Milnor's result about exotic spheres?
- What does the Atiyah-Singer index theorem connect, and why has it proven useful in physics as well as mathematics?
- Why could Stephen Smale prove the Poincaré conjecture in dimensions five and higher decades before it was solved in lower dimensions?
- What did Sergei Novikov prove about the rational Pontryagin classes of a smooth manifold?
- What is higher algebraic K-theory, and how does it connect topology, algebra, and number theory?
- What does Thurston's geometrisation conjecture propose about decomposing three-dimensional manifolds, and who eventually proved it?
- How did Vaughan Jones discover a new knot invariant while working on an unrelated problem in operator algebras?
- What areas of mathematics did Timothy Gowers, Maxim Kontsevich, and Curtis McMullen each contribute to?
- How did Vladimir Voevodsky's homotopy theory for algebraic varieties let him prove Milnor's conjecture?
- What does the Poincaré conjecture claim about closed three-dimensional manifolds?
- How did Grigori Perelman use Ricci flow and a surgery procedure to overcome the obstacle that had stopped Richard Hamilton?
- Why is the Poincaré conjecture the only Millennium Prize Problem solved to date?
- What did Grigori Perelman prove using Richard Hamilton's Ricci flow technique?
- Why did Perelman decline both the Fields Medal and the Clay Millennium Prize?
- How did Raymer and Smith use knot theory to classify the tangles that formed when they tumbled string in a box?
- How did string length and tumbling time affect the number and complexity of knots that formed?
- How does topology explain why some material properties change abruptly rather than gradually?
- What did James Maynard and Maryna Viazovska each prove about prime gaps and sphere packing respectively?
- What does it mean for a nonlinear wave equation's solution to 'blow up' in finite time?