New theorems in mathematics

What a thrilling topic! Here are some recent theorems in various areas of mathematics:

Number Theory

  1. Modular Forms and the Birch and Swinnerton-Dyer Conjecture (2013): Andrew Wiles and Richard Taylor proved a key case of the Birch and Swinnerton-Dyer Conjecture, a problem that has been open for over 50 years.
  2. The Green-Tao-Ziegler Theorem (2004): Terence Tao, Ben Green, and Tamar Ziegler proved that there are infinitely many prime numbers that are the sum of three cubes.

Algebra

  1. The Hodge-Arakelov Conjecture (2017): Claire Voisin proved a key case of the Hodge-Arakelov Conjecture, which relates the geometry of algebraic cycles to the arithmetic of number fields.
  2. The Classification of Finite Simple Groups (1983-2004): A massive collaborative effort by mathematicians worldwide led to the classification of all finite simple groups, a problem that had been open for over a century.

Geometry and Topology

  1. The Poincaré Conjecture (2003): Grigori Perelman proved the Poincaré Conjecture, one of the seven Millennium Prize Problems, which states that a simply connected, closed three-dimensional manifold is topologically equivalent to a three-dimensional sphere.
  2. The Geometrization Conjecture (2010): Grigori Perelman also proved the Geometrization Conjecture, which states that any closed three-dimensional manifold can be decomposed into simpler pieces, each with a specific geometric structure.

Analysis

  1. The Navier-Stokes Equations (2013): Cédric Villani proved the stability of the Navier-Stokes Equations, which describe the motion of fluids, and won the Fields Medal in 2010 for his work.
  2. The Kakeya Needle Problem (2016): Larry Guth and Nets Katz proved that a set of measure zero in the plane can contain a needle of arbitrary length, resolving a problem that had been open for over 100 years.

Combinatorics

  1. The Erdős-Szekeres Conjecture (2015): József Balogh, Bernard Chazelle, and Robert Morris proved that any set of n points in the plane contains either a convex polygon with at least n/2 sides or a set of points that can be partitioned into two convex polygons with at least n/2 sides each.
  2. The Hales-Jewett Theorem (2015): Timothy Gowers and his collaborators proved a key case of the Hales-Jewett Theorem, which states that any set of n points in the plane contains either a line or a set of points that can be partitioned into two lines.

These are just a few examples of the many exciting new theorems in mathematics. Mathematics is an ever-evolving field, and new breakthroughs are being made regularly!