Condensed mathematics

Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that similar analogies in the individual fields are instead different expressions of the same concepts (similar to the way in which different human languages can express the same thing).

Concept

Topology plays a crucial part in number theory, functional analysis, and geometry. Topologically, a coffee cup with a handle has the same topology as a doughnut, while one without has the same topology as a sphere.

Many mathematical objects have a topology — a way of measuring the proximity of the object's parts. These include topological abelian groups and topological vector spaces. Functions typically can be represented in spaces with an infinite number of dimensions. The set of real numbers has the topology of a straight line, while p-adic number systems have a fractal topology.

Subgoals

Parts of the unification program are to replace topological spaces by “condensed sets", turn functional analysis into a branch of commutative algebra, and various types of analytic geometry into algebraic geometry. Condensed sets form condensed abelian groups into an abelian category, which satisfies the constraints of homological algebra.

Approximately, topological space X is replaced by the functor that takes a profinite set S to the maps from S to X. For every compact Hausdorff space X there is a surjection from a profinite set onto X. For example, one such surjection is the decimal expansion from the profinite set of sequences in {0,1,2,3,4,5,6,7,8,9} onto the interval [0, 1].

History

In 2018, German number theorist and Fields Medal winner Peter Scholze and Dustin Clausen realized that conventional topology exposed incompatibilities across geometry, functional analysis and p-adic numbers — and became convinced that alternative foundations could bridge those gaps. The pair announced their effort to rebuild mathematics in 2019.

In 2020 Scholze authored an involved proof that was critical to the theory. He asked other mathematicians led by Johan Commelin to help him verify its correctness. Over a 6-month period the group verified the proof using the proof assistant Lean.

External links

• Scholze, Peter (2020-12-05). "Liquid tensor experiment". Xena. Retrieved 2021-06-21.
• Scholze, Peter (2019). "Lectures on Condensed Mathematics" (PDF).
• Scholze, Peter (2020). "Lectures on Analytic Geometry" (PDF).
• Pstragowski, Piotr Tadeusz (2020-11-09). "Masterclass in Condensed Mathematics". www.math.ku.dk. Retrieved 2021-06-21.