Condensed mathematics
Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead different expressions of the same concepts (similarly to the way in which different human languages can express the same thing).[1]
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.[1][2]
Subgoals
Parts of the unification program are to replace topological spaces by “condensed sets",[3][4] turn functional analysis into a branch of commutative algebra, and various types of analytic geometry into algebraic geometry.[5] Condensed sets form condensed abelian groups into an abelian category, which satisfies the constraints of homological algebra.[3]
Approximately, a topological space X is replaced by the functor that takes a profinite set S to the set of continuous 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].[3]
History
In 2018, German number theorist and Fields Medal winner Peter Scholze and Canadian/American mathematician 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.[1]
In 2020 Scholze authored an involved proof that was critical to the theory. He asked other mathematicians led by Johan Commelin to provide a formalized and verified proof of this result.[4] Over a 6-month period the group verified the proof using the proof assistant [[Lean (proof assistant) |Lean]].[1]
References
- ^ a b c d Castelvecchi, Davide (2021-06-18). "Mathematicians welcome computer-assisted proof in 'grand unification' theory". Nature. doi:10.1038/d41586-021-01627-2.
- ^ Ferrari, S. (December 2019). "Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions". Infinite Dimensional Analysis, Quantum Probability and Related Topics. 22 (04): 1950026. arXiv:1510.08283. doi:10.1142/s0219025719500267. ISSN 0219-0257.
- ^ a b c Kionke, Steffen (17 December 2020). "Condensed mathematics – SPP 2026". Retrieved 2021-06-21.
- ^ a b Scholze, Peter (2020-12-05). "Liquid tensor experiment". Xena.
{{cite web}}: CS1 maint: url-status (link) - ^ "condensed mathematics in nLab". ncatlab.org. Retrieved 2021-06-21.
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.