Mathematical Concepts Essential for understanding Space

This list of spacial concepts will more likely help the foxes rather than the hedgehogs. Foxes like fox a top down approach to understanding the universe including its mathematical underpinings. This is the approach to mathematics taken by Terence Tao, Fields medalist, and professor of mathematics among others.

Mathematics has evolved over time. It is "of this world" which means it has evolved in the context of our haptic sensory limits and then progressively through our instrumentation enabled experimentation.

Successive definitions of space have given us new tools for understanding the nature of reality and for computational progress.

Linear algebra, vector space and tensors feature in the foundations of AI and Machine Learning. These mathematical concepts occur in a continum of expanded understanding of reality.

Space   Wiki: Space (Mathematics)                Spatial scale   Wiki: Spatial scale      Wiki: Orders of Magnitude of Length      Cosmic distance ladder       Distance measures (cosmology)    Euclidean Space   Wiki: Euclidean Space                Cartesian Space   Cartesian Space      Wiki: Analytic geometry          Minkowski Space Minkowski space is the main mathematical description of spacetime in the absence of gravitation.   Wiki: Minkowski Space      Wiki: Non-Euclidean Geometry      YT:Classroom Aid - Minkowski Space-Time (3.55 min)    Vector Space (also called a linear space) dual vector space (or just dual space for short)   Wiki: Vector Space      Wiki: Tensor      Wiki: Tensor      Wiki: Lie algebra      Wiki: Dual vector space (aka dual space)    Hilbert Space   Wiki: Hilbert_space      What is a Hilbert Space? (May 23, 2025 )      Hilbert Space Kernel Methods for Machine Learning: Background and Foundations (May 23, 2025 )    Newtonian Space and Time In Newtonian physics, space and time had their independent identities.   Wiki: Absolute space and time                Einstein Space Time - Special Relativity Special relativity is the theory of a fixed, flat space-time, without gravity   Wiki: Special Relativity                Minkowski Spacetime Minkowski space-time is flat, static and infinite.   Wiki:Minkowski spacetime only applies in special relativity.      Quantamagazine:Sean Carroll: How to Think About Relativity?      YT: Quanta Magazine: Vijay Balasubramanian, a physicist at the University of Pennsylvania, takes us on a journey through space-time (September 25, 2024)    Riemannian geometry   Wiki: Riemannian Geometry          Einstein Space Time - General Relativity General relativity is the theory of dynamic space-time, in which curvature gives rise to gravity.   Wiki: General Relativity                  YT: How We Mapped The Infinite Universe With Jim Al-Khalili (June 19, 2025) (59 min)

Topological Space

	Wiki: Topological Space

Quantum spacetime

	Wiki: Quantum spacetime
	Wiki: Mathematical formulation of quantum mechanics
	Wiki: Quantum mechanics
	Quantum field theory
	Simons Foundation: Quanta Magazine: The accelerating effort to understand the mathematics of quantum field theory will have profound consequences for both math and physics.(June 10, 2021)
	YT: Harvard CMSA Algebraic quantum field theory (Langlands)(February 8, 2024)
	YT: Machine-learned flows for QCD: Numerical lattice quantum field theory is one of our important tools for investigating quantum chromodynamics machine learning techniques may provide algorithmic advances that could open access to calculations which are presently intractable(QCD)
	YT: Can Something Come From Nothing? Exploring The Universe's Origins With Jim Al-Khalili (June 26, 2025) (59 min)

Macrosopic Scale