The notion of space in mathematics. Matilde Marcolli
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1 The notion of space in mathematics Matilde Marcolli general audience lecture at Revolution Books Berkeley 2009
2 Space is always decorated with adjectives (like numbers: integer, rational, real, complex) Linear space Topological space Metric space Projective space Measure space Noncommutative space 1
3 Space is a kind of structure Often (not always) a set (points of space) with some relational properties Operations: adding vectors, cutting and pasting, measuring size, intersections and unions Proximity relation: neighborhoods, closeness, convergence, distance Smooth Topo- Hierarchy of structures: logical Measure space Symmetries: transformations preserving all the structure (very different symmetries for different kinds of spaces) Telling spaces apart: invariants (numerical, algebraic structures, discrete) 2
4 Philosophical digression: Absolute vs. Relational view of space - Relational/Transformational viewpoint: Heraclitus, Democritus, Descartes, Leibniz, Bergson - Absolute view of space: Eleatic school (Parmenides, Zeno), Aristotle, Kant, Newton, Comte Mathematical reconciliation of philosophical dichotomy: - Felix Klein (Erlangen Program 1872): emphasis on transformation groups 3
5 Linear spaces (or vector spaces): set of vectors composition of vectors with dilations and Examples: straight lines, planes,... Classical mechanics: equilibrium of forces Dimension: Maximal number of linearly independent vectors 4
6 Projective spaces: non-zero vectors up to scaling: (identify v = (x 1, x 2, x 3 ) and λv = (λx 1, λx 2, λx 3 ), scaled by nonzero λ) Renaissance perspective drawings 1-dimensional real projective space = circle 1-dimensional complex projective space = sphere 5
7 More interesting shapes: 2-dimensional real projective space: Identifying diametrically opposite points on the boundary of a disk 6
8 Different kinds of numbers (fields) different kinds of projective spaces Finite projective spaces (discrete versus continuum in geometry) Relational properties: lines through given points, lines intersecting, planes containing lines,... 7
9 Topological spaces formalize the relation of being near a point (qualitative: does not quantify how near) Open condition: stable under small variations (close condition: being on the border of two regions) Transformations: continuous deformations a donut is topologically the same as a cup of coffee 8
10 Knots and links Topologically different: cannot be deformed one into the other without cutting - Invariants of knots 9
11 Topology of knots and DNA Topoisomerases enzymes act on the topology: unknotting DNA prior to replication 10
12 Nice topological spaces: triangulations Essential to computer graphics Graphs: simplest class of piecewise linear spaces 11
13 Examples of graphs: San Francisco subway system Moscow subway system 12
14 The most interesting graph of today: the world wide web Methods of topology for internet connectivity 13
15 More examples of topological spaces: - Sphere: - Torus: - Klein bottle: - Real projective plane: 14
16 How to distinguish topological spaces? Euler characteristic χ = #Faces #Edges + #Vertices is a topological invariant - Sphere: χ = 2, orientable - Real projective space: χ = 1, non-orientable - Klein bottle: χ = 0, non-orientable - Torus: χ = 0, orientable - Genus g surface: χ = 2 2g, orientable 15
17 Orientability Max Bill: Möbius band sculpture Maurits Cornelis Escher: Möbius band 16
18 Smooth spaces (or smooth manifolds): Topological spaces locally indistinguishable from a vector space Example: the Earth from ground level looks flat Tangent space Local coordinates: number of independent parameters describing a physical system - Dimension from tangent space (linear space) 17
19 Local versus global properties: locally like flat space (linear space) but globally: nontrivial topology View from inside a 3-torus (Jeff Weeks The shape of space ) 18
20 Smooth space Topological space but beware... Exotic smoothness: 4-dimensional flat space has infinitely many different smooth structures (Donaldson) - small: contained inside ordinary flat space - large: do not fit in ordinary space Dimension 3 and 4 are the most complicated!! Poincaré conjecture (Perelman): there is only one type of 3-dimensional sphere Smooth 4-dimensional sphere?? Unknown 19
21 Exotic smoothness can affect our understanding of the distant universe (gravitational lensing) passing through a small exotic space changes lensing 20
22 What detects exotic smoothness? Not topological invariants (Euler characteristic etc) Different properties of particle physics! Compare solutions of equations of motion for elementary particles: - Donaldson invariants (1980s) from electroweak forces - Seiberg Witten invariants (1990s) from string theory 21
23 Metric spaces topological space where can measure distance between points (Not just near but how near) Voronoi cells: points closer to one of the centers Metric space topological space but not all topological spaces can be metric 22
24 Unit ball: distance one from a point Sergels Torg Stockholm: unit ball in distance d((x, y), (0, 0)) = (x 4 + y 4 ) 1/4 23
25 Smooth spaces can be metric: Riemannian manifolds General Relativity, spacetime Lorentzian metric: light cones 24
26 What kind of space is space? (3-dimensional section of spacetime) Metric properties (positive/negative curvature) related to cosmological constant The problem of Cosmic topology Dodecahedral universe: Poincaré sphere 25
27 Searching for dodecahedral topology in the cosmic microwave background radiation Trying to match sides of polyhedron 26
28 Singular spaces Algebraic varieties: polynomial equations yx(x 2 + y z) = 0 (If polynomial homogeneous: inside projective spaces) Singularities: black holes, big bang, gravitational lensing 27
29 Measure spaces and fractals Measure the size of regions of space: area, volume, length Also measuring probability of an event Quantum mechanics, observables (theory of von Neumann algebras) 28
30 Dimension: Hausdorff dimension (real number) Sierpinski carpet: dimension log 3 log (union of three copies scaled down by a factor of two) Fractal: dimension not an integer Mandelbrot (1980s) 29
31 Transformations of measure spaces Anything that preserves measure of sets even if it cuts and rearranges pieces Non-measurable sets: Banach-Tarski paradox (cut ball in finitely many pieces and reassemble them by rotating and translating into a ball twice as big) Property of group of transformations 30
32 Noncommutative spaces (Connes 1980s) Quantum mechanics: Heisenberg uncertainty principle: positions and velocities do not commute (cannot be measured simultaneously) x v Quotients (gluing together points) of topological/smooth/metric/measure spaces noncommutative spaces Models for particle physics 31
33 Examples of noncommutative spaces: Space of Penrose tilings Quasicrystals 32
34 Do we need all these notions of space? Yes: interplay of different structures - Topological spaces can be smooth in different ways or not at all (exotic smoothness). - Topological spaces acquire a new notion of dimension when seen as measure spaces (fractals). - Riemannian manifolds (like spacetime) can be locally isometric but globally different due to topology (cosmic topology). - Different physics on different spaces. 33
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