Miniconference on the Mathematics of Computation 2018 CMS Winter Meeting The New World of Infinite Random Geometric Graphs Anthony Bonato Ryerson University

Happy Birthday, Robert!

Graphs in normed spaces fix a normed space: S eg: 1 ≤ p ≤ ∞; ℓpd : Rd with Lp-norm p < ∞: 𝑥 𝑝 = 𝑛 𝑥 𝑛 𝑝 1/𝑝 p = ∞: 𝑥 𝑝 = max 𝑛 𝑥 𝑛 V: set of points in S E: adjacency determined by relative distance Infinite random geometric graphs - Anthony Bonato

Random geometric graphs Infinite random geometric graphs - Anthony Bonato

Local Area Random Graph (LARG) model parameters: p in (0,1) a normed space S V: a countable set in S E: if || u – v || < 1, then uv is an edge with probability p Infinite random geometric graphs - Anthony Bonato

Geometric existentially closed (g.e.c.) ∀ A, B finite ∀𝛿<1 ∃z 𝛿 1 ∀ x Infinite random geometric graphs - Anthony Bonato

LARG graphs almost surely g.e.c. 1-geometric graph: g.e.c. and 1-threshold: adjacency only may occur if distance < 1 Theorem (BJ,11) With probability 1, and for any fixed p, LARG generates 1-geometric graphs. proof analogous to Erdős-Rényi result for the Rado graph R 1-geometric graphs “look like” R in their unit balls, but can have diameter > 2 Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Geometrization lemma in some settings, graph distance approximates the space’s metric geometry Lemma (BJ,11) If G = (V,E) is a 1-geometric graph and V is dense, then 𝑑 𝐺 𝑥,𝑦 = 𝑑 𝑥,𝑦 +1. graph distance integrally-approximates metric distance Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Step-isometries S and T normed spaces, f: S → T is a step-isometry if 𝑑 𝑆 𝑥,𝑦 = 𝑑 𝑇 𝑓(𝑥),𝑓(𝑦) restriction of notion of isometry remove floors captures integer distances only in R equivalent to: int(x) = int(f(x)) frac(x) < frac(y) iff frac(f(x)) < frac((y)) Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Example: ℓ∞ V: dense countable set in R E: LARG model integer distance free (IDF) set pairwise ℓ∞ distance non-integer dense sets contain idf dense sets “random” countable dense sets are idf Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Categoricity countable V is Rado if the LARG graphs on it are isomorphic with probability 1 Theorem (BJ,11) Dense idf sets in ℓ∞d are Rado for all d > 0. new class of infinite graphs GRd which are unique limit objects of random graph processes in normed spaces Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Sketch of proof for d = 1 back-and-forth build partial isomorphism from V = V(t) and W = W(t) to be a step-isomorphism via induction add v not in V, and go-forth (back similar) a = max{frac(f(u)): u ∈ V, frac(u) < frac(v)}, b = min{frac(f(u)): u ∈ V, frac(u) > frac(v)} a < b, as fractional parts distinct by idf want f(v) to satisfy: int(f(v)) = int(v) frac(f(v)) ∈ [a,b) I = (int(v) + a, int(v) + b) choose vertex in 𝐼∩𝑊 (using density) will maintain step-isometry in (IS) use g.e.c. to find f(v) in co-domain correctly joined to W. Infinite random geometric graphs - Anthony Bonato

The new world

Infinite random geometric graphs - Anthony Bonato Properties of GRd symmetry: step-isometric isomorphisms of finite induced subgraphs extend to automorphisms indestructible locally R, but infinite diameter Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Dimensionality equilateral dimension D of normed space: maximum number of points equal distance p = ∞: D = 2d points of hypercube p = 1: Kusner’s conjecture: D = 2d proven only for d ≤ 4 equilateral clique number of a graph, ω3: max |A| so that A has all vertices of graph distance 3 apart Theorem (BJ,15) ω3(GRd) = 2d. if d ≠ d’, then GRd ≄ GRd’ Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Euclidean distance Lemma (BJ,11) In ℓ22, every step-isometry is an isometry. countable dense V is strongly non-Rado if any two such LARG graphs on V are with probability 1 not isomorphic Corollary (BJ,11) All countable dense sets in ℓ22 are strongly non-Rado. non-trivial proof, but ad hoc Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Honeycomb metric Theorem (BJ,12) Almost all countable dense sets R2 with the honeycomb metric are strongly non-Rado. Infinite random geometric graphs - Anthony Bonato

Enter functional analysis Miniconference on the Mathematics of Computation Enter functional analysis (Balister,Bollobás,Gunderson,Leader,Walters,18) Let S be finite-dimensional normed space not isometric to ℓ∞d . Then almost all countable dense sets in S are strongly non-Rado. proof uses functional analytic tools: ℓ∞-decomposition Mazur-Ulam theorem properties of extreme points in normed spaces Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato ℓ∞d are special spaces ℓ∞d are the only finite-dimensional normed spaces where almost all countable sets are Rado interpretation: ℓ∞d is the only space whose geometry is approximated by graph structure Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Questions classify which countable dense sets are Rado in ℓ∞d same question, but for finite-dimensional normed spaces. what about infinite dimensional spaces? Infinite random geometric graphs - Anthony Bonato

Infinitely many parallel universes

Classical Banach spaces C(X): continuous function on a compact Hausdorff space X eg: C[0,1] ℓ∞ bounded sequences c: convergent sequences c0: sequences convergent to 0 Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Separability a normed space is separable if it contains a countable dense set C[0,1], c, and c0 are separable  ℓ∞ and ω1 are not separable Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Hierarchy c c0 Banach-Mazur C(X) Infinite random geometric graphs - Anthony Bonato

Graphs on sequence spaces fix V a countable dense set in c LARG model defined analogously to the finite dimensional case NB: countably infinite graph defined over infinite-dimensional space Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Rado sets in c and c0 Theorem (BJ,Quas,18+): Almost all countable sets in c are Rado. Theorem (BJQ,18+): Almost all countable sets in c0 are Rado. Ideas of proof: almost all sets are dense and idf like ℓ∞d, but more machinery to deal with the fractional parts of limits of images in back-and-forth argument Infinite random geometric graphs - Anthony Bonato

The curious geometry of sequence spaces

Geometric structure: c vs c0 c vs c0 are isomorphic as vector spaces not isometrically isomorphic: c contains extreme points eg: (1,1,1,1, …) unit ball of c0 contains no extreme points Infinite random geometric graphs - Anthony Bonato

Interpolating the space from the graph Theorem (BJQ,18+) Suppose V and W are Banach spaces with dense sets X and Y. If G and H are the 1-geometric graphs on X and Y (resp) and are isomorphic, then there is a surjective isometry from V to W. hidden geometry: if we know LARG graphs almost surely, then we can recover the Banach space! Idea - use Dilworth’s theorem: δ-surjective ε-isometries of Banach spaces are uniformly approximated by genuine isometries Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Graph structure: c vs c0 Theorem (BJQ,18+) The graphs G(c) and G(c0) are not isomorphic to any GRd. G(c) and G(c0) are non-isomorphic. Infinite random geometric graphs - Anthony Bonato

Continuous functions

Dense sets in C[0,1] (BJQ,18+) piecewise linear functions and polynomials almost all sets are smoothly dense Brownian motion path functions almost all sets are IC-dense Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Isomorphism in C[0,1] Theorem (BJQ,18+) Smoothly dense sets give rise to a unique isotype of LARG graphs: GR(SD). Almost surely IC-sets give rise to a unique isotype of LARG graphs: GR(ICD). Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Non-isomorphism Theorem (BJQ,18+) The graphs GR(SD) and GR(ICD) are non-isomorphic. Idea: Dilworth’s theorem and Banach-Stone theorem: isometries on C[0,1] induce homeomorphisms on [0,1] Infinite random geometric graphs - Anthony Bonato

Infinite random geometric graphs - Anthony Bonato Questions “almost all” countable sets in C[0,1] are Rado? need a suitable measure of random continuous function explicit representations? automorphism groups? which Banach spaces have Rado sets? program: interplay of graph structure and the geometry of Banach spaces Infinite random geometric graphs - Anthony Bonato

Contact Web: http://www.math.ryerson.ca/~abonato/ Blog: https://anthonybonato.com/ @Anthony_Bonato https://www.facebook.com/anthony.bonato.5