1. Tilings, Geometric Representations, and Discrete Analytic Functions László Lovász Microsoft Research lovasz@microsoft.com

2. Harmonic functions Every non-constant function has at least 2 poles.

3. Example 1: Random walks S 2 3 1 f(v)= E (f(Z v )) Z v : (random) point where random walk from v hits S v 0 v 1 f(v)= P ( random walk from v hits t before s) s t

4. Example 2: Electrical networks 0 v 1 f(v)= electrical potential s t

5. Example 3: Rubber bands f(v)= position of nodes 0 1

6. “Dictionary” of quantities from electrical networks, random walks, and static:

7. Analytic functions on the lattice (Ferrand 1944, Duffin 1956) z z+1 z+i z+i+1

8. “z”“z” 0 12 3 2+i 2-i 1+i 1-i-i i 2i 3i 1+2i 4 3+i 3-i

9. 0 14 9 3+4i 3-4i 2i -2i -4 -9 -3+4i 16 8+6i 8-6i

10. 0 16 19 2+7i -1+i i -i-i -6i -19i -7-2i 44 13+17i -1-i2-7i -10+10i 1-i 1+i

12. 3 3 3 3 2 2 2 5 4 1 10 Brooks-Smith-Stone-Tutte 1940 0 3 4 5 6 7 9

13. 3 3 3 3 2 2 2 5 4 1 10

14. Rotation-free circulation: Discrete holomorphic forms on surfaces Mercat 2001 Weighted version:

15. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 17 Grid on the torus

16. 11 14 6 5 9 3 4 2 1 3 1 11 16 13 2 14 9 11 27 16 5 0 12 4 5 14 0 -6 -3 -2 25 23 10 8 9

17. A : null-homologous circulationsdim( A ) = #faces-1= f-1 B : potentials dim( B ) = #nodes-1= n-1 C : rotation-free circulations dim( C ) = #edges- (#nodes-1)- (#faces-1) = 2·genus mutually orthogonal Discrete Hodge decomposition

18. “Analytic function cannot vanish on a large set’’ “Large” is not size: 0 0 0 00 0 2 1 1 planar piece separating set of size < 2g

19. f : nonzero rotation-free circulation G ’: subgraph where f does not vanish U : connected subgraph such that f vanishes on all edges incident with U “Analytic function cannot vanish on a large set’’ U can be separated from G ’ by at most 4g-3 nodes. Benjamini-L

20. Main (easy) topological lemma:

21. 2 1 1 0 2 3 (assume no other 0’s) Proof of Theorem u

22. Which sets of edges can be supports of rotation-free circulations? -  weighting of the edges - for weights =1 - for some weighting Can be characterized using matroid theory LL-Schrijver Combinatorially analytic maps:

23. incidence vector of edge uv projection of  uv on C If g>0 and G is 3-connected and simple, then  uv  0 for every edge uv. u v qp If g=0, then  uv =0 for every edge uv.

24. ee e

25. incidence vector of edge uv projection of  uv on C If g>0 and G is 3-connected and simple, then  uv  0 for every edge uv. u v qp If g>0 and G is 3-connected and simple, then  nowhere-0 rotation-free circulation on G.

26. A strange identity: If G is a 3-connected map, then

27. Toroidal maps: analytic functions, straight line embeddings, rubber bands, square tilings

28. Universal cover map

29. 18 5 12 7 2 9 15 4 5 1 6 18 5 12 7 2 9 15 4 5 1 6 18 15 6 18 15 6 44 5 2 5 2 18 15 18 15 7 9 5 1 7 9 5 1 g=1 : Two linearly independent rotation-free circulations

30. 44 5 2 5 2 16 21 20 5 0 7 18 5 12 7 2 9 15 4 5 1 6 49 54 53 38 33 40 18 5 12 7 2 9 15 4 5 1 6 63 48 18 15 6 30 15 18 15 6 2617 7 9 5 1 5950 7 9 5 1 Two linearly independent harmonic functions

31. Two coordinates  periodic embedding in the plane  embedding in the torus This gives an embedding in the torus

32. Horizontal coordinate is nowhere-zero flow  nondegenerate squares Edge  square horizontal length  size

33. R. Kenyon

35. Which other properties of analytic functions have discrete analogues? Multiplication of analytic (meromorphic) functions? Merkat: weight the edges, “critical” weighting Riemann-Roch? (Cai)

36. Global information from local observation “Can you hear the shape of a drum?” - Observe a graph locally (a single node, a neighborhood of a node, a “window”) - There is a local random process on the graph (random walk, heat bath,...) - Infer global properties of the graph

37. The graph, with the window:

38. The Noisy Circulator: -2 4 3 1 -2 2 3 1 -2.4 1.6 -2.4 2.6 0.6 (face balancing) 1.32.3 (edge excitation) frequency 1 frequency p (node balancing) -3 3 -2 2 0

39. If no edge excitation occurs, then the weighting converges to a rotation-free flow. dimension of rotation-free flows =2g dimension of rotation-free flows in window =2g if edge excitations occur infrequently: random vectors from a (2g) -dimensional subspace with errors

40. If p<N -c, then observing the Noisy Circulator for N c /p steps, we can determine g with high probability.