1. Contributed Talk at the International Workshop on VISUALIZATION and MATHEMATICS 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares Math&Media Laboratory, Department of Mathematics, PUC-Rio Visualizing Forman's discrete vector field

2. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 2 VisMath 2002 Outline of the Talk Differential Morse theory Forman’s discrete Morse theory Discrete gradient vector field Critical cells and topology Hypergraphs & hypertrees Algorithm description Applications

3. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 3 VisMath 2002 Differential Morse Theory The topology of a differentiable manifold is very closely related to the critical points of a real smooth map defined on it.

4. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 4 VisMath 2002 Gradient Vector Field f: X->R real differentiable map defined on a differential manifold X. Critical points: x  X such that  f(x) = 0. f Morse function: critical points are not degenerated. V Morse vector field iff there exists a Morse function f such that V=  f.

5. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 5 VisMath 2002 Index of a critical point: number of negative eigenvalues of the Hessian matrix of f Critical points depends on the Morse vector field used Handlebody decomposition: each critical point is associated to a handle in the decomposition Critical Points

6. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 6 VisMath 2002 Forman’s Discrete Morse Theory (1995) General CW-complex Combinatorial structure Free of geometric embedding Tool for computational topology and geometry

7. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 7 VisMath 2002 Cell Complexes (1) A cell  of dimension k is a space homeomorphic to the ball of dimension k. A cell complex K is a collection of cells such that every intersection of the closures of two cells of K is also a cell of K.

8. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 8 VisMath 2002 Cell Complexes (2) In Forman’s theory, cells are given by their incidences and not as a geometric realization. Here we will consider only finite cell complexes

9. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 9 VisMath 2002 Hasse Diagram Simple oriented graph built out of K: Nodes represent the cells of K Links connect cells to their faces of codimension 1

10. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 10 VisMath 2002 Discrete Gradient Vector Field (1) A discrete gradient vector field V is an acyclic pairing in the Hasse diagram: Acyclic : inverting the orientation of the links between paired cells do not create circuit.

11. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 11 VisMath 2002 V discrete gradient vector field : V: disjoint pairs of cells { ,  } with  face of  V(  )= , V(  )=0 V is acyclic: there is no trivial closed V-path V-path:  0,  0,...,  r,  r such that V(  p )=  p face of  p+1,  p+1   p Discrete Gradient Vector Field (2)

12. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 12 VisMath 2002 Critical Cells (1) Critical cells of V: unpaired cells, i.e. unmatched nodes in the Hasse diagram. Index of a critical cell is its dimension

13. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 13 VisMath 2002 Critical Cells (2) Depend on the gradient vector field used. Empty discrete gradient vector field: every cell is critical. Optimal discrete gradient vector field  Minimum number of critical cells.

14. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 14 VisMath 2002 Relation to Topology (1) Morse Inequalities The cell complex deformation retracts along the discrete gradient field

15. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 15 VisMath 2002 We can build a cell complex L out of the only critical cells of K; and a homotopy from K to L. For polyhedra, the minimal number of critical cells is a topological invariant. Relation to Topology (2)

16. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 16 VisMath 2002 Optimum Complexity Optimum in each level of the Hasse diagram leads to the global optimum. Reaching the optimum inside a level is a MAX SNP hard problem. MAX SNP hard : polynomial approximations can be arbitrary far from the optimum. (Eğecioğlu, 1995)(Hachimori, 2000)

17. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 17 VisMath 2002 Hypergraphs and the Hasse diagram A level of a Hasse diagram is a hypergraph. A level of the Hasse diagram of a discrete gradient vector field is a hypertree.

18. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 18 VisMath 2002 Hypergraphs Regular links connect two nodes Hyperlinks connect 1 or more than 3 nodes. Orientation: each link has exactly one source node, and possibly many destination nodes. Regular components: connected components of the graphs with the only regular links.

19. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 19 VisMath 2002 Hypertrees Every node is the source of at most one hyperlink Every regular component has the source node of at most one hyperlink There is an orientation with no circuits

20. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 20 VisMath 2002 Algorithm Processing each level in the Hasse diagram separately, from the highest dimension to the lowest one: –first level is a normal graph –other levels are hypergraphs

21. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 21 VisMath 2002 Example: EdgeBreaker on a Torus (1) C R L S S* E

22. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 22 VisMath 2002 Example: EdgeBreaker on a Torus (2)

23. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 23 VisMath 2002 Example: EdgeBreaker on a Torus (3: dual tree)

24. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 24 VisMath 2002 Example: EdgeBreaker on a Torus (4: graph)

25. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 25 VisMath 2002 Algorithm Outline Successively for each level (hypergraph) (n,n-1), (n-1,n-2),..., (3,2), (2,1): Orienting the remaining links of the 0-1 graph 1 - Selecting links to build a hypertree 2 - Orienting links to define the discrete gradient vector field

26. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 26 VisMath 2002 Algorithm: Constructing Hypertrees Kind of a greedy algorithm: spanning tree of the regular components adding one boundary link to each regular component if any adding hyperlinks which do not create circuit with some priority.

27. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 27 VisMath 2002 Algorithm: Orienting links Regular components are oriented tree. The root node is critical or incident to the hyperlink. Root nodes of the regular components are the source of the hyperlinks.

28. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 28 VisMath 2002 Discrete Gradient Field Visualization (1)

29. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 29 VisMath 2002 Discrete Gradient Field Visualization (2)

30. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 30 VisMath 2002 Discrete Gradient Field Visualization (3)

31. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 31 VisMath 2002 Visualizing Abstract Complex

32. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 32 VisMath 2002 Topological Decomposition

33. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 33 VisMath 2002 Next Steps Tools to understand the topology of 3-manifolds: –auditory discrete Morse theory –visual investigation of 3-manifolds –graphical navigation guided by the topology Topology consistent morphing Conditions for reaching optimality in polynomial time Applications to volumetric compression

34. Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio Classical Morse Theory Gradient Vector Field Critical Points Forman’s Theory Cell Complexes Hasse Diagram Discrete Gradient Field Critical cells & topology Critical Cells Relation to Topology Optimal Complexity Hypergraph, Hypertree Algorithm EdgeBreaker Example Outline Applications Discrete Gradient Field Visualization Visualizing Abstract Complex Topological Decomposition 34 VisMath 2002 Morse Inequalities m(k) : number of critical points of index k  (k) : k-th Betti number Strong Morse Inequalities  (k) -  (k-1) + …   (0)  m(k) - m(k-1) + …  m(0) Weak Morse Inequalities  (k)  m(k) Euler Characteristic  (n) -  (n-1) + …   (0) = m(n) - m(n-1) + …  m(0)