Geometry, Combinatorics, and Algorithms (GeCoaL)
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1 Geometry, Combinatorics, and Algorithms (GeCoaL) Éric Colin de Verdière École normale supérieure, CNRS, Paris May 12, 2011
2 Main Research Area Communities computational geometry/topology discrete algorithms Philosophy introducing (classical but powerful) tools from algebraic and combinatorial topology to solve algorithmic (and combinatorial) problems with some (low-dimensional) geometric flavor
3 People Team members current: myself (CR CNRS), Arnaud de Mesmay: pre-doc student, PhD starting Sep (w.h.p.) past: Michel Pocchiola and student, till Sep future: w.h.p., Claire Mathieu (a.k.a. Claire Kenyon) hired as DR1 CNRS here in Oct (may delay arrival) Main coauthors and other collaborators ENS: Grégory Ginot (Math dept), Pierre Pansu (Math dept), Jean Ponce (DI, ANR Triangles) France: Francis Lazarus (Grenoble), Xavier Goaoc (INRIA Nancy), Monique Teillaud (INRIA Sophia) world: Sergio Cabello (Slovenia), Alexander Schrijver (CWI, Amsterdam), Jeff Erickson (UIUC), Bojan Mohar (SFU, Vancouver), Yusu Wang (Ohio State)
4 People Team members current: myself (CR CNRS), Arnaud de Mesmay: pre-doc student, PhD starting Sep (w.h.p.) past: Michel Pocchiola and student, till Sep future: w.h.p., Claire Mathieu (a.k.a. Claire Kenyon) hired as DR1 CNRS here in Oct (may delay arrival) Main coauthors and other collaborators ENS: Grégory Ginot (Math dept), Pierre Pansu (Math dept), Jean Ponce (DI, ANR Triangles) France: Francis Lazarus (Grenoble), Xavier Goaoc (INRIA Nancy), Monique Teillaud (INRIA Sophia) world: Sergio Cabello (Slovenia), Alexander Schrijver (CWI, Amsterdam), Jeff Erickson (UIUC), Bojan Mohar (SFU, Vancouver), Yusu Wang (Ohio State)
5 Visibility, Teaching Service for conferences recent PC memberships: SOCG 2011 (ACM Symposium on Computational Geometry) SODA 2009 (ACM-SIAM Symposium on Discrete Algorithms) CTIC 2008 (Computational Topology in Image Context) organization: French CG week 2010 (55 participants) Recent invited talks SIAM Conference on Discrete Mathematics 2010 French Graphs and Algorithms meeting 2010 jury of 2 PhD theses (2009, 2010) Teaching MPRI2 (pre-doc students): Algorithms for embedded graphs student supervision (1st year math students on topics in algorithms and complexity)
6 Visibility, Teaching Service for conferences recent PC memberships: SOCG 2011 (ACM Symposium on Computational Geometry) SODA 2009 (ACM-SIAM Symposium on Discrete Algorithms) CTIC 2008 (Computational Topology in Image Context) organization: French CG week 2010 (55 participants) Recent invited talks SIAM Conference on Discrete Mathematics 2010 French Graphs and Algorithms meeting 2010 jury of 2 PhD theses (2009, 2010) Teaching MPRI2 (pre-doc students): Algorithms for embedded graphs student supervision (1st year math students on topics in algorithms and complexity)
7 Current Research Themes
8 Algorithms for Curves on Surfaces Shortest graphs/curves with given topological properties cutting a surface into a disk using short curves [ESA 10] splitting a surface into nontrivial parts [SOCG 06, CGTA] shortening a curve up to deformation [SODA 06, SICOMP] Motivations in computer graphics handles arise naturally, or from scanning 3d models parameterization: texture mapping, remeshing, compression,... geometric modelling, CAGD Techniques (Courtesy J.M. Favreau) algebraic topology (homotopy/homology, covering spaces) (combinatorial) hyperbolic geometry algorithmic tools
9 Algorithmic Aspects of Topological Graph Theory Shortest nontrivial cycles in surface-embedded graph edgewidth: length of shortest nontrivial cycle useful structural properties if large enough ( local planarity ) crucial building block for other algorithms (planar crossing numbers, isomorphism test,... )
10 Algorithmic Aspects of Topological Graph Theory Shortest nontrivial cycles in surface-embedded graph edgewidth: length of shortest nontrivial cycle useful structural properties if large enough ( local planarity ) crucial building block for other algorithms (planar crossing numbers, isomorphism test,... ) Results: edgewidth computation efficient approx./output-sensitive algorithms [SOCG 10, CGTA?] case of weighted, directed graph [SOCG 10] Other miscellaneous results on graphs on surfaces testing existence of cycle with given top. properties [submitted] bound on number of irreducible triangulations of surfaces with boundary [submitted]
11 Algorithms for Planar Problems Measure of similarity of curves in the plane: Fréchet distance widely explored, application-relevant two variations: in presence of obstacles [SOCG 08, CGTA] alternate measure [in progress] tools: homotopy, covering spaces, shortest paths in geometric settings Shortest disjoint paths in planar graphs hard problem in general special case tractable in polynomial time [STACS 08, TALG] tools: combinatorial optimization, maxand min-cost flows, cuts, (topology)
12 Algorithms for Planar Problems Measure of similarity of curves in the plane: Fréchet distance widely explored, application-relevant two variations: in presence of obstacles [SOCG 08, CGTA] alternate measure [in progress] tools: homotopy, covering spaces, shortest paths in geometric settings t 2 Shortest disjoint paths in planar graphs hard problem in general special case tractable in polynomial time [STACS 08, TALG] tools: combinatorial optimization, maxand min-cost flows, cuts, (topology) t 1 t 4 s 4 s 3 s 1 s 2 t 3
13 Combinatorial Geometry, with (Avanced) Topology Helly s theorem, baby version Let F be a family of convex sets in the plane. If any triple of elements in F intersects, then the whole family F intersects. Two generalizations, in R d : For every G F, the intersection of G 1 is the disjoint union of at most k convex sets, or 2 has trivial topology (empty or point) Our result encompasses both generalizations [submitted] tools: advanced algebraic topology (spectral sequences, nerve theorems) unifies several existing results in geometric transversal theory
14 Future of the Team
15 First Objective: Reach Critical Size (!) [In my personal view!] New head of the team: Claire Mathieu Team full prof. at Brown general field of interest: combinatorial optimization, approx. & randomized algorithms (FOCS, STOC, SODA) main recent direction: algorithms for planar graphs Inspired by approximation algorithms in geometric contexts algorithms in general, with slant on geometric/topological problems Benefits for the department algorithms: broad, exciting field topic very suitable for ENS students (strong math background) Claire wants to teach!
16 Scientific Objectives: Algorithms and Topology [as far as I am concerned!] Short term: Graphs on surfaces, again... Algorithms: planar graphs and graphs on surfaces... Computational topology problems for curves and surfaces in 3d...
17 Scientific Objectives: Algorithms and Topology [as far as I am concerned!] Short term: Graphs on surfaces, again... Algorithms: planar graphs and graphs on surfaces... Computational topology problems for curves and surfaces in 3d...
18 Scientific Objectives: Algorithms and Topology [as far as I am concerned!] Short term: Graphs on surfaces, again... Algorithms: planar graphs and graphs on surfaces... Computational topology problems for curves and surfaces in 3d...
19
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