Geometry Day

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An extended version of the Geometry Seminar, Tuesday Jan 13 (9:00-14:45), Malá Strana, S11, Prague.

• PROGRAM:
  • 9:00-9:45 Roman Nedela: Surfaces supporting maps of given type
  • 9:45-10:15 Tomáš Kaiser: Colouring normal quadrangulations of projective spaces
  • 10:15-10:45 coffee break
  • 10:45-11:25 Pavel Valtr: Polygons with many critical visibilities
  • 11:30-12:30 open problem session
  • 13:00-14:00 lunch at Ananta restaurant
  • 14:00-14:45 Todor Antić: Unbent collections of planar and non planar graphs
  • (abstracts below)

• Roman Nedela (University of West Bohemia, Plzeň): Surfaces supporting maps of given type:
  • Abstract: A map is a 2-cell decomposition of a compact connected surface without boundary. Usually maps are described by 2-cell embeddings of connected graphs. A map is of type (p,q), if every face is bounded by a walk of length p, and every vertex is of degree q. Due to the uniformisation theory a topological map can be make geometric by introducing a geometry of a Riemann surface such that the edges of the embedded graph become geodesic lines and angles at each vertex are equally distributed. Given (p,q) the problem of characterising surfaces (both orientable and non-orientable) supporting maps of type (p,q) was solved in [1]. In my talk I present a solution of a more general problem of characterising hypermaps of given type (p,q,r). A motivation for this research comes from coding theory. Joint work with S. Gyurki, J. Siagiova and J. Siran.
    
    [1] A. L. Edmonds, J. H. Ewing and R. S. Kulkarni, Regular tessellations of surfaces and (p, q, 2)-triangle groups, Annals Math. 116 (1982), 113-132.
• Tomáš Kaiser (University of West Bohemia, Plzeň): Colouring normal quadrangulations of projective spaces:
  • Abstract: Youngs proved that every non-bipartite quadrangulation of the projective plane RP^2 is 4-chromatic. Kaiser and Stehlik generalised the notion of a quadrangulation to higher dimensions and extended Youngs' theorem by proving that every non-bipartite quadrangulation of the d-dimensional projective space RP^d with d>=2 has chromatic number at least d+2. On the other hand, Hachimori et al. defined another kind of high-dimensional quadrangulation, called a normal quadrangulation. They proved that if a non-bipartite normal quadrangulation G of RP^d with any d>=2 satisfies a certain geometric condition, then G is 4-chromatic, and asked whether the geometric condition can be removed from the result. In this paper, we give a negative solution to their problem for the case d=3, proving that there exist 3-dimensional normal quadrangulations of RP^3 whose chromatic number is arbitrarily large. Moreover, we prove that no normal quadrangulation of RP^d with any d>=2 has chromatic number 3.
    Joint work with On-Hei Solomon Lo, Atsuhiro Nakamoto, Yuta Nozaki, and Kenta Ozeki.
• Pavel Valtr (Charles University, Praha): Polygons with many critical visibilities:
  • Abstract: For any n>3 we construct a polygon P with n vertices such that there are at least c.n.logn pairs {a,b} of its vertices with the property that the straight-line segment ab contains exactly one vertex of P different from a,b, and all other points of the segment lie in the interior of P.
    Joint work with Maria Saumell.
• Todor Antić (Charles University, Praha): Unbent collections of planar and non planar graphs:
  • Abstract: Recently, there has been growing interest in the graph drawing community in representing graphs by multiple drawings, so that each drawing highlights a different aspect of the information which the graph represents. I will present a take on this recent trend in the context of orthogonal drawings of graphs. An orthogonal drawing of a max-degree 4 graph G is a drawing of G with edges represented by sequences of horizontal and vertical segments. An unbent collection of a graph G is a collection of orthogonal drawings of G such that each edge is represented by a single segment (so with no bends) in at least one drawing. We will see several results about sizes of such collections in the case where G is planar or not and on the total number of bends in an unbent collection of G when G is planar.
    This is part of joint works with Giuseppe Liotta, Tomáš Masařík, Giacomo Ortali, Matthias Pfretzschner, Peter Stumpf, Alexander Wolff, Johannes Zink and Therese Beidl.

Everybody is welcome to attend.

Organizers: Martin Balko, Maria Saumell, and Pavel Valtr