43 research outputs found
Face enumeration on simplicial complexes
Let be a closed triangulable manifold, and let be a
triangulation of . What is the smallest number of vertices that can
have? How big or small can the number of edges of be as a function of
the number of vertices? More generally, what are the possible face numbers
(-numbers, for short) that can have? In other words, what
restrictions does the topology of place on the possible -numbers of
triangulations of ?
To make things even more interesting, we can add some combinatorial
conditions on the triangulations we are considering (e.g., flagness,
balancedness, etc.) and ask what additional restrictions these combinatorial
conditions impose. While only a few theorems in this area of combinatorics were
known a couple of decades ago, in the last ten years or so, the field simply
exploded with new results and ideas. Thus we feel that a survey paper is long
overdue. As new theorems are being proved while we are typing this chapter, and
as we have only a limited number of pages, we apologize in advance to our
friends and colleagues, some of whose results will not get mentioned here.Comment: Chapter for upcoming IMA volume Recent Trends in Combinatoric
On stacked triangulated manifolds
We prove two results on stacked triangulated manifolds in this paper: (a)
every stacked triangulation of a connected manifold with or without boundary is
obtained from a simplex or the boundary of a simplex by certain combinatorial
operations; (b) in dimension , if is a tight connected
closed homology -manifold whose th homology vanishes for ,
then is a stacked triangulation of a manifold.These results give
affirmative answers to questions posed by Novik and Swartz and by Effenberger.Comment: 11 pages, minor changes in the organization of the paper, add
information about recent result
Tight triangulations of closed 3-manifolds
It is well known that a triangulation of a closed 2-manifold is tight with
respect to a field of characteristic two if and only if it is neighbourly; and
it is tight with respect to a field of odd characteristic if and only if it is
neighbourly and orientable. No such characterization of tightness was
previously known for higher dimensional manifolds. In this paper, we prove that
a triangulation of a closed 3-manifold is tight with respect to a field of odd
characteristic if and only if it is neighbourly, orientable and stacked. In
consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for
fields of odd characteristic.
Next let be a field of characteristic two. It is known that, in
this case, any neighbourly and stacked triangulation of a closed 3-manifold is
-tight. For triangulated closed 3-manifolds with at most 71
vertices or with first Betti number at most 188, we show that the converse is
true. But the possibility of an -tight non-stacked triangulation on
a larger number of vertices remains open. We prove the following upper bound
theorem on such triangulations. If an -tight triangulation of a
closed 3-manifold has vertices and first Betti number , then
. Equality holds here if and only if all
the vertex links of the triangulation are connected sums of boundary complexes
of icosahedra.Comment: 21 pages, 1 figur
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
Geometric and Algebraic Combinatorics
The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture
Thirty-five years and counting
It has been 35 years since Stanley proved that f-vectors of boundaries of
simplicial polytopes satisfy McMullen's conjectured g-conditions. Since then
one of the outstanding questions in the realm of face enumeration is whether or
not Stanley's proof could be extended to larger classes of spheres. Here we
hope to give an overview of various attempts to accomplish this and why we feel
this is so important. In particular, we will see a strong connection to
f-vectors of manifolds and pseudomanifolds. Along the way we have included
several previously unpublished results involving how the g-conjecture relates
to bistellar moves and small g_2, the topology and combinatorics of stacked
manifolds introduced independently by Bagchi and Datta, and Murai and Nevo, and
counterexamples to over optimistic generalizations of the g-theorem.Comment: 29 page
Real Analysis, Quantitative Topology, and Geometric Complexity
Contents
1 Mappings and distortion
2 The mathematics of good behavior much of the time, and the BMO frame of
mind
3 Finite polyhedra and combinatorial parameterization problems
4 Quantitative topology, and calculus on singular spaces
5 Uniform rectifiability
Appendices
A Fourier transform calculations
B Mappings with branching
C More on existence and behavior of homeomorphisms
D Doing pretty well with spaces which may not have nice coordinates
E Some simple facts related to homologyComment: 161 pages, Latex2