430 research outputs found
On the simple connectedness of hyperplane complements in dual polar spaces
Let be a dual polar space of rank \geq 4 be a hyperplane of
and be the complement of \Delta\Delta points, then is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings
Simple maps, Hurwitz numbers, and Topological Recursion
We introduce the notion of fully simple maps, which are maps with non
self-intersecting disjoint boundaries. In contrast, maps where such a
restriction is not imposed are called ordinary. We study in detail the
combinatorics of fully simple maps with topology of a disk or a cylinder. We
show that the generating series of simple disks is given by the functional
inversion of the generating series of ordinary disks. We also obtain an elegant
formula for cylinders. These relations reproduce the relation between moments
and free cumulants established by Collins et al. math.OA/0606431, and implement
the symplectic transformation on the spectral curve in
the context of topological recursion. We conjecture that the generating series
of fully simple maps are computed by the topological recursion after exchange
of and . We propose an argument to prove this statement conditionally to
a mild version of symplectic invariance for the -hermitian matrix model,
which is believed to be true but has not been proved yet.
Our argument relies on an (unconditional) matrix model interpretation of
fully simple maps, via the formal hermitian matrix model with external field.
We also deduce a universal relation between generating series of fully simple
maps and of ordinary maps, which involves double monotone Hurwitz numbers. In
particular, (ordinary) maps without internal faces -- which are generated by
the Gaussian Unitary Ensemble -- and with boundary perimeters
are strictly monotone double Hurwitz numbers
with ramifications above and above .
Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this
implies an ELSV-like formula for these Hurwitz numbers.Comment: 66 pages, 7 figure
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Towards the horizons of Tits's vision -- on band schemes, crowds and F1-structures
This text is dedicated to Jacques Tits's ideas on geometry over F1, the field
with one element. In a first part, we explain how thin Tits geometries surface
as rational point sets over the Krasner hyperfield, which links these ideas to
combinatorial flag varieties in the sense of Borovik, Gelfand and White and
F1-geometry in the sense of Connes and Consani. A completely novel feature is
our approach to algebraic groups over F1 in terms of an alteration of the very
concept of a group. In the second part, we study an incidence-geometrical
counterpart of (epimorphisms to) thin Tits geometries; we introduce and
classify all F1-structures on 3-dimensional projective spaces over finite
fields. This extends recent work of Thas and Thas on epimorphisms of projective
planes (and other rank 2 buildings) to thin planes.Comment: 29 page
Mini-Workshop: Amalgams for Graphs and Geometries
[no abstract available
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