60 research outputs found
Topological complexity of the relative closure of a semi-Pfaffian couple
Gabrielov introduced the notion of relative closure of a Pfaffian couple as
an alternative construction of the o-minimal structure generated by
Khovanskii's Pfaffian functions. In this paper, use the notion of format (or
complexity) of a Pfaffian couple to derive explicit upper-bounds for the
homology of its relative closure.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti
numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve
Topology of definable Hausdorff limits
Let be a set definable in an o-minimal expansion of the
real field, be its projection, and assume that the non-empty
fibers are compact for all and uniformly bounded,
{\em i.e.} all fibers are contained in a ball of fixed radius If
is the Hausdorff limit of a sequence of fibers we give an
upper-bound for the Betti numbers in terms of definable sets
explicitly constructed from a fiber In particular, this allows to
establish effective complexity bounds in the semialgebraic case and in the
Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative
closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian
functions in a way that is adapted to complexity problems. Our results can be
used to estimate the Betti numbers of a relative closure in the
special case where is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and
notations in an attempt to be clearer, references adde
Quantitative study of semi-Pfaffian sets
We study the topological complexity of sets defined using Khovanskii's
Pfaffian functions, in terms of an appropriate notion of format for those sets.
We consider semi- and sub-Pfaffian sets, but more generally any definable set
in the o-minimal structure generated by the Pfaffian functions, using the
construction of that structure via Gabrielov's notion of limit sets. All the
results revolve around giving effective upper-bounds on the Betti numbers (for
the singular homology) of those sets.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame
topology, spectral sequences, Morse theory.Comment: Author's PhD thesis. Approx. 130 pages, no figure
Recommended from our members
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
Knots, links, and long-range magic
We study the extent to which knot and link states (that is, states in 3d
Chern-Simons theory prepared by path integration on knot and link complements)
can or cannot be described by stabilizer states. States which are not classical
mixtures of stabilizer states are known as "magic states" and play a key role
in quantum resource theory. By implementing a particular magic monotone known
as the "mana" we quantify the magic of knot and link states. In particular, for
Chern-Simons theory we show that knot and link states are generically
magical. For link states, we further investigate the mana associated to
correlations between separate boundaries which characterizes the state's
long-range magic. Our numerical results suggest that the magic of a majority of
link states is entirely long-range. We make these statements sharper for torus
links.Comment: 36 pages; 64 knots; 34 links; v2 to match published versio
Recommended from our members
Reelle Algebraische Geometrie
This workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
Wallpaper Fermions and the Nonsymmorphic Dirac Insulator
Recent developments in the relationship between bulk topology and surface
crystal symmetry have led to the discovery of materials whose gapless surface
states are protected by crystal symmetries. In fact, there exists only a very
limited set of possible surface crystal symmetries, captured by the 17
"wallpaper groups." We show that a consideration of symmetry-allowed band
degeneracies in the wallpaper groups can be used to understand previous
topological crystalline insulators, as well as to predict new examples. In
particular, the two wallpaper groups with multiple glide lines, and
, allow for a new topological insulating phase, whose surface spectrum
consists of only a single, fourfold-degenerate, true Dirac fermion. Like the
surface state of a conventional topological insulator, the surface Dirac
fermion in this "nonsymmorphic Dirac insulator" provides a theoretical
exception to a fermion doubling theorem. Unlike the surface state of a
conventional topological insulator, it can be gapped into topologically
distinct surface regions while keeping time-reversal symmetry, allowing for
networks of topological surface quantum spin Hall domain walls. We report the
theoretical discovery of new topological crystalline phases in the AB
family of materials in SG 127, finding that SrPb hosts this new
topological surface Dirac fermion. Furthermore, (100)-strained AuY and
HgSr host related topological surface hourglass fermions. We also
report the presence of this new topological hourglass phase in
BaInSb in SG 55. For orthorhombic space groups with two glides, we
catalog all possible bulk topological phases by a consideration of the allowed
non-abelian Wilson loop connectivities, and we develop topological invariants
for these systems. Finally, we show how in a particular limit, these
crystalline phases reduce to copies of the SSH model.Comment: Final version, 6 pg main text + 29 pg supplement, 6 + 13 figure
Entanglement in Many-Body Systems
The recent interest in aspects common to quantum information and condensed
matter has prompted a prosperous activity at the border of these disciplines
that were far distant until few years ago. Numerous interesting questions have
been addressed so far. Here we review an important part of this field, the
properties of the entanglement in many-body systems. We discuss the zero and
finite temperature properties of entanglement in interacting spin, fermionic
and bosonic model systems. Both bipartite and multipartite entanglement will be
considered. At equilibrium we emphasize on how entanglement is connected to the
phase diagram of the underlying model. The behavior of entanglement can be
related, via certain witnesses, to thermodynamic quantities thus offering
interesting possibilities for an experimental test. Out of equilibrium we
discuss how to generate and manipulate entangled states by means of many-body
Hamiltonians.Comment: 61 pages, 29 figure
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