3,366 research outputs found
The moduli space of matroids
In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.Comment: 83 page
Quantum energy inequalities and local covariance II: Categorical formulation
We formulate Quantum Energy Inequalities (QEIs) in the framework of locally
covariant quantum field theory developed by Brunetti, Fredenhagen and Verch,
which is based on notions taken from category theory. This leads to a new
viewpoint on the QEIs, and also to the identification of a new structural
property of locally covariant quantum field theory, which we call Local
Physical Equivalence. Covariant formulations of the numerical range and
spectrum of locally covariant fields are given and investigated, and a new
algebra of fields is identified, in which fields are treated independently of
their realisation on particular spacetimes and manifestly covariant versions of
the functional calculus may be formulated.Comment: 27 pages, LaTeX. Further discussion added. Version to appear in
General Relativity and Gravitatio
The Absolute Relativity Theory
This paper is a first presentation of a new approach of physics that we
propose to refer as the Absolute Relativity Theory (ART) since it refutes the
idea of a pre-existing space-time. It includes an algebraic definition of
particles, interactions and Lagrangians. It proposed also a purely algebraic
explanation of the passing of time phenomenon that leads to see usual
Euler-Lagrange equations as the continuous version of the
Knizhnik-Zamolodchikov monodromy. The identification of this monodromy with the
local ones of the Lorentzian manifolds gives the Einstein equation
algebraically explained in a quantized context. A fact that could lead to the
unification of physics. By giving an algebraic classification of particles and
interactions, the ART also proposes a new branch of physics, namely the Mass
Quantification Theory, that provides a general method to calculate the
characteristics of particles and interactions. Some examples are provided. The
MQT also predicts the existence of as of today not yet observed particles that
could be part of the dark matter. By giving a new interpretation of the weak
interaction, it also suggests an interpretation of the so-called dark energy
Rational motivic path spaces and Kim's relative unipotent section conjecture
We initiate a study of path spaces in the nascent context of "motivic dga's",
under development in doctoral work by Gabriella Guzman. This enables us to
reconstruct the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of Kim's
relative unipotent section conjecture into several smaller conjectures with a
homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that
the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine's t-structure for mixed Tate
motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to
exposition throughout. 52 page
Orientation theory in arithmetic geometry
This work is devoted to study orientation theory in arithmetic geometric
within the motivic homotopy theory of Morel and Voevodsky. The main tool is a
formulation of the absolute purity property for an \emph{arithmetic cohomology
theory}, either represented by a cartesian section of the stable homotopy
category or satisfying suitable axioms. We give many examples, formulate
conjectures and prove a useful property of analytical invariance. Within this
axiomatic, we thoroughly develop the theory of characteristic and fundamental
classes, Gysin and residue morphisms. This is used to prove Riemann-Roch
formulas, in Grothendieck style for arbitrary natural transformations of
cohomologies, and a new one for residue morphisms. They are applied to rational
motivic cohomology and \'etale rational -adic cohomology, as expected by
Grothendieck in \cite[XIV, 6.1]{SGA6}.Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference
in the Tata Institute. Thanks a lot goes to the referee for his enormous work
(more than 100 comments) which was of great help. Among these corrections, he
indicated to me a sign mistake in formula (3.2.14.a) which was very hard to
detec
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