217 research outputs found
Tightness and efficiency of irreducible automorphisms of handlebodies
Among (isotopy classes of) automorphisms of handlebodies those called
irreducible (or generic) are the most interesting, analogues of pseudo-Anosov
automorphisms of surfaces. We consider the problem of isotoping an irreducible
automorphism so that it is most efficient (has minimal growth rate) in its
isotopy class. We describe a property, called tightness, of certain invariant
laminations, which we conjecture characterizes this efficiency. We obtain
partial results towards proving the conjecture. For example, we prove it for
genus two handlebodies. We also show that tightness always implies efficiency.
In addition, partly in order to provide counterexamples in our study of
properties of invariant laminations, we develop a method for generating a class
of irreducible automorphisms of handlebodies.Comment: This is the version published by Geometry & Topology on 4 March 200
Enhanced Koszulity in Galois cohomology
Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro--groups, Galois cohomology is a Koszul algebra. In particular, we show that for all known cases where the maximal pro--quotient of the absolute Galois group is finitely generated, Galois cohomology is universally Koszul. Assuming the Elementary Type conjecture, this gives us infinitely many refinements of the Bloch-Kato Conjecture. We moreover obtain several unconditional results. Lastly, we show that all forms of enhanced Koszulity are preserved under certain natural operations, which generalizes results that were only known to hold in the commutative case
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Applied Harmonic Analysis and Data Science (hybrid meeting)
Data science has become a field of major importance for science and technology
nowadays and poses a large variety of
challenging mathematical questions.
The area
of applied harmonic analysis has a significant impact on such problems by providing methodologies
both for theoretical questions and for a wide range of applications
in signal and image processing and machine learning.
Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018,
this workshop focused
on several exciting novel directions such as mathematical theory of
deep learning, but also reported progress on long-standing open problems in the field
Gaussian optimizers and other topics in quantum information
The main topic of this thesis is the proof of two fundamental entropic
inequalities for quantum Gaussian channels. Quantum Gaussian channels model the
propagation of electromagnetic waves through optical fibers and free space in
the quantum regime, and provide the mathematical model to determine the maximum
rates achievable by quantum communication devices for communication and quantum
key distribution. The first inequality proven in this thesis is the quantum
Entropy Power Inequality, which provides a lower bound to the output entropy of
the beam-splitter or of the squeezing in terms of the entropies of the two
inputs. The second inequality states that Gaussian states minimize the output
entropy of the one-mode quantum Gaussian attenuator among all the input states
with a given entropy, and was a longstanding conjecture in quantum
communication theory. The generalization of this inequality to the multimode
attenuator would determine its triple trade-off region and its capacity region
for broadcast communication to two receivers. The thesis contains further
results in Gaussian quantum information, quantum statistical mechanics and
relativistic quantum information. The most important of these results concerns
the Eigenstate Thermalization Hypothesis (ETH). The ETH is an assumption in
quantum statistical mechanics stating that the eigenstates of the Hamiltonian
of a system+bath compound look as thermal states if we can access only the
system. We prove that the ETH must hold if the system thermalizes for any
initial product state of the system+bath compound with a well-defined
temperature.Comment: PhD thesis defended on 12 Sep 2016, supervisor: Prof. Vittorio
Giovannett
Ramifications of Lineland
A non-technical overview on gravity in two dimensions is provided.
Applications discussed in this work comprise 2D type 0A/0B string theory, Black
Hole evaporation/thermodynamics, toy models for quantum gravity, for numerical
General Relativity in the context of critical collapse and for solid state
analogues of Black Holes. Mathematical relations to integrable models,
non-linear gauge theories, Poisson-sigma models, KdV surfaces and
non-commutative geometry are presented.Comment: 45 pages, 3 eps figures, proceedings contribution to 5th Workshop on
Quantization, Dualities & Integrable Systems in Denizli, Turkey; v2: added
refs. and a comment on phase transitions: v3: minor cosmetic chang
The Dialogical Entailment Task
In this paper, a critical discussion is made of the role of entailments in the so-called New Paradigm of psychology of reasoning based on Bayesian models of rationality (Elqayam & Over, 2013). It is argued that assessments of probabilistic coherence cannot stand on their own, but that they need to be integrated with empirical studies of intuitive entailment judgments. This need is motivated not just by the requirements of probability theory itself, but also by a need to enhance the interdisciplinary integration of the psychology of reasoning with formal semantics in linguistics. The constructive goal of the paper is to introduce a new experimental paradigm, called the Dialogical Entailment task, to supplement current trends in the psychology of reasoning towards investigating knowledge-rich, social reasoning under uncertainty (Oaksford and Chater, 2019). As a case study, this experimental paradigm is applied to reasoning with conditionals and negation operators (e.g. CEM, wide and narrow negation). As part of the investigation, participants’ entailment judgments are evaluated against their probability evaluations to assess participants’ cross-task consistency over two experimental sessions
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