18,078 research outputs found
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Crystals, instantons and quantum toric geometry
We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.Comment: 33 pages, 5 figures; Contribution to the proceedings of "Geometry and
Physics in Cracow", Jagiellonian University, Cracow, Poland, September 21-25,
2010. To be published in Acta Physica Polonica Proceedings Supplemen
The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics from Neural Data
We develop of a line of work initiated by Curto and Itskov towards
understanding the amount of information contained in the spike trains of
hippocampal place cells via topology considerations. Previously, it was
established that simply knowing which groups of place cells fire together in an
animal's hippocampus is sufficient to extract the global topology of the
animal's physical environment. We model a system where collections of place
cells group and ungroup according to short-term plasticity rules. In
particular, we obtain the surprising result that in experiments with spurious
firing, the accuracy of the extracted topological information decreases with
the persistence (beyond a certain regime) of the cell groups. This suggests
that synaptic transience, or forgetting, is a mechanism by which the brain
counteracts the effects of spurious place cell activity
Mixed Reality Architecture: a dynamic architectural topology
Architecture can be shown to structure patterns of co-presence and in turn to be
structured itself by the rules and norms of the society present within it. This two-way
relationship exists in a surprisingly stable framework, as fundamental changes to
buildings are slow and costly. At the same time, change within organisations is
increasingly rapid and buildings are used to accommodate some of that change. This
adaptation can be supported by the use of telecommunication technologies, overcoming
the need for co-presence during social interaction. However, often this results in a loss
of accountability or ‘civic legibility’, as the link between physical location and social
activity is broken. In response to these considerations, Mixed Reality Architecture
(MRA) was developed. MRA links multiple physical spaces across a shared 3D virtual
world. We report on the design of MRA, including the key concept of the Mixed Reality
Architectural Cell, a novel architectural interface between architectural spaces that are
remote to each other. An in-depth study lasting one year and involving six office-based
MRACells, used video recordings, the analysis of event logs, diaries and an interview
survey. This produced a series of ethnographic vignettes describing social interaction
within MRA in detail. In this paper we concentrate on the topological properties of MRA.
It can be shown that the dynamic topology of MRA and social interaction taking place
within it are fundamentally intertwined. We discuss how topological adjacencies across
virtual space change the integration of the architectural spaces that MRA is installed in.
We further reflect on how the placement of MRA technology in different parts of an
office space (deep or shallow) impacts on the nature of that particular space. Both the
above can be shown to influence movement through the building and social interaction
taking place within it. These findings are directly relevant to new buildings that need to
be designed to accommodate organisational change in future but also to existing
building stock that might be very hard to adapt. We are currently expanding the system
to new sites and are planning changes to the infrastructure of MRA as well as its
interactional interface
Topological Schemas of Memory Spaces
Hippocampal cognitive map---a neuronal representation of the spatial
environment---is broadly discussed in the computational neuroscience literature
for decades. More recent studies point out that hippocampus plays a major role
in producing yet another cognitive framework that incorporates not only
spatial, but also nonspatial memories---the memory space. However, unlike
cognitive maps, memory spaces have been barely studied from a theoretical
perspective. Here we propose an approach for modeling hippocampal memory spaces
as an epiphenomenon of neuronal spiking activity. First, we suggest that the
memory space may be viewed as a finite topological space---a hypothesis that
allows treating both spatial and nonspatial aspects of hippocampal function on
equal footing. We then model the topological properties of the memory space to
demonstrate that this concept naturally incorporates the notion of a cognitive
map. Lastly, we suggest a formal description of the memory consolidation
process and point out a connection between the proposed model of the memory
spaces to the so-called Morris' schemas, which emerge as the most compact
representation of the memory structure.Comment: 24 pages, 8 Figures, 1 Suppl. Figur
Mirror Maps in Chern-Simons Gauge Theory
We describe mirror symmetry in N=2 superconformal field theories in terms of
a dynamical topology changing process of the principal fiber bundle associated
with a topological membrane. We show that the topological symmetries of
Calabi-Yau sigma-models can be obtained from discrete geometric transformations
of compact Chern-Simons gauge theory coupled to charged matter fields. We
demonstrate that the appearence of magnetic monopole-instantons, which
interpolate between topologically inequivalent vacua of the gauge theory,
implies that the discrete symmetry group of the worldsheet theory is realized
kinematically in three dimensions as the magnetic flux symmetry group. From
this we construct the mirror map and show that it corresponds to the
interchange of topologically non-trivial matter field and gauge degrees of
freedom. We also apply the mirror transformation to the mean field theory of
the quantum Hall effect. We show that it maps the Jain hierarchy into a new
hierarchy of states in which the lowest composite fermions have the same
filling fractions.Comment: 40 pages LaTeX, 4 postscript files, uses psfig.sty; minor textual
changes, typos corrected, references adde
Characterizing the Shape of Activation Space in Deep Neural Networks
The representations learned by deep neural networks are difficult to
interpret in part due to their large parameter space and the complexities
introduced by their multi-layer structure. We introduce a method for computing
persistent homology over the graphical activation structure of neural networks,
which provides access to the task-relevant substructures activated throughout
the network for a given input. This topological perspective provides unique
insights into the distributed representations encoded by neural networks in
terms of the shape of their activation structures. We demonstrate the value of
this approach by showing an alternative explanation for the existence of
adversarial examples. By studying the topology of network activations across
multiple architectures and datasets, we find that adversarial perturbations do
not add activations that target the semantic structure of the adversarial class
as previously hypothesized. Rather, adversarial examples are explainable as
alterations to the dominant activation structures induced by the original
image, suggesting the class representations learned by deep networks are
problematically sparse on the input space
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