289,040 research outputs found
Topological methods for complex-analytic Brauer groups
Using methods from algebraic topology and group cohomology, I pursue
Grothendieck's question on equality of geometric and cohomological Brauer
groups in the context of complex-analytic spaces. The main result is that
equality holds under suitable assumptions on the fundamental group and the
Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf
manifolds, and complex-analytic surfaces.Comment: 20 pages, minor corrections, to appear in Topolog
Coupling methods for random topological Markov chains
We apply coupling techniques in order to prove that the transfer operators
associated with random topological Markov chains and non-stationary shift
spaces with the big images and preimages-property have a spectral gap.Comment: 17 page
Statistical Methods in Topological Data Analysis for Complex, High-Dimensional Data
The utilization of statistical methods an their applications within the new
field of study known as Topological Data Analysis has has tremendous potential
for broadening our exploration and understanding of complex, high-dimensional
data spaces. This paper provides an introductory overview of the mathematical
underpinnings of Topological Data Analysis, the workflow to convert samples of
data to topological summary statistics, and some of the statistical methods
developed for performing inference on these topological summary statistics. The
intention of this non-technical overview is to motivate statisticians who are
interested in learning more about the subject.Comment: 15 pages, 7 Figures, 27th Annual Conference on Applied Statistics in
Agricultur
Data-driven network alignment
Biological network alignment (NA) aims to find a node mapping between
species' molecular networks that uncovers similar network regions, thus
allowing for transfer of functional knowledge between the aligned nodes.
However, current NA methods do not end up aligning functionally related nodes.
A likely reason is that they assume it is topologically similar nodes that are
functionally related. However, we show that this assumption does not hold well.
So, a paradigm shift is needed with how the NA problem is approached. We
redefine NA as a data-driven framework, TARA (daTA-dRiven network Alignment),
which attempts to learn the relationship between topological relatedness and
functional relatedness without assuming that topological relatedness
corresponds to topological similarity, like traditional NA methods do. TARA
trains a classifier to predict whether two nodes from different networks are
functionally related based on their network topological patterns. We find that
TARA is able to make accurate predictions. TARA then takes each pair of nodes
that are predicted as related to be part of an alignment. Like traditional NA
methods, TARA uses this alignment for the across-species transfer of functional
knowledge. Clearly, TARA as currently implemented uses topological but not
protein sequence information for this task. We find that TARA outperforms
existing state-of-the-art NA methods that also use topological information,
WAVE and SANA, and even outperforms or complements a state-of-the-art NA method
that uses both topological and sequence information, PrimAlign. Hence, adding
sequence information to TARA, which is our future work, is likely to further
improve its performance
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