3,364 research outputs found
Computing simplicial representatives of homotopy group elements
A central problem of algebraic topology is to understand the homotopy groups
of a topological space . For the computational version of the
problem, it is well known that there is no algorithm to decide whether the
fundamental group of a given finite simplicial complex is
trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex that is simply connected (i.e., with
trivial), compute the higher homotopy group for any given .
%The first such algorithm was given by Brown, and more recently, \v{C}adek et
al.
However, these algorithms come with a caveat: They compute the isomorphism
type of , as an \emph{abstract} finitely generated abelian
group given by generators and relations, but they work with very implicit
representations of the elements of . Converting elements of this
abstract group into explicit geometric maps from the -dimensional sphere
to has been one of the main unsolved problems in the emerging field
of computational homotopy theory.
Here we present an algorithm that, given a~simply connected space ,
computes and represents its elements as simplicial maps from a
suitable triangulation of the -sphere to . For fixed , the
algorithm runs in time exponential in , the number of simplices of
. Moreover, we prove that this is optimal: For every fixed , we
construct a family of simply connected spaces such that for any simplicial
map representing a generator of , the size of the triangulation of
on which the map is defined, is exponential in
Hochschild homology invariants of K\"ulshammer type of derived categories
For a perfect field of characteristic and for a finite dimensional
symmetric -algebra K\"ulshammer studied a sequence of ideals of the
centre of using the -power map on degree 0 Hochschild homology. In joint
work with Bessenrodt and Holm we removed the condition to be symmetric by
passing through the trivial extension algebra. If is symmetric then the
dual to the K\"ulshammer ideal structure was generalised to higher Hochschild
homology in earlier work. In the present paper we follow this program and
propose an analogue of the dual to the K\"ulshammer ideal structure on the
degree Hochschild homology theory also to not necessarily symmetric
algebras
Interpretable statistics for complex modelling: quantile and topological learning
As the complexity of our data increased exponentially in the last decades, so has our
need for interpretable features. This thesis revolves around two paradigms to approach
this quest for insights.
In the first part we focus on parametric models, where the problem of interpretability
can be seen as a “parametrization selection”. We introduce a quantile-centric
parametrization and we show the advantages of our proposal in the context of regression,
where it allows to bridge the gap between classical generalized linear (mixed)
models and increasingly popular quantile methods.
The second part of the thesis, concerned with topological learning, tackles the
problem from a non-parametric perspective. As topology can be thought of as a way
of characterizing data in terms of their connectivity structure, it allows to represent
complex and possibly high dimensional through few features, such as the number of
connected components, loops and voids. We illustrate how the emerging branch of
statistics devoted to recovering topological structures in the data, Topological Data
Analysis, can be exploited both for exploratory and inferential purposes with a special
emphasis on kernels that preserve the topological information in the data.
Finally, we show with an application how these two approaches can borrow strength
from one another in the identification and description of brain activity through fMRI
data from the ABIDE project
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Scl, sails and surgery
We establish a close connection between stable commutator length in free
groups and the geometry of sails (roughly, the boundary of the convex hull of
the set of integer lattice points) in integral polyhedral cones. This
connection allows us to show that the scl norm is piecewise rational linear in
free products of Abelian groups, and that it can be computed via integer
programming. Furthermore, we show that the scl spectrum of nonabelian free
groups contains elements congruent to every rational number modulo
, and contains well-ordered sequences of values with ordinal type
. Finally, we study families of elements in free groups
obtained by surgery on a fixed element in a free product of Abelian groups
of higher rank, and show that \scl(w(p)) \to \scl(w) as .Comment: 23 pages, 4 figures; version 3 corrects minor typo
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