Papers:
 Peter Bubenik and Pawel Dlotko. A
persistence landscapes toolbox for topological
statistics. Available at arXiv:1501.00179
[cs.CG].
We give efficient algorithms for computing and manipulating
persistence landscapes. As an example we compute mean
landscapes in low degrees for points sampled uniformly from
various spheres, calculate the distances between them, and
show that they are statistically significant.

Violeta KovacevNikolic, Peter
Bubenik, Dragan Nikolić, and Giseon Heo. Using
cycles in high dimensional data to analyze protein
binding, submitted. Available at arXiv:1412.1394
[stat.ME].
We show that persistent homology is able to differentiate
between open and closed forms of the maltosebinding
protein, a large biomolecule consisting of 370 amino acid
residues. We also observe that the majority of active site
residues and allosteric pathway residues are located in
the vicinity of the most persistent loop in the
corresponding filtered VietorisRips complex.
 Peter Bubenik, Vin de Silva and Jonathan Scott. Metrics
for generalized persistence modules, Foundations
of Computational Mathematics, in press. Also available
at arXiv:1312.3829
[math.AT].
We consider the question of defining interleaving metrics on
generalized persistence modules over arbitrary preordered
sets. Our constructions are functorial, which implies a form
of stability for these metrics. We describe a large class of
examples, inverseimage persistence modules, which occur
whenever a topological space is mapped to a metric space.
Several standard theories of persistence and their stability
can be described in this framework. This includes the
classical case of sublevelset persistent homology. We
introduce a distinction between `soft' and `hard' stability
theorems. While our treatment is direct and elementary, the
approach can be explained abstractly in terms of monoidal
functors.
 Peter Bubenik. Statistical
topological data analysis using persistence landscapes,
Journal of Machine
Learning Research, in press. Also available at arXiv:1207.6437
[math.AT].
I deﬁne a new descriptor for persistent homology, which I
call the persistence landscape, for the purpose of
facilitating statistical inference. This descriptor may be
thought of as an embedding of the usual descriptors,
barcodes and persistence diagrams, into a space of
functions. The persistence landscape is a piecewise linear
function. The linear structure of the function space allows
simple and fast calculations. In fact the function space is
a separable Banach space and so has a nice probability
theory. For examples, I calculate mean landscapes for random
geometric complexes, random clique complexes and Gaussian
random fields.
 Peter Bubenik and Jonathan A. Scott. Categorification
of persistent homology, Discrete
and Computational Geometry, 51
(2014), no. 3, pp. 600627. Also available at arXiv:1205.3669
[math.AT].
We redevelop persistent homology (topological persistence)
from a categorical point of view. The main objects of study
are diagrams, indexed by the poset of real numbers, in some
target category. The set of such diagrams has an
interleaving distance, which we show generalizes the
previouslystudied bottleneck distance. To illustrate the
utility of this approach, we greatly generalize previous
stability results for persistence, extended persistence, and
kernel, image and cokernel persistence. We give a natural
construction of a category of interleavings of these
diagrams, and show that if the target category is abelian,
so is this category of interleavings.
 Yuliy Baryshnikov, Peter Bubenik, and Matthew Kahle. Mintype
Morse theory for conﬁguration spaces of hard spheres.
International
Mathematics Research Notices, 2014
(2014), no. 9, pp. 25772592. Also available at arXiv:1108.3061
[math.AT].
Hard spheres are important models for matter in statistical
physics. Surprisingly, the answers to a number of basic
topological questions of the configuration space of hard
spheres are unknown. We develop a Morse theory for studying
the homotopy theory of this configuration space. The
critical points and critical submanifolds in this theory
correspond to mechanically balanced configurations of
spheres.
 Peter Bubenik. Simplicial
models for concurrency. Electronic
Notes in Theoretical Computer Science, 283
(2012) pp. 312. Also available at arXiv:1011.6599
[cs.DC].
I give a simplicial model for concurrent programs, and a
simplicial model for the possible executions from one state
to another. The latter use necklaces of simplices, an idea
used by Jacob Lurie in his study of higher dimensional
category theory. It should be possible to give a more
economical cubical version of these models.
 Peter Bubenik and Leah Gold. Graph
products of spheres, associative graded algebras and
Hilbert series. Mathematische
Zeitschrift, 268
(2011), pp. 821–836. Also available at arXiv:0901.4493
[math.AT].
Given a ﬁnite, simple, vertex–weighted graph, we construct a
graded associative (noncommutative) algebra, whose
generators correspond to vertices and whose ideal of
relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of
this algebra is the inverse of the clique polynomial of the
graph. Using this result it easy to recognize if the ideal
is inert, from which strong results on the algebra follow.
Noncommutative Gr¨obner bases play an important role in our
proof.
There is an interesting application to toric topology. This
algebra arises naturally from a partial product of spheres,
which is a special case of a generalized moment–angle
complex. We apply our result to the loop–space homology of
this space.
 Peter Bubenik, Gunnar Carlsson, Peter T. Kim and ZhiMing
Luo. Statistical
topology via Morse theory, persistence, and
nonparametric estimation. Algebraic
Methods in Statistics and Probability II. Contemporary
Mathematics, 516
(2010), 7592. Also available at arXiv:0908.3688
[math.ST].
Given data of the form (x_1, y_1), ..., (x_N, y_N) where the
x_i are points on a manifold, we assume that the data is
generated by y_i = f(x_i) + epsilon_i, where f is some
unknown function (from some fixed class of functions, e.g.
Lipschitz) and epsilon_i is Gaussian noise with mean 0 and
some fixed variance. We would like to use the data to
recover the persistent homology of the lower excursion sets
of f. Using a stability theorem for persistent homology, we
can do this by estimating f with respect to the supremum
norm. We do this by triangulating the manifold and filtering
the triangulation using an estimator obtained by smoothing
the data using kernels. The persistent homology of this
filtered simplicial complex is the desired estimate of the
persistent homology of the sublevel sets of f. This
construction is asymptotically optimal, with specified rate
and constant, in the minimax sense.
 Moo K. Chung, Peter Bubenik and Peter T. Kim. Persistence
Diagrams of Cortical Surface Data.
Information Processing in Medical Imaging 2009. Lecture
Notes in Computer Science, 5636
(2009), 386397.
We apply the techniques of the paper "Statistical topology
via Morse theory, persistence and nonparametric estimation"
to brain imaging data. In this case, the manifold is the
brain cortex, and the unknown function is the cortical
thickness. We use the resulting persistence diagrams to
differentiate between the autistic and control subjects.
 Peter Bubenik. Models and van
Kampen theorems for directed homotopy theory. Homology,
Homotopy and Applications, 11
(2009), 185202. Also available at arXiv:0810.4164
[math.AT].
In order to model concurrent parallel computing, I study
topological spaces with a distinguished set of paths, called
directed paths. The topological space models the state space
of the system, and the directed paths model the execution
paths. To reduce to the essentially different executions, we
reduce to homotopy classes of directed paths. To reduce the
size of the state space we apply future retracts and past
retracts. We also prove some theorems for doing these
constructions in a piecebypiece fashion.
More technically, since the directed paths are generally not
reversible, the directed homotopy classes of directed paths
do not assemble into a groupoid, and there is no direct
analog of the fundamental group. However, they do assemble
into a category, called the fundamental category. I define
models of the fundamental category, such as the fundamental
bipartite graph, and minimal extremal models which are shown
to generalize the fundamental group. In addition, I prove
van Kampen theorems for subcategories, retracts, and models
of the fundamental category.
 Peter Bubenik. Separated
Lie
models the homotopy Lie algebra. Journal
of Pure and Applied Algebra 212
(2008), no.2, 401410. Also available at arXiv:math/0406405
[math.AT].
For a simply connected topological space X there is a
differential graded Lie algebra, called the Quillen model,
which determines the rational homotopytype of X. Its
homology is isomorphic to the rational homotopy groups of
the loop space on X, called the homotopy Lie algebra. In
this paper I show that for spaces with finite (rational) LS
category, these can be assumed to satisfy a condition I call
separated, which is useful for calculations of the homotopy
Lie algebra. This separated condition implies the free
condition introduced in "Free and semiinert cell
attachments".
The separated Lie models give a nice characterization of the
rational homotopy Lie algebra in the case where the "top"
differential creates at least two new homology classes: the
radical is contained in previous homology and the rational
homotopy Lie algebra contains a free Lie algebra on two
generators. So it satisfies the AvramovFelix conjecture.
Any rational space constructed using a sequence of cell
attachments of length N is equivalent to a space constructed
using a sequence of free cell attachments of length N+1.
This is shown by proving a similar result for differential
graded Lie algebras (dgLs). As a results, one obtains a
method for calculating the homotopy Lie algebra, and the
homology of certain dgLs.
 Peter Bubenik and Peter T. Kim.
A
statistical approach to persistent homology.
Homology, Homotopy,
and Applications, 9
(2007), No. 2, pp.337362. Also available at arXiv:math/0607634
[math.AT].
Consider a finite set of points randomly sampled from a
Riemannian manifold according to some unknown probability
distribution. We define two filtered complexes with which we
can calculate the persistent homology of a probability
distribution. Using these two filtrations we calculate the
persistent homology of several directional densities. Using
statistical estimators we can recover the persistent
homology of the underlying distribution from the sample, for
certain examples.
We also show that the classical theory of spacings can be
used to calculate the exact expectations of the persistent
homology of samples from the uniform distribution on the
circle, together with their asymptotic behavior.
Some nice topological aspects in this paper include the
following. The Morse filtrations of the von MisesFisher,
Watson, and Bingham distributions correspond to different CW
structures of the nsphere. Similarly, the matrix von Mises
distribution corresponds to a relative CW structure on
SO(3), where this last decomposition is obtained using the
Hopf fibration S^0 > S^3 > RP^3.
 Peter Bubenik and John Holbrook. Densities
for
Random Balanced Sampling, Journal
of
Multivariate Analysis 98
(2007) pp. 350369. Also available at arXiv:math/0608737
[math.ST].
A random balanced sample (RBS) is a multivariate
distribution with $n$ components $X_k$, each uniformly
distributed on $[1,1]$, such that the sum of these
components is precisely 0. The corresponding vectors $\vec
X$ lie in an $(n1)$dimensional polytope $M(n)$.
We present new methods for the construction of such RBS via
densities over $M(n)$ and these apply for arbitrary $n$.
While simple densities had been known previously for small
values of $n$ ($2,3,4$), for larger $n$ the known
distributions with large support were fractal distributions
(with fractal dimension asymptotic to $n$ as $n\to\infty$).
Applications of RBS distributions include sampling with
antithetic coupling to reduce variance, isolation of
nonlinearities, and goodness of fit testing. We also show
that the previously known densities (for $n\leq4$) are in
fact the only solutions in a natural and very large class of
potential RBS densities. It follows that the new methods
lead in another direction entirely.
 Peter Bubenik and Krzysztof Worytkiewicz. A
model category structure for local pospaces.
Homology,
Homotopy and Applications, 8
(2006), pp. 263292. Also available at arXiv:math/0506352
[math.AT].
Local pospaces are topological spaces together with a local
partialorder. Maps between these spaces are continuous maps
which respect the order. Some computer scientists use local
pospaces to model concurrent systems and would like a good
theory of equivalences for these spaces.
In this paper we construct a homotopy theory for these
spaces. Spaces which are trivial or nearly trivial in the
classical undirected case can be highly nontrivial when
homotopies are forced to respect directions. Our aim is to
construct a (Quillen) model category for local pospaces as
a framework for a directed homotopy theory. This is
technically difficult because local pospaces are not closed
under colimits, which is a necessary condition in a model
category. Our method is inspired by the construction of
Voevodsky's A1homotopy theory and Dugger's universal
homotopy theories. We pass to the simplicial presheaf
category and apply Jardine's model structure. We analyze the
weak equivalences in this model category. Finally one uses
the right context (see "Context for models of concurrency")
and localizes with respect to a class of equivalences  for
example, the dihomotopy equivalences. Further work will
analyze the weak equivalences in this model category.
 Peter Bubenik. Context
for models of concurrency. Electronic
Notes in Theoretical Computer Science, 230
(2009), 321. Preliminary version in Proceedings of the
Workshop on Geometry and Topology in Concurrency and
Distributed Computing, BRICS
Notes NS042, pp.3349). Also available at arXiv:math/0608733
[math.AT].
In this paper I examine many simple examples of
partiallyordered spaces which model concurrent systems. It
would be useful to be able to replace a given model with a
simpler one. What one wants is a good notion of a directed
homology equivalence. However, I show that the usual
definition of directed homotopy equivalence is too coarse.
To solve this problem I introduce the notion of context.
More precisely, I show that dihomotopy equivalences are best
defined in the category of pospaces under a pospace A
where the choice of A depends on the 'pastings' that one
would like to consider.
 Peter Bubenik. Free
and
semiinert cell attachments. Transactions
of the American Mathematical Society 357
(2005) pp. 45334553. Also available at arXiv:math/0312387
[math.AT].
In this paper I give some new results on the cell attachment
problem, which was perhaps first studied by J.H.C. Whitehead
around 1940: If one attaches one or more cells to a
topological space, what is the effect on the homology of the
loop space, and on the homotopytype?I introduce the free
and semiinert conditions under which I determine the loop
space homology as a module and as an algebra respectively.
Under a further condition I determine the homotopytype of
the space.
 Peter Bubenik.
Cell attachments and the homology of loop spaces and
differential graded algebras.
Thesis, University of Toronto, 2003. Available at arXiv:math/0601421
[math.AT].
 A summary of my thesis: in
pdf, ps and
dvi formats
