By studying spaces of flow graphs in a closed oriented manifold, we equip the Morse complex with the operations of an open topological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello, K. Fukaya and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the chain complex of the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncompactified moduli spaces of higher genus pseudoholomorphic curves.

In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem', see Alon et al. (Proc. Amer. Math. Soc., 2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and Lasoń (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of $\mathbb{R}^d$ such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lasoń, we show that for every collection $\mu_1,\ldots,\mu_{2d-1}$ of $2d-1$ continuous, signed locally finite measures on $\mathbb{R}^d$, there exist two nontrivial axis-aligned $d$-dimensional cuboids (rectangular parallelepipeds) $C_1$ and $C_2$ such that $\mu_i(C_1)=\mu_i(C_2)$ for each $i\in\{1,\ldots,2d-1\}$. We also show by examples that the bound $2d-1$ cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.

We show that for any $k> 1$, stratified sets of finite complexity are insufficient to realize all homology classes of codimension $k$ in all smooth manifolds. We also prove a similar result concerning smooth generic maps whose double-point sets are co-oriented.

We compute a primary cohomological obstruction to the existence of an equipartition for $j$ mass distributions in $\mathbb{R}^d$ by two hyperplanes in the case $2d-3j = 1$. The central new result is that such an equipartition always exists if $d=6\cdot 2^k +2$ and $j=4\cdot 2^k+1$ which for $k=0$ reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. The theorem follows from a Borsuk-Ulam type result claiming the non-existence of a $\mathbb{D}_8$-equivariant map $f \colon S^{d}\times S^d\rightarrow S(W^{\oplus j})$ for an associated real $\mathbb{D}_8$-module $W$. This is an example of a genuine combinatorial geometric result which involves $\mathbb{Z}/4$-torsion in an essential way and cannot be obtained by the application of either Stiefel-Whitney classes or cohomological index theories with $\mathbb{Z}/2$ or $\mathbb{Z}$ coefficients. The method opens a possibility of developing an ``effective primary obstruction theory'' based on $G$-manifold complexes, with applications in geometric combinatorics, discrete and computational geometry, and computational algebraic topology.

We describe a new general method for the fibrewise extension of a given endofunctor on the category of pointed topological spaces to the category of fibrewise pointed spaces. We derive some properties of the construction and show how it can be profitably used to build the Whitehead-Ganea framework for the fibrewise Lusternik-Schnirelmann category and the topological complexity.

Let $K$ be a simplicial complex and suppose that $K$ collapses onto $L$. Define $n$ to be $1$ less than the minimum number of collapsible sets it takes to cover $L$. Then the discrete geometric Lusternik-Schnirelmann category of $K$ is the smallest $n$ taken over all such $L$. In this paper, we give an algorithm which yields an upper bound for the discrete geometric category. We show our algorithm is correct and give several bounds for the discrete geometric category of well-known simplicial complexes. We show that the discrete geometric category of the dunce cap is $2$, implying that the dunce cap is ``further" from being collapsible than Bing's house whose discrete geometric category is $1$.

Hadwiger integrals employ the intrinsic volumes as measures for integration of real-valued functions. We provide a formula for the expected values of Hadwiger integrals of Gaussian-related random fields. The expected Hadwiger integrals of random fields are both theoretically interesting and potentially useful in applications such as sensor networks, image processing, and cell dynamics. Furthermore, combining the expected integrals with a functional version of Hadwiger's theorem, we obtain expected values of more general valuations on Gaussian-related random fields.

This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one dimensional case. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the covering ought to be good, which holds if the diameter of the covering elements does not exceed a certain size), but no a priori requirements on their distribution. An upper bound for the coverage probability is also obtained as a consequence of the concentration inequality. The techniques rely on a formulation of the coverage criteria in terms of the Euler characteristic of the nerve complex associated to the random covering.

We give a lower bound for the Lusternik-Schnirelmann category of compact exceptional Lie groups by computing the module category weight through analyzing several Eilenberg-Moore type spectral sequences.

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced in [I. Basabe, J. González, Y.B. Rudyak and D. Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres, arXiv: 1009.1851v5], [D. Tamaki, Cellular Stratified Spaces I. Face Categories and Classifying Spaces, arXiv:1106.3772] by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.

We study pureness and fullness of invariant complex structures on nilmanifolds. We prove that in dimension six, apart from the complex torus, there exist only two non-isomorphic complex structures satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the product of two almost complex manifolds which are pure and full is not necessarily full.

We introduce the space of equivariant local maps and present the full proof of the splitting theorem for the set of otopy classes of such maps in the case of a representation of a compact Lie group.

We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches.

For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at http://www.pawelpilarczyk.com/combtop/.

The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $\pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.