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This paper aims to use Cartan's original method in proving Theorem A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the…

In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…

We define the notion of an $\mathcal{RO}(G)$-graded Tambara functor and prove that any $G$-spectrum with norm multiplication gives rise to such an $\mathcal{RO}(G)$-graded Tambara functor.

We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and…

We develop a new method in the computation of equivariant homotopy, which is based on the splitting of cofiber sequences associated to universal spaces in the category of equivariant spectra. In particular, we use this method to compute the…

Interpreting the syzygy theorem for tame modules over posets in the setting of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning the existence of stratifications of…

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo \wedge bo$ and $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$. These splittings helped make it feasible to do computations using the $bo$- and $BP…

We obtain a combinatorial expression for the coefficients of the boundary map of real isotropic and odd orthogonal Grassmannians providing a natural generalization of the formulas already obtained for Lagrangian and maximal isotropic…

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary…

We show that normalized cacti form an $\infty$-operad in the form of a dendroidal space satisfying a weak Segal condition. To do this, we introduce a new topological operad of bracketed trees and an enrichment of the dendroidal category…

The classical Dold-Kan correspondence is known to admit a categorification in the form of an equivalence between the $\infty$-categories of $2$-simplicial stable $\infty$-categories and connective chain complexes of stable…

We construct Lie algebras of derivations (and identify their geometrical realization) whose Maurer-Cartan sets provide moduli spaces describing the classes of homotopy types of rational spaces sharing either the same homotopy Lie algebra,…

We construct a dga to computing the cohomology of ordered configuration spaces on an algebraic variety with vanishing Euler characteristic. It follows that the $k$-th Betti number of $Conf(C,n)$ ($C$ is an elliptic curve) grows as a…

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that…

Given an arrangement of subtori of arbitrary codimension in a torus, we compute the cohomology groups of the complement. Then, using the Leray spectral sequence, we describe the multiplicative structure on the graded cohomology. We also…

We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez-Dolan homotopy cardinality, by constructing a unique additive common generalization after restriction to an odd prime p. This is…

We prove that the comparison map from $G$-fixed points to $G$-homotopy fixed points, for the $G$-fold smash power of a bounded below spectrum $B$, becomes an equivalence after $p$-completion if $G$ is a finite $p$-group and $H_*(B; F_p)$ is…

Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of…

A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if A is a finite abelian p-group of rank r, then any finite A-space X which is acyclic in the nth Morava K-theory with n at least r…

We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus $g$: stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $Sp_{2g}(\mathbb{Z})$-representations…