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For $n\in\{2^t-3,2^t-2,2^t-1\}$ ($t\ge3$) we study the cohomology algebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-dimensional subspaces of $\mathbb R^n$. A complete description of…

Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants…

We prove an analogue of the Gabriel--Quillen embedding theorem for exact $\infty$-categories, giving rise to a presentable version of Klemenc's stable envelope of an exact $\infty$-category. Moreover, we construct a symmetric monoidal…

Based on the work of Dundas, Lindenstrauss and Richter we compute the topological Andr\'e-Quillen homology with reduced coefficients for Eilenberg-MacLane spectra such as $H\mathbb{Z}$ and $H\mathbb{Z}/p^n$. The case of $H\mathbb{F}_p$ was…

Dilute Temperley--Lieb algebras are variants of Temperley--Lieb algebras arising in statistical mechanics in the study of solvable lattice models. In this paper we prove that the (co)homology of dilute Temperley--Lieb algebras vanishes in…

Given an unknown $\mathbb{R}^n$-valued function $f$ on a metric space $X$, can we approximate the persistent homology of $f$ from a finite sampling of $X$ with known pairwise distances and function values? This question has been answered in…

This paper presents, with explanatory details, the handle decompositions, fundamental groups and homology groups of 3-manifolds, including some knot complements. Hence, along this paper, when the word manifold appears it is implicit that…

We introduce generalizations of global equivariant spectra which encode globally equivariant cohomology theories equipped with additional transfers, such as the deflation maps present in equivariant topological $K$-theory. We call these…

We prove a splitting result in global equivariant homotopy theory that is a simultaneous refinement of the Segal--Becker splitting and its `Real' and equivariant generalizations, and of the explicit Brauer induction of Boltje and Symonds.…

We calculate the heights of Stiefel--Whitney classes of the canonical vector bundle over the oriented Grassmannians $\widetilde G_{n,4}\cong SO(n)/(SO(4)\times SO(n-4))$ in the cases $n\in\{2^t-2,2^t-1,2^t,2^t+1\}$, $t\ge4$. Using some…

We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a M\"obius band, the two-point finite…

The motivic hit problem asks for a minimal set of generators of $H^{*,*}(BV_n;\mathbb{F}_2)$ as a module over the motivic Steenrod algebra. For the distinguished degrees $d=k+2d_1$ with $d_1=(n-1)(2^k-1)$, Kameko constructed a top layer…

Given a finite simple connected graph $\Gamma$, the graphical configuration space $\mathrm{Conf}_{\Gamma}(X)$ is the space of collections of points in $X$ indexed by the vertices of $\Gamma$, where points corresponding to adjacent vertices…

Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989),…

We define a properad $Y^{(n)}_\infty$ that encodes $n$-pre-Calabi--Yau algebras with vanishing copairing. These algebras include chains on the based loop space of any space $X$ endowed with a fundamental class $[X]$ such that $(X,[X])$…

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic…

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres using a…

Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some…

We construct worst-case examples for the standard reduction algorithm for computing persistent homology. Our constructions are similar to the worst-case examples introduced by Morozov, but we replace the single-triangle arrangement with a…

Classifying the stratospheric polar vortex provides predictability for surface weather on extended-range timescales. However, providing a scientifically sound classification is challenging: all the definitions proposed in over 60 years of…