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We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $SL_n(\mathbb{Z}, p^m)$ in degrees $* < p-1$, for all large enough $n$, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the…

We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and…

In this article, we consider extended tame persistence commutative differential graded algebras (CDGAs) associated with relative Sullivan algebras. In particular, if the relative Sullivan algebra is a model for a map between spaces, then…

We introduce a commutative product of degree $-n$ on the homology $H_\ast(X)$ of an $n$-dimensional special cubical set $X$ and lift it on the free loop homology $H_\ast(\Lambda M)$ for $M=|X|$ to be the geometric realization. These…

We establish and explore the correspondence between positive functionals and cocycles in higher unitary cohomology. We generalize the classical cocycle version of the Gelfand-Naimark-Segal construction to higher degrees and apply it to…

We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete…

We establish that the dioperad $Y^{(n)}$, encoding bialgebras with a product of degree zero, a coproduct of degree $(1-n)$ and a rank three cyclic tensor, which satisfy a deformed version of the balanced infinitesimal bialgebra condition,…

Parameterized stable homotopy theory organizes local systems of spectra over homotopy types, governed by a "yoga" of six functors. To provide semantics for the recently developed Linear Homotopy Type Theory (LHoTT), good model categories of…

We show that the Morse complex of a compact Lie monoid can be given the structure of an $f$-bialgebra, a chain-level version of bialgebras introduced in [CHM24]; and that this assignment defines an $\infty$-functor. As a consequence, we…

We give a direct proof that the proalgebraic graded Grothendieck-Teichm\"uller group $\mathsf{GRT}_{\mathbb{K}}$ is isomorphic to the group of automorphisms of the prounipotent cyclic operad of parenthesized ribbon chord diagrams based on…

In this paper, we present a new qualitative extension of the Hopf theorem (and a generalization of Borsuk-Ulam theorem), concerning continuous maps $f$ from a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We remove the…

In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set…

In this work, we obtained separation results via codimension-1 maps to generalized manifolds. More specifically, we proved results that allow us to estimate the number of connected components of the complement of the image of such maps.

The well known Joker $\mathcal{A}(1)$-module of Adams and Priddy is known to be realisable as the cohomology of a $1$-connected space. By attaching an extra cell we obtain an $8$-dimensional Poincar\'e duality space whose mod~$2$ cohomology…

Recently, Das defined a new type of algebras, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras appearing naturally in differential geometry. We show that the nonsymmetric operad of Yamaguti algebras…

Using the language of moperads -- monoids in the category of right modules over an operad -- we reinterpret the Alekseev--Enriquez--Torossian construction of Kashiwara--Vergne (KV) solutions from associators. We show that any equivalence…

The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane,…

Given a map $B\to B\mathrm{Top}(n)$ of spaces, one can define a version $\mathbb{E}_{B}$ of the little cubes operad, whose construction is due to Lurie. We show that $\mathbb{E}_{B}$ enjoys the universal property that, for every…

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. In this paper, first,…

The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the…