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An even Artin group is a group which has a presentation with relations of the form ( s t ) n = ( t s ) n with n ≥ 1 . With a group G we associate a Lie Z -algebra TG r ( G ) . This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TG r ( G ) . By means of this presentation we obtain information about the diagram of G. We then prove an isomorphism criterion for Coxeter matrices that ensures that the diagram of G is uniquely determined by this information. Let d ≥ 2 . We show that, if two even Artin groups G and H having presentations with relations of the form ( s t ) d k = ( t s ) d k with k ≥ 0 are such that TG r ( G ) ≃ TG r ( H ) , then G and H have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups G and H such that TG r ( G ) ≃ TG r ( H ) .

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a very useful combinatorial tool in graph isomorphism testing. We address the applicability of $k$-WL to recognition of graph properties. Let $G$ be an input graph with $n$ vertices. We show that, if $n$ is prime, then vertex-transitivity of $G$ can be seen in a straightforward way from the output of 2-WL on $G$ and on the vertex-individualized copies of $G$. However, if $n$ is divisible by 16, then $k$-WL is unable to distinguish between vertex-transitive and non-vertex-transitive graphs with $n$ vertices as long as $k=o(\sqrt n)$. Similar results are obtained for recognition of arc-transitivity., 30 pages, 2 figures. This paper supersedes Section 5 in the first version of arXiv:2002.04590. The 3rd version is a thorough revision of the paper

Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof uses homotopy Gerstenhaber algebras in an essential way. In particular, we show that the normalized singular cochains on the classifying space of a torus are formal as a homotopy Gerstenhaber algebra., 52 pages; new Sections 2.2 (Notation) and 13 (Examples), appendix expanded, minor changes

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) The number of isomorphism classes of semisymmetric idempotent Latin squares of order $n$ equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order $n+1$, and (2) Suppose $A$ and $B$ are totally symmetric Latin squares of order $n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[x^m,z^n][y,z]\rangle$. By investigating cohomology jump loci of irreducible ${\rm GL}(2,\mathbb{C})$-character varieties, we show: if $m,m'\ne 0$, $n,n'>0$ and $G_{m',n'}\cong G_{m,n}$, then $m=m',n=n'$; if $m,m',n,n'>0$ and $H_{m',n'}\cong H_{m,n}$, then $m'=m, n'=n$., 18 pages

Let A be a commutative augmented ring and I be its augmentation ideal. This paper shows that the sequence { I n / I n + 1 } becomes stationary up to isomorphism. The result yields stability in the associated graded ring of A along I.

We prove that the spaces $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{D}_{\operatorname{poly}}^{\bullet}\big)$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to an $L_\infty$ isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair $(L,A)$. Consequently, both $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{T}_{\operatorname{poly}}^{\bullet})$ and $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{D}_{\operatorname{poly}}^{\bullet})$ admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold)., Comment: [v2] 50 pages, paper was expanded; [v1] Paper arXiv:1605.09656v1 was expended and split into two papers. The first part is arXiv:1605.09656v2. The second part is the present paper. A new result addressing uniqueness of the constructed structures has been added

A skew brace is a triplet $(A,\cdot,\circ)$, where $(A,\cdot)$ and $(A,\circ)$ are groups such that the brace relation $x\circ (y\cdot z) = (x\circ y)\cdot x^{-1}\cdot (x\circ z)$ holds for all $x,y,z\in A$. In this paper, we study the number of finite skew braces $(A,\cdot,\circ)$, up to isomorphism, such that $(A,\cdot)$ and $(A,\circ)$ are both isomorphic to $T^n$ with $T$ non-abelian simple and $n\in\mathbb{N}$. We prove that it is equal to the number of unlabeled directed graphs on $n+1$ vertices, with one distingusihed vertex, and whose underlying undirected graph is a tree. In particular, it depends only on $n$ and is independent of $T$., slightly revised based on referee's comments

We enumerate the 15768 15768 perfect groups of order up to 2 ⋅ 10 6 2\cdot 10^6 , up to isomorphism, thus also completing the missing cases reported by Holt and Plesken [Perfect groups, The Clarendon Press, Oxford University Press, New York, 1989]. The work supplements the by now well-understood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups.

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines p-nilpotence of the group if a certain condition is met.

A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed., Comment: Replaces 'Finite axiomatizability for profinite groups I: group theory', adding significant additional material. New version has better proofs of bi-intepretability

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo $p\gg 0$ from an algebraic Lie algebra $\mathfrak{g},$ such that $\mathfrak{g}$ has no nontrivial semi-invariants in $Sym(\mathfrak{g})$ and $Sym(\mathfrak{g})^{\mathfrak{g}}$ is a polynomial algebra. As an application, we solve the derived isomorphism problem of enveloping algebras for the above class of Lie algebras., Comment: Significantly revised, final version. To appear in Journal of Algebra

In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras. Funding for open access charge: Universidad de Málaga / CBUA. Spanish Ministerio de Ciencia e Innovación through the project PID2019-104236GB-I00. Junta de Andalucía through the projects FQM-336 and UMA18-FEDERJA-119, all of them with FEDER funds. Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant number 307910/2020-2 and Capes-PrInt grant number 88881.310538/2018-01 - Brazil.

Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra isomorphism between the cohomology $H^*(\mathbb{G}_{n,\ell},\mathbb{Z})$ of $\mathbb{G}_{n,\ell}$ and a natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$ cyclotomic nilHecke algebra $H_{\ell,n}^{(0)}=\langle\psi_1,\cdots,\psi_{n-1},y_1,\cdots,y_n\rangle$. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class $(a_1,\cdots,a_{n})$ coincides with the basis element $b_{\mathbf{\lambda}}$ constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where $0\leq a_1\leq a_2\leq\cdots\leq a_{n}\leq \ell-n$ with $a_i\in\mathbb{Z}$ for each $i$, $\mathbf{\lambda}$ is the $\ell$-multipartition of $n$ associated to $(\ell+1-(a_{n}+n), \ell+1-(a_{n-1}+n-1),\cdots,\ell+1-(a_1+1))$. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian $\mathbb{G}_{\ell-n,\ell}$ and the $b_{\lambda}$'s basis of the natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of $H_{\ell,n}^{(0)}$ is also obtained. As an application, we obtain a second version of Giambelli formula for Schubert classes., Comment: corrected a number of typos and added one corollary in the last section

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism., 28 pages; added several references, final version to appear in Algebr. Represent. Theory

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left invariant deformations $\{M_t\}_{t\in B}$ of a compact complex manifold $M=(\Gamma\setminus G, J)$ where $G$ is a Lie group, $\Gamma$ a sublattice and $J$ a left invariant complex structure, the set of all $t\in B$ such that the Dolbeault cohomology on $M_t$ may be computed by left invariant tensor fields is an analytic open subset of $B$.

In this paper, we address the additivity of $n$-multiplicative isomorphisms and $n$-multiplicative derivations on Gamma rings. We proved that, if $\M$ is a $\Gamma$-ring satisfying the some conditions, then any $n$-multiplicative isomorphism $\left(\varphi, \phi\right)$ of $\M$ onto an arbitrary gamma ring is additive., Comment: 10 pages

The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work., Comment: arXiv admin note: text overlap with arXiv:1905.13051

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra B by a Lie algebra X corresponds to a Lie algebra morphism B→Der(X) from B to the Lie algebra Der(X) of derivations on X .In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field K ,in such a way that these generalized derivations characterize the K -algebra actions. We prove that the answer is no, as soon as the field K is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from 2 as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms gl(V) as a representing object for the representations on a vector space V ., Authors are listed in alphabetical order, info:eu-repo/semantics/published

All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described. If $M$ is an irreducible Leibniz representation of $\mathfrak{sl}_2$, by Weyl's result the left action on $M$ as a Lie algebra representation decomposes into a direct sum of irreducible Lie representations of $\mathfrak{sl}_2$. Hence, the problem of description reduces to the study of the right action. In case the number of such irreducible Lie representations is two, up to a Leibniz algebra representation isomorphism there are exactly two types of irreducible Leibniz representations.

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by and shed light on a proposed construction of a cryptographic protocol for multiparty non-interactive key exchange., Comment: 16 pages, to appear in Algebra and Number Theory

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $\beta_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $\beta_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $\beta_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes., Comment: Mostly minor revisions in v2. The biggest change is that we now have a complete proof that the isomorphism on homology in the gas regime is induced by the inclusion map

In this paper, we construct for higher twists that arise from cohomotopy classes, the Chern character in higher twisted K-theory, that maps into higher twisted cohomology. We show that it gives rise to an isomorphism between higher twisted K-theory and higher twisted cohomology over the reals. Finally we compute spherical T-duality in higher twisted K-theory and higher twisted cohomology in very general cases., Comment: 40 pages, revised

Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Mar\'oti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the algorithm, both with and without the use of CFSG., Comment: 42 pages; v2: clarified reference; v3: added reference; v4: minor corrections; v5: correction to main result; v6: correction to main result and general reorganization, as part of PhD thesis

Simons and Sullivan constructed a model of differential K-theory, and showed that the differential K-theory functor fits into a hexagon diagram. They asked whether, like the case of differential characters, this hexagon diagram uniquely determines the differential K-theory functor. This article provides a partial affirmative answer to their question: For any fixed compact manifold, the differential K-theory groups are uniquely determined by the Simons–Sullivan diagram up to an isomorphism compatible with the diagonal arrows of the hexagon diagram. We state a necessary and sufficient condition for an affirmative answer to the full question. This approach further yields an alternative proof of a weaker version of Simons and Sullivan’s results concerning axiomatization of differential characters. We further obtain a uniqueness result for generalised differential cohomology groups. The proofs here are based on a recent work of Pawar.

From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversally oriented Riemannien foliation, we can associate a manifold, called basique manifold, which is \'equiped with an action of orthogonal group, El Kacimi-Alaoui has shown how to associate a transversally elliptic basic differential operator an operator on a vector bundle, called useful bundle, over the basique manifold. The idea is to obtain the desired cohomologic formula from r\'esultats about the operator on the useful bundle. This thesis is a first step in this direction. While the Riemannien foliation is Killing, Goertsches et T\"oben have remarked that there exists a naturel cohomologic isomorphism between the equivariant basique cohomology of the Killing foliation and the equivariant cohomology of the basique manifold. The principal result of this thesis is the geometric realisation of the cohomologic isomorphism by Chern characters under some hypoth\`eses., Comment: Phd thesis, in French. Universit\'e de Montpellier (Nov 2017)

In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom. In the second part of the paper, we give an alternative definition of these maps by using holomorphic curve method, provided that the symplectic cobordisms are Lefschetz fibration satisfying certain nice conditions. Under additionally certain monotonicity assumptions, we show that these two definitions are equivalent., The preview title of this paper is "Cobordism Maps on PFH induced by Lefschetz Fibration over Higher Genus Base". According to referees'suggestion, I change it to be "On cobordism maps on periodic Floer homology". In this version, the exposition of the paper is improved. Besides, I correct some mistakes about the composition rule and restate Theorem 1 for a general setting

A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively, $\mathcal{V}$) is a non-abelian (respectively, a non-locally-finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism relation and elementary equivalence on the set of finitely generated groups in $\mathcal{UV}$., Comment: Some typos are fixed. To appear in the Journal of Group Theory

This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in $kk$. We show that under very mild assumptions on $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ depends only on the isomorphism classes of the cokernels of the matrix $I-A_E$ and of its transpose, which are respectively the $kk$ groups $KH^1(L(E))=kk_{-1}(L(E),\ell)$ and $KH_0(L(E))=kk_0(\ell,L(E))$. Hence if $L(E)$ and $L(F)$ are unital Leavitt path algebras such that $KH_0(L(E))\cong KH_0(L(F))$ and $KH^1(L(E))\cong KH^1(L(F))$ then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and K\"unneth theorems of Rosenberg and Schochet., Comment: 26 pages. Version 2 has a few minor changes

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles., Comment: 29 pages. Contains material from first version of arXiv:1806.01908. Section 4 revised and expanded. Figure added

In this work, we compute the $0$th cohomology group of a complex of groups of cobordism-framed correspondences, and prove the isomorphism to Milnor $K$-groups. An analogous result for common framed correspondences has been proved by A. Neshitov in his paper "Framed correspondences and the Milnor---Witt $K$-theory". Neshitov's result is, at the same time, a computation of the homotopy groups $\pi_{i,i}(S^0)(Spec(k)).$ This work could be used in the future as basis for computing homotopy groups $\pi_{i,i}(MGL_{\bullet})(Spec(k))$ of the spectrum $MGL_{\bullet}.$, Comment: 18 pages

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with proper morphism $f$ and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth correspondence}) gives rise to an enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$ of proper-smooth correspondences. In this paper we extend the well-known theories of characteristic classes of singular varieties such as Baum-Fulton-MacPherson's Riemann-Roch (abbr. BFM-RR) and MacPherson's Chern class transformation and so on to this enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$. In order to deal with local complete intersection (abbr. $\ell.c.i.$) morphism instead of smooth morphism, in a similar manner we consider an enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$ of \emph{proper-$\ell.c.i.$} zigzags and extend BFM-RR to this enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$. We also consider an enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ of proper-smooth correspondences $(X \xleftarrow f M \xrightarrow g Y; E)$ equipped with complex vector bundle $E$ on $M$ (such a correspondence is called \emph{a cobordism bicycle of vector bundle}) and we extend BFM-RR to this enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ as well., 35 pages, comments are welcome; to appear in Fundamenta Mathematicae

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$., Comment: 11 pages

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{{ev}})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\widehat{S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\widehat{S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $\widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions., Comment: Accepted version, 85 pages

We establish various results including the following: if $1, Comment: 11 pp; corrected errors including an error in the second application which now leads to weaker bounds. Details are given at the end of the paper

We provide three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them \(Z(\mathcal J)\), \(Z(\mathcal S^2)\) and \(Z(\mathcal T_s^2)\). The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, \(Z(\mathcal S^2)\) and \(Z(\mathcal T_s^2)\) are not asymptotically Hilbertian. Moreover, the space \(Z(\mathcal T_s^2)\) is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987-2001, 2012). This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its \(n\)-dimensional subspaces to \(\ell_2^n\) grows to infinity as slowly as we wish when \(n\to \infty\).

We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with "the calculation 1+1=2 on a 4D abacus", using the study of homomorphic expansions (aka universal finite type invariants) for ribbon 2-knots, and the relationship between the corresponding associated graded space of arrow diagrams and universal enveloping algebras. This complements the results of the first author, Le and Thurston, where similar arguments using a "3D abacus" and the Kontsevich Integral were used to derive Duflo's theorem for metrized Lie algebras; and results of the first two authors on finite type invariants of w-knotted objects, which also imply a relation of 2-knots with Duflo's theorem in full generality, though via a lengthier path., Comment: 21 pages

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M., Comment: 25 pages. This version includes a minor revision to Theorem 3.3

A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate their isomorphism classes. We discuss not only an algorithm to test the superspeciality but also an algorithm to test isomorphisms for Howe curves. Our algorithms are much more efficient than conventional ones proposed by the authors so far for general canonical curves. We show the existence of a superspecial Howe curve in characteristic $7, Comment: 18 pages. Magma codes used to obtain the main results will be appear at the website of the first author

A twin building consists of two buildings that are twinned by a codistance function. We prove that the local structure of a twin building uniquely determines the two buildings up to isomorphism. This has been known for twin buildings satisfying a technical condition (co)., 14 pages

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$., 31 pages, Final version, To appear in Annali della Scuola Normale Superiore di Pisa

The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur-positivity of skew Macdonald polynomials with parameters $(q,q^2)$ as well as combinatorial formulas for spherical function values., Major revision. v1 Theorem 4.7 is incorrect and replaced with v2 Theorem 4.9. The product defined in Section 4.4 is replaced with a module structure. The application to the structure constants for Macdonald polynomials is replaced with an application to the Schur expansion of skew Macdonald Polynomials. The dependence on character sheaves is removed and the connection explained in a new section

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $${\mathbb {X}}$$ X equipped with a continuous function $$f: {\mathbb {X}}\rightarrow \mathbb {R}$$ f : X → R . We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $$\mathbb {R}$$ R . We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $$({\mathbb {X}}, f)$$ ( X , f ) when it is applied to points randomly sampled from a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of $$({\mathbb {X}}, f)$$ ( X , f ) , a constructible $$\mathbb {R}$$ R -space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $$({\mathbb {X}},f)$$ ( X , f ) to the mapper of a super-level set of a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

A countable semigroup is $$\aleph _0$$ ℵ 0 -categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the $$\aleph _0$$ ℵ 0 -categoricity of semigroups. Our main results are a complete classification of $$\aleph _0$$ ℵ 0 -categorical orthodox completely 0-simple semigroups, and descriptions of the $$\aleph _0$$ ℵ 0 -categorical members of certain classes of strong semilattices of semigroups.

We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups., Comment: 14 pages

If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$, where $\text{gp}(M)$ is the Grothendieck group of $M$. Let $\mathcal{C}$ be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid $M$ in $\mathcal{C}$ are connected to the combinatorics and geometry of its conic hull $\text{cone}(M) \subseteq \mathbb{F} \otimes_{\mathbb{Z}} \text{gp}(M)$. First, we show that the submonoids of $M$ determined by the faces of $\text{cone}(M)$ account for all divisor-closed submonoids of $M$. Then we appeal to the geometry of $\text{cone}(M)$ to characterize whether $M$ is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in $\mathcal{C}$. Along the way, we determine the cones that can be realized by monoids in $\mathcal{C}$ and by finitary monoids in $\mathcal{C}$., 40 pages, 4 figures; to appear in Linear Algebra and Its Applications

On-line social networks, such as in Facebook and Twitter, are often studied from the perspective of friendship ties between agents in the network. Adversarial ties, however, also play an important role in the structure and function of social networks, but are often hidden. Underlying generative mechanisms of social networks are predicted by structural balance theory, which postulates that triads of agents, prefer to be transitive, where friends of friends are more likely friends, or anti-transitive, where adversaries of adversaries become friends. The previously proposed Iterated Local Transitivity (ILT) and Iterated Local Anti-Transitivity (ILAT) models incorporated transitivity and anti-transitivity, respectively, as evolutionary mechanisms. These models resulted in graphs with many observable properties of social networks, such as low diameter, high clustering, and densification. We propose a new, generative model, referred to as the Iterated Local Model (ILM) for social networks synthesizing both transitive and anti-transitive triads over time. In ILM, we are given a countably infinite binary sequence as input, and that sequence determines whether we apply a transitive or an anti-transitive step. The resulting model exhibits many properties of complex networks observed in the ILT and ILAT models. In particular, for any input binary sequence, we show that asymptotically the model generates finite graphs that densify, have clustering coefficient bounded away from 0, have diameter at most 3, and exhibit bad spectral expansion. We also give a thorough analysis of the chromatic number, domination number, Hamiltonicity, and isomorphism types of induced subgraphs of ILM graphs.

We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.

Assume that $\mathbb F$ is an algebraically closed with characteristic $0$. The Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb F$-algebra generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} is central in $\mathfrak{BI}$. In this paper we classify the finite-dimensional irreducible $\mathfrak{BI}$-modules up to isomorphism. As we will see the elements $X,Y,Z$ are not always diagonalizable on finite-dimensional irreducible $\mathfrak{BI}$-modules., Comment: The paper is to correct the main result of Communications in Algebra 44 (2016), 919-943; the paper arXiv:1910.11446 is to correct the main result of Communications in Algebra 47 (2019), 1869-1891