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For a finite dimensional algebra |$A$|⁠ , the notion of |$g$| -fan |$\Sigma (A)$| is defined from two-term silting complexes of |$\textsf{K}^{\textrm{b}}(\textsf{proj} A)$| in the real Grothendieck group |$K_{0}(\textsf{proj} A)_{\mathbb{R}}$|⁠. In this paper, we discuss the theory of shards to |$\Sigma (A)$|⁠ , which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of |$\textsf{mod}A$| and the set of shards of |$\Sigma (A)$| for |$g$| -finite algebra |$A$|⁠. Moreover, we show that the semistable region of a brick of |$\textsf{mod}A$| is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of |$\textsf{mod}A$|⁠. [ABSTRACT FROM AUTHOR]

We classify twistings of Grothendieck's differential operators on a smooth variety |$X$| in prime characteristic |$p$|⁠. We prove that isomorphism classes of twistings are in bijection with |$H^{2}(X,\mathbb{Z}_{p}(1))$|⁠ , the degree 2, weight 1 syntomic cohomology of |$X$|⁠. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed. [ABSTRACT FROM AUTHOR]

We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring |$R$|⁠. The weak equivalences are given by: (1) an |$R$| -linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an |$R$| -homology equivalence between universal covers, and (3) |$R$| -homology equivalences. Analogously, for any field |${\mathbb{F}}$|⁠ , we construct three model structures on the category of connected simplicial cocommutative |${\mathbb{F}}$| -coalgebras. The weak equivalences in this context are (1 ′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2 ′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3 ′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3 ′), we prove that, when |${\mathbb{F}}$| is algebraically closed, the simplicial |${\mathbb{F}}$| -coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when |${\mathbb{F}}$| is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of |${\mathbb{F}}$|⁠. [ABSTRACT FROM AUTHOR]

We show that isomorphism classes of indecomposable |$\tau $| -rigid pairs over |$\Pi _{n}$|⁠ , the preprojective algebra of |$A_{n}$|⁠ , are in bijection with internal |$n$| -simplices in the prism |$\Delta _{n} \times \Delta _{1}$|⁠ , the product of an |$n$| -simplex with a 1-simplex. We show further that this induces a bijection between triangulations of |$\Delta _{n} \times \Delta _{1}$| and basic support |$\tau $| -tilting pairs over |$\Pi _{n}$| such that bistellar flips of triangulations correspond to mutations of support |$\tau $| -tilting pairs. These bijections are shown to be compatible with the known bijections involving the symmetric group. [ABSTRACT FROM AUTHOR]

We study the representation theory of the Soergel calculus algebra |$ {{ \tilde {A}_w^{{\mathbb {C}}}}}:= \mbox {End}_{ {{\mathcal {D}}}_{(W,S)}} (\underline {w}) $| over |$ {\mathbb {C}}$| in type |$ \tilde {A}_1$|⁠. We generalize the recent isomorphism between the nil-blob algebra |$ {{\mathbb {N}\mathbb {B}}_n}$| and a diagrammatically defined subalgebra |$ {A}_w^{{\mathbb {C}}}$| of |${{ \tilde {A}_w^{{\mathbb {C}}}}}$| to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for |$ \tilde {A}_1$|⁠. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of |$ {{\mathbb {N}\mathbb {B}}_n}$|⁠ , we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module |$ \Delta _w(v) $| for |$ {{ \tilde {A}_w^{{\mathbb {C}}}}} $|⁠. The entries of the diagonalized matrices turn out to be products of roots for |$ \tilde {A}_1$|⁠. We use this to study Jantzen-type filtrations of |$ \Delta _w(v) $| for |$ {{ \tilde {A}_w^{{\mathbb {C}}}}}$|⁠. We show that, at an enriched Grothendieck group level, the corresponding sum formula has terms |$ \Delta _w(s_{\alpha }v)[ l(s_{\alpha }v)- l(v)] $|⁠ , where |$ [ \cdot ] $| denotes grading shift. [ABSTRACT FROM AUTHOR]

Let |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| be operator algebras with |$c_{0}$| -isomorphic diagonals and let |${{\mathcal{K}}}$| denote the compact operators. We show that if |${{\mathcal{A}}}\otimes{{\mathcal{K}}}$| and |${{\mathcal{B}}}\otimes{{\mathcal{K}}}$| are isometrically isomorphic, then |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| are isometrically isomorphic. If the algebras |${{\mathcal{A}}}$| and |${{\mathcal{B}}}$| satisfy an extra analyticity condition a similar result holds with |${{\mathcal{K}}}$| being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have |$K_{0}$| -groups isomorphic to |${{\mathbb{Z}}}$|⁠. This has implications in the study of stable isomorphisms between various semicrossed products. [ABSTRACT FROM AUTHOR]

We construct for every connected locally finite graph |$\Pi $| the quantum automorphism group |$\operatorname{QAut} \Pi $| as a locally compact quantum group. When |$\Pi $| is vertex transitive, we associate to |$\Pi $| a new unitary tensor category |${\mathcal{C}}(\Pi)$| and this is our main tool to construct the Haar functionals on |$\operatorname{QAut} \Pi $|⁠. When |$\Pi $| is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs |$\Pi $|⁠ , |$\Pi ^{\prime}$| and prove that this implies monoidal equivalence of |$\operatorname{QAut} \Pi $| and |$\operatorname{QAut} \Pi ^{\prime}$|⁠. [ABSTRACT FROM AUTHOR]

Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra |${{\mathcal {V}}}$|⁠ , we have a surjective homomorphism of differential algebras |$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$|⁠ ; equivalently, the singular support of |${{\mathcal {V}}}$| is a closed subscheme of the arc space of the associated scheme |$X_{{{\mathcal {V}}}}$|⁠. We give many new examples of classically free vertex algebras (i.e. this map is an isomorphism), including |$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$| for all positive integers |$n$| and |$k$|⁠. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular |${{\mathcal {W}}}$| -algebra of |${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$| at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras. [ABSTRACT FROM AUTHOR]

We prove a structural result in mirror symmetry for projective Calabi–Yau (CY) manifolds. Let |$X$| be a connected symplectic CY manifold, whose Fukaya category |$\mathcal {F}(X)$| is defined over some suitable Novikov field |${\mathbb {K}}$|⁠ ; its mirror is assumed to be some smooth projective |${\mathbb {K}}$| -variety |$Y$| that is "maximally degenerating". Suppose that some split-generating subcategory of (a |$\textsf {dg}$| enhancement of) |$D^bCoh(Y)$| embeds into |$\mathcal {F}(X)$|⁠ : we call this hypothesis "core homological mirror symmetry". We prove that the embedding extends to an equivalence of categories, |$D^{b}Coh(Y) \simeq D^{\pi }(\mathcal {F}(X))$|⁠ , using Abouzaid's split-generation criterion. The results only depend on certain formal properties of the Fukaya category, which have been established in certain cases but not in complete generality. The appendix, which can be read on its own, proves a result about Hochschild cohomology for schemes: the compatibility of the global Hochschild–Kostant–Rosenberg isomorphism with Kodaira–Spencer deformation theory. [ABSTRACT FROM AUTHOR]

In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue |$1$| can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space. Further, we show a formula relating the Alexander module's Hodge numbers to those of finite covers of the base space, under some assumptions. Dedicated to the memory of Georgia Benkart [ABSTRACT FROM AUTHOR]

Let |$Z\to X$| be a closed immersion of smooth affine schemes over a field |$k$|⁠ , and let |$X^h_Z$| denote the henselisation of |$X$| along |$Z$|⁠. We prove that |$E(X^h_Z)\simeq E(Z)$| for every additive presheaf |$E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$| on the stable motivic homotopy category over |$k$| that is |$l_\varepsilon $| -torsion or |$l$| -torsion, where |$l\in \mathbb Z$| is coprime to |$\operatorname {char} k$|⁠ , and |$l_\varepsilon =\sum _{i=1}^n \langle (-1)^i \rangle $|⁠. More generally, the isomorphism holds for any homotopy invariant |$l_\varepsilon $| -torsion or |$l$| -torsion linear |$\sigma $| -quasi-stable framed additive presheaf on |$\textrm {Sm}_{k}$|⁠. This generalises the result known earlier for local schemes. We prove the above isomorphism by constructing (stable) |${\mathbb {A}}^1$| -homotopies of motivic spaces via algebraic geometry. To achieve this, we replace Quillen's trick with an alternative and more general construction that provides relative curves required in our setting. [ABSTRACT FROM AUTHOR]

In this paper, we first establish an |$L^2$| -type Dolbeault isomorphism for logarithmic differential forms by Hrmander's |$L^2$| estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Kähler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Girbau's vanishing theorem, Le Potier's vanishing theorem, and a version of the Kawamata–Viehweg vanishing theorem. [ABSTRACT FROM AUTHOR]

Let |$\Theta _n = (\theta _1, \dots , \theta _n)$| and |$\Xi _n = (\xi _1, \dots , \xi _n)$| be two lists of |$n$| variables, and consider the diagonal action of |${{\mathfrak {S}}}_n$| on the exterior algebra |$\wedge \{ \Theta _n, \Xi _n \}$| generated by these variables. Jongwon Kim and the 2nd author defined and studied the fermionic diagonal coinvariant ring |$FDR_n$| obtained from |$\wedge \{ \Theta _n, \Xi _n \}$| by modding out by the ideal generated by the |${{\mathfrak {S}}}_n$| -invariants with vanishing constant term. On the other hand, the 2nd author described an action of |${{\mathfrak {S}}}_n$| on the vector space with basis given by noncrossing set partitions of |$\{1,\dots ,n\}$| using a novel family of skein relations that resolve crossings in set partitions. We give an isomorphism between a natural Catalan-dimensional submodule of |$FDR_n$| and the skein representation. To do this, we show that set partition skein relations arise naturally in the context of exterior algebras. Our approach yields an |${{\mathfrak {S}}}_n$| -equivariant way to resolve crossings in set partitions. We use fermions to clarify, sharpen, and extend the theory of set partition crossing resolution. [ABSTRACT FROM AUTHOR]

A singular foliation |$\mathcal {F}$| gives a partition of a manifold |$M$| into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space |$M / \mathcal {F}$| and that of the basic differential forms on |$M$|⁠. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when |$\mathcal {F}$| is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of |$M$|⁠ , and, as a special case of the latter, when |$\mathcal {F}$| is induced by a linearizable Lie groupoid or is a singular Riemannian foliation. [ABSTRACT FROM AUTHOR]

We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero, that is, a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in étale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations. [ABSTRACT FROM AUTHOR]

Let |$X$| be a perfectoid space with tilt |$X^\flat $|⁠. We build a natural map |$\theta :\Pic X^\flat \to \lim \Pic X$| where the (inverse) limit is taken over the |$p$| -power map and show that |$\theta $| is an isomorphism if |$R = \Gamma (X,\sO _X)$| is a perfectoid ring. As a consequence, we obtain a characterization of when the Picard groups of |$X$| and |$X^\flat $| agree in terms of the |$p$| -divisibility of |$\Pic X$|⁠. The main technical ingredient is the vanishing of higher derived limits of the unit group |$R^*$|⁠ , whence the main result follows from the Grothendieck spectral sequence. [ABSTRACT FROM AUTHOR]

We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel –Ekholm–Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the 1st Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist. [ABSTRACT FROM AUTHOR]

We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The |$k$| -th MD homotopy group is a group |$MDH^b_{\bullet }$| for any |$b\in [1,\infty ]$| together with homomorphisms |$MD\pi ^b\to MD\pi ^{b^{\prime}}$| for any |$b\geq b^{\prime}$|⁠. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For |$b=1$|⁠ , it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for |$b=\infty $|⁠ , the |$MD$| -homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case. [ABSTRACT FROM AUTHOR]

For any triple of positive integers |$A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$| and |$c \in{{\mathbb{C}}}^*$|⁠ , cusp polynomial |${ f_{A^\prime }} = x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$| is known to be mirror to Geigle–Lenzing orbifold projective line |${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$|⁠. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial |${ f_{A^\prime }}$| turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of |${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$|⁠. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any |$G$| —a symmetry group of a cusp polynomial |${ f_{A^\prime }}$|⁠ , we introduce the Frobenius manifold of a pair |$({ f_{A^\prime }},G)$| and show that it is isomorphic to the Frobenius manifold of the Gromov–Witten theory of Geigle–Lenzing weighted projective line |${{\mathbb{P}}}^1_{A,\Lambda }$|⁠ , indexed by another set |$A$| and |$\Lambda $|⁠ , distinct points on |${{\mathbb{C}}}\setminus \{0,1\}$|⁠. For some special values of |$A^{\prime}$| with the special choice of |$G$| it happens that |${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb{P}}}^1_{A,\Lambda }$|⁠. Combining our mirror symmetry isomorphism for the pair |$(A,\Lambda)$|⁠ , together with the "usual" one for |$A^{\prime}$|⁠ , we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta–function. [ABSTRACT FROM AUTHOR]

Let |$G$| be a reductive complex Lie group acting holomorphically on |$X=\mathbb{C}^n$|⁠. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on |$\mathbb{C}^n$| such that the |$G$| -action becomes linear. Equivalently, is there a |$G$| -equivariant biholomorphism |$\Phi \colon X\to V$| where |$V$| is a |$G$| -module? There is an intrinsic stratification of the categorical quotient |$X/\!\!/G$|⁠ , called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of |$G$|⁠. Suppose that there is a |$\Phi $| as above. Then |$\Phi $| induces a biholomorphism |${\varphi }\colon X/\!\!/G\to V/\!\!/G$| that is stratified, that is, the stratum of |$X/\!\!/G$| with a given label is sent isomorphically to the stratum of |$V/\!\!/G$| with the same label. The counterexamples to the Linearization Problem construct an action of |$G$| such that |$X/\!\!/G$| is not stratified biholomorphic to any |$V/\!\!/G$|⁠. Our main theorem shows that, for a reductive group |$G$| with |$\dim G\leq 1$|⁠ , the existence of a stratified biholomorphism of |$X/\!\!/G$| to some |$V/\!\!/G$| is not only necessary but also sufficient for linearization. In fact, we do not have to assume that |$X$| is biholomorphic to |$\mathbb{C}^n$|⁠ , only that |$X$| is a Stein manifold. [ABSTRACT FROM AUTHOR]

By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics. [ABSTRACT FROM AUTHOR]

Let |${\mathfrak{g}}$| be a symmetrizable Kac–Moody algebra with associated Yangian |$Y_\hbar{\mathfrak{g}}$| and Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$|⁠. An elementary result of fundamental importance to the theory of Yangians is that, for each |$c\in{\mathbb{C}}$|⁠ , there is an automorphism |$\tau _c$| of |$Y_\hbar{\mathfrak{g}}$| corresponding to the translation |$t\mapsto t+c$| of the complex plane. Replacing |$c$| by a formal parameter |$z$| yields the so-called formal shift homomorphism |$\tau _z$| from |$Y_\hbar{\mathfrak{g}}$| to the polynomial algebra |$Y_\hbar{\mathfrak{g}}[z]$|⁠. We prove that |$\tau _z$| uniquely extends to an algebra homomorphism |$\Phi _z$| from the Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$| into the |$\hbar $| -adic closure of the algebra of Laurent series in |$z^{-1}$| with coefficients in the Yangian |$Y_\hbar{\mathfrak{g}}$|⁠. This induces, via evaluation at any point |$c\in{\mathbb{C}}^\times $|⁠ , a homomorphism from |$\textrm{D}Y_\hbar{\mathfrak{g}}$| into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of |$\textrm{D}Y_\hbar{\mathfrak{g}}$| and |$Y_\hbar{\mathfrak{g}}$| and, as a corollary, we find that the Yangian |$Y_\hbar{\mathfrak{g}}$| can be realized as a degeneration of the Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$|⁠. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for |$\textrm{D}Y_\hbar{\mathfrak{g}}$| applicable when |${\mathfrak{g}}$| is of finite type or of simply laced affine type. [ABSTRACT FROM AUTHOR]

Associated with each finite subgroup |$\Gamma $| of |${\textrm {SL}}_2({\mathbb {C}})$| there is a family of noncommutative algebras |$O_\tau (\Gamma)$| quantizing |${\mathbb {C}}^2/\!\!/\Gamma $|⁠. Let |$G_\Gamma $| be the group of |$\Gamma $| -equivariant automorphisms of |$O_\tau $|⁠. In [ 16 ], one of the authors defined and studied a natural action of |$G_\Gamma $| on certain quiver varieties associated with |$\Gamma $|⁠. He established a |$G_\Gamma $| -equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of |$O_\tau $| -ideals. In this paper we prove that the action of |$G_\Gamma $| on the quiver variety is transitive when |$\Gamma $| is a cyclic group. This generalizes an earlier result due to Berest and Wilson who showed the transitivity of the automorphism group of the 1st Weyl algebra on the Calogero–Moser spaces. Our result has two important implications. First, it confirms the Bocklandt–Le Bruyn conjecture for cyclic quiver varieties. Second, it will be used to give a complete classification of algebras Morita equivalent to |$O_\tau (\Gamma),$| thus answering the question of Hodges. At the end of the introduction we explain why the result of this paper does not extend when |$\Gamma $| is not cyclic. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present paper, we show that there are at most finitely many isomorphism classes of hyperbolic polycurves (i.e. successive extensions of families of hyperbolic curves) over certain types of fields whose étale fundamental group is isomorphic to a prescribed profinite group. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate |$(+1)$| -shifted Poisson structures in the context of differential geometry. The relevant notion is that of |$(+1)$| -shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of |$(+1)$| -shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a |${\mathbb{Z}}$| -graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack |${\mathfrak{X}}$|⁠. It turns out that |$(+1)$| -shifted Poisson structures on |${\mathfrak{X}}$| correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex |$T_{\mathfrak{X}}$| and the cotangent complex |$L_{\mathfrak{X}}$| of a differentiable stack |${\mathfrak{X}}$| in terms of any Lie groupoid |$\Gamma{\rightrightarrows } M$| representing |${\mathfrak{X}}$|⁠. They correspond to a homotopy class of 2-term homotopy |$\Gamma$| -modules |$A[1]\rightarrow TM$| and |$T^{\vee } M\rightarrow A^{\vee }[-1]$|⁠ , respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a |$(+1)$| -shifted Poisson structure on a differentiable stack |${\mathfrak{X}}$| defines a morphism |${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the number |$B(p,g)$| of isomorphism classes of abelian varieties over a prime field |$\mathbb{F}_p$| of dimension |$g$| has a lower bound |$p^{\frac{1}{2}g^2(1+o(1))}$| as |$g \rightarrow \infty$|⁠. This is the 1st nontrivial result on the lower bound of |$B(p,g)$|⁠. We also improve the upper bound |$2^{34g^2}p^{\frac{69}{4} g^2 (1+o(1))}$| of |$B(p,g)$| given by Lipnowski and Tsimerman [ 7 ] to |$p^{\frac{45}{4} g^2(1+o(1))}$|⁠. [ABSTRACT FROM AUTHOR]

Let |$(X_1,\dots ,X_m)$| be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential |$V$|⁠ , and let |$(S_1,\dots ,S_m)$| be a free semicircular family. For |$k < m$|⁠ , we show that conditional expectations and conditional non-microstates free entropy given |$X_1$|⁠ , ..., |$X_k$| arise as the large |$N$| limit of the corresponding conditional expectations and entropy for the |$N \times N$| random matrix models associated to |$V$|⁠. Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism |$\mathrm{W}^*(X_1,\dots ,X_m) \to \mathrm{W}^*(S_1,\dots ,S_m)$| that maps |$\mathrm{W}^*(X_1,\dots ,X_k)$| to |$\mathrm{W}^*(S_1,\dots ,S_k)$| for each |$k = 1, \dots , m$| and that also witnesses the Talagrand inequality for the law of |$(X_1,\dots ,X_m)$| relative to the law of |$(S_1,\dots ,S_m)$|⁠. [ABSTRACT FROM AUTHOR]

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph and the corresponding quotient graph. [ABSTRACT FROM AUTHOR]

In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of |$\mathrm{C}^*$| -correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C |$^*$| -correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of |$\mathrm{C}^*$| -correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana's injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid |$\mathrm{C}^*$| -correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg. [ABSTRACT FROM AUTHOR]

We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph complex. This result may have applications in theory of multi-vector fields |$T_{\textrm{poly}}^{\geq 1}$| of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types. [ABSTRACT FROM AUTHOR]

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these "big" mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms. [ABSTRACT FROM AUTHOR]

We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber–Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restriction maps, or deforming both. We make our computations precise in the case of quartic hypersurfaces, and compute explicit dimensions in many examples. [ABSTRACT FROM AUTHOR]

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case, our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products. [ABSTRACT FROM AUTHOR]

The (tree) amplituhedron |$\mathcal{A}_{n,k,m}$| is the image in the Grassmannian |$\text{Gr}_{k,k+m}$| of the totally nonnegative part of |$\text{Gr}_{k,n}$|⁠, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in |$\mathcal{N}=4$| supersymmetric Yang–Mills theory. When |$k+m=n$|⁠, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when |$k=1$|⁠, the amplituhedron is a cyclic polytope. While the case |$m=4$| is most relevant to physics, the amplituhedron is an interesting mathematical object for any |$m$|⁠. In this article, we study it in the case |$m=1$|⁠. We start by taking an orthogonal point of view and define a related "B-amplituhedron" |$\mathcal{B}_{n,k,m}$|⁠, which we show is isomorphic to |$\mathcal{A}_{n,k,m}$|⁠. We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron |$\mathcal{A}_{n,k,1}$| using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that |$\mathcal{A}_{n,k,1}$| can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that |$\mathcal{A}_{n,k,1}$| is homeomorphic to a ball. [ABSTRACT FROM AUTHOR]

We introduce, for every |$\mathbb{Z}$| -graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of |$\mathbb{Z}$| -graded manifolds and we extend the Emmrich–Weinstein theorem to the context of |$\mathbb{Z}$| -graded manifolds. [ABSTRACT FROM AUTHOR]

For |$\Gamma$| a finite, connected metric graph, |$\Gamma^n_{\textbf{r}}$| is the space of configurations of |$n$| points in |$\Gamma$| with a restraint parameter |$\textbf{r}$| dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this article, we study the homotopy, homeomorphism, and isotopy types of |$\{\Gamma^n_{\textbf{r}}\}$| over the space of parameters |$\textbf{r}$| and provide a polynomial upper bound (in the number of edges of |$\Gamma$|⁠) for the number of isotopy types. [ABSTRACT FROM AUTHOR]

We study the category of finite-dimensional representations for a basic classical Lie superalgebra g = g0 ⊕ g1. For the ortho-symplectic Lie superalgebra g = osp(1,2n), we show that various objects in that category admit a fusion flag, that is, a sequence of graded g0[t]-modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite-dimensional irreducible g-modules, truncated Weyl modules and Demazure type modules. This result shows that the character of these types of representations can be expressed in terms of characters of fusion products and we prove that the graded multiplicities are given by products of q-binomial coefficents. Moreover, we establish a presentation for these types of fusion products in terms of generators and relations of the enveloping algebra. [ABSTRACT FROM AUTHOR]

We construct a klt del Pezzo surface which is not globally F-split, over any algebraically closed field of positive characteristic. [ABSTRACT FROM AUTHOR]

The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space Ω*X of X provides an A∞ quasi-equivalence between the dg category of flat Z-graded connections on X and the dg category of representations of C∙(Ω*X), the dg algebra of singular chains on Ω*X. [ABSTRACT FROM AUTHOR]

To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, étale, Hodge, etc.) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of differential graded (dg) categories. This leads to new additive invariants of dg categories, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism. [ABSTRACT FROM AUTHOR]

The only unital C*-algebras that admit elimination of quantifiers in continuous logic in the language of unital C*-algebras are C,C², C(Cantor space) and M2(C). We also prove that the theory of C*-algebras does not have model companion and show that the theory of Mn(On+1) is not ∀∃-axiomatizable for any n ≥ 2. [ABSTRACT FROM AUTHOR]

Let G be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each θ-group (G0, g1) and a vector X ∈ g1 a flat G-connection ∇X on the trivial G-bundle on P1−{0,∞}, generalizing the construction of Frenkel and Gross in [5]. In this paper, we study the local monodromies of the flat G-connection ∇X and compute the de Rham cohomology of ∇X with value in the adjoint representation of G. In particular, we show that in many cases the connection ∇X is cohomologically rigid. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple and short proof of the uniqueness of generic representations in an L-packet for a quasi-split connected classical group over a non-archimedean local field. [ABSTRACT FROM AUTHOR]

We prove a structure theorem for relative coarse space morphisms from smooth Deligne- Mumford stacks, showing that each such map can be decomposed in terms of root stack and canonical stack morphisms. We explain how our result can be understood locally in terms of pseudo-reflections. Lastly, we give an application to equivariant K-theory, give several examples illustrating our result, and pose some related questions. [ABSTRACT FROM AUTHOR]

Any Calabi-Yau three-fold X with infinite fundamental group admits an étale Galois covering either by an abelian three-fold or by the product of a K3 surface and an elliptic curve.Wecall X of type A in the former case and of type K in the latter case. In this article, we provide the full classification of Calabi-Yau three-folds of type K, based on Oguiso and Sakurai's work [25]. Together with a refinement of their result on Calabi-Yau threefolds of type A, we finally complete the classification of Calabi-Yau three-folds with infinite fundamental group. [ABSTRACT FROM AUTHOR]

We establish the isomorphism OX⋊r α G ≃ OX⋊r α G for any non-degenerate C∗-correspondence (X, C) and any discrete group G acting on (X, C). In the case where (X, C) is hyperrigid, a similar formula is also established for the full crossed product. These results are obtained by studying the C∗-envelopes and the hyperrigidity of a related class of non-selfadjoint crossed products that were introduced recently by the author and Chris Ramsey. [ABSTRACT FROM AUTHOR]

Easy quantum groups form a class of compact matrix quantum groups which are completely determined by set partitions. They have been defined in 2009 by Banica and Speicher in the orthogonal case (as quantum subgroups of Wang's free orthogonal quantum group O+ n ). We extend their definition to the unitary case (as quantum subgroups of Wang's free unitary quantum group U+ n ) using colored partitions. In the free case (i.e., restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. We show how the quantum groups associated with them can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups. [ABSTRACT FROM AUTHOR]

We study the classical limit of a tensor product of Kirillov-Rmodules over a quantum loop algebra, and show that it is realized from the classical limits of the tensor factors using the notion of fusion products. In the process of the proof, we also give defining relations of the fusion product of the (graded) classical limits of Kirillov- Reshetikhin modules. [ABSTRACT FROM AUTHOR]

Recently, Boutonnet, Chifan, and Ioana proved that McDuff's family of continuum many pairwise non-isomorphic separable II1 factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are non-isomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories. [ABSTRACT FROM AUTHOR]