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Lasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface |$X$| implies that all the finite dimensional linear representations of the fundamental group |$\pi _1(X)$| are finite. In this article, we generalize Lasell and Ramachandran's result to the case of |$\pi _1-small$| divisors on quasiprojective varieties. We also study |$\pi _1-small$| curves and hyperbolicity properties of smooth projective surfaces of general type with infinite fundamental groups. [ABSTRACT FROM AUTHOR]

We construct relative moduli spaces of semistable pairs on a family of projective Deligne–Mumford stacks. We define moduli stacks of stable orbifold Pandharipande–Thomas pairs on stacks of expanded degenerations and pairs and then show they are separated and proper Deligne–Mumford stacks of finite type. As an application, we present the degeneration formula for the absolute and relative orbifold Pandharipande–Thomas invariants. [ABSTRACT FROM AUTHOR]

We prove that for the dyadic maximal operator |$\textrm {M}$| and every locally integrable function |$f\in L^1_{{\textrm {loc}}}(\mathbb R^d)$| with bounded variation, also |$\textrm {M} f$| is locally integrable and |$\mathop {\textrm {var}}\textrm {M} f\leq C_d\mathop {\textrm {var}} f$| for any dimension |$d\geq 1$|⁠. It means that if |$f\in L^1_{{\textrm {loc}}}(\mathbb R^d)$| is a function whose gradient is a finite measure then so is |$\nabla \textrm {M} f$| and |$\|\nabla \textrm {M} f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$|⁠. We also prove this for the local dyadic maximal operator. [ABSTRACT FROM AUTHOR]

We show that every smooth projective curve over a finite field |$k$| admits a finite tame morphism to the projective line over |$k$|⁠. Furthermore, we construct a curve with no such map when |$k$| is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama–Yasuda, and Anbar–Tutdere. [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of |$ U(\textrm {End}(\Bbbk ^r)[x,y]) $| and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank. [ABSTRACT FROM AUTHOR]

In this paper, we improve the finiteness constant for the finiteness principles for |$C^m({\mathbb{R}}^n,{\mathbb{R}}^D)$| and |$C^{m-1,1}({\mathbb{R}}^n,{\mathbb{R}}^D)$| selection proven in [ 19 ] and extend the more general shape fields finiteness principle to the vector-valued case. [ABSTRACT FROM AUTHOR]

We initiate a detailed analysis of |$C^{\ast }$| -diagonals in classifiable |$C^{\ast }$| -algebras, answering natural questions concerning topological properties of their spectra and uniqueness questions. Firstly, we construct |$C^{\ast }$| -diagonals with connected spectra in all classifiable stably finite |$C^{\ast }$| -algebras, which are unital or stably projectionless with continuous scale. Secondly, for classifiable stably finite |$C^{\ast }$| -algebras with torsion-free |$K_0$| and trivial |$K_1$|⁠ , we further determine the spectra of the |$C^{\ast }$| -diagonals up to homeomorphism. In the unital case, the underlying space turns out to be the Menger curve. In the stably projectionless case, the space is obtained by removing a non-locally-separating copy of the Cantor space from the Menger curve. Thirdly, we show that each of our classifiable |$C^{\ast }$| -algebras has continuum many pairwise non-conjugate such Menger manifold |$C^{\ast }$| -diagonals. [ABSTRACT FROM AUTHOR]

Certain rigid irregular -connections constructed by the 1st-named author are related via pullbacks along a finite covering and Fourier transform to rigid local systems on a punctured projective line. This kind of property was first observed by Katz for hypergeometric connections and used by Sabbah and Yu to compute irregular Hodge filtrations for hypergeometric connections. This strategy can also be applied to the aforementioned -connections and we compute jumping indices and dimensions for their irregular Hodge filtrations. [ABSTRACT FROM AUTHOR]

We explore the finite-dimensional part of the non-commutative Choquet boundary of an operator algebra. In other words, we seek finite-dimensional boundary representations. Such representations may fail to exist even when the underlying operator algebra is finite dimensional. Nevertheless, we exhibit mechanisms that detect when a given finite-dimensional representation lies in the Choquet boundary. Broadly speaking, our approach is topological and requires identifying isolated points in the spectrum of the -envelope. This is accomplished by analyzing peaking representations and peaking projections, both of which being non-commutative versions of the classical notion of a peak point for a function algebra. We also connect this question with the residual finite dimensionality of the -envelope and to a stronger property that we call -liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite dimensional. [ABSTRACT FROM AUTHOR]

We study the Bergman metric of a finite ball quotient |$\mathbb{B}^n/\Gamma $|⁠ , where |$n \geq 2$| and |$\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$| is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if |$\Gamma $| is trivial, that is, when the ball quotient |$\mathbb{B}^n/\Gamma $| is the unit ball |${\mathbb{B}}^n$| itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric. [ABSTRACT FROM AUTHOR]

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual étale fundamental group |$\pi _1^{\acute{\mathrm{e}}{\mathrm{t}}}$| defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite étale covers. We prove exactness of the general homotopy sequence for the pro-étale fundamental group, i.e. that for a geometric point |$\bar{s}$| on |$S$| and a flat proper morphism |$X \rightarrow S$| of finite presentation whose geometric fibres are connected and reduced, the sequence |$ \pi _1^{{\mathrm{pro}}\acute{\mathrm{e}}{\mathrm{t}}}(X_{\bar{s}}) \rightarrow \pi _1^{{\mathrm{pro}}\acute{\mathrm{e}}{\mathrm{t}}}(X) \rightarrow \pi _1^{{\mathrm{pro}}\acute{\mathrm{e}}{\mathrm{t}}}(S) \rightarrow 1 $| is "nearly exact." This generalizes a theorem of Grothendieck from finite étale covers to geometric coverings. We achieve the proof by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization in this setting. [ABSTRACT FROM AUTHOR]

In this paper, we consider the |$L^2$| -critical fractional Schrödinger equation |$iu_t-|D|^{\beta }u+|u|^{2\beta }u=0$| with initial data |$u_0\in H^{\beta /2}(\mathbb{R})$| and |$\beta $| close to |$2$|⁠. We show that if the initial data have negative energy and slightly supercritical mass, then the solution blows up in finite time. We also give a specific description for the blow-up dynamics. This is an extension of the works of F. Merle and P. Raphaël for |$L^2$| -critical Schrödinger equations. However, the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required. [ABSTRACT FROM AUTHOR]

Let |$G$| be a complex semisimple group. In this note, we give a method to classify |$\mathbb Q$| -Fano compactifications of |$G$|⁠. We will prove that there are only finitely many |$\mathbb Q$| -Fano |$G$| -compactifications that admit (singular) Kähler–Einstein metrics. As an application, this improves a former result in [ 19 ]. [ABSTRACT FROM AUTHOR]

We consider a locally finite (Radon) measure on |$ {\operatorname{SO}}^+(d,1)/ \Gamma $| invariant under a horospherical subgroup of |$ {\operatorname{SO}}^+(d,1) $| where |$ \Gamma $| is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting |$ 1 $| -parameter diagonalizable flow (geodesic flow). [ABSTRACT FROM AUTHOR]

Let |$f:X\to X $| be a dominant self-morphism of an algebraic variety. Consider the set |$\Sigma _{f^{\infty }}$| of |$f$| -periodic subvarieties of small dynamical degree (SDD), the subset |$S_{f^{\infty }}$| of maximal elements in |$\Sigma _{f^{\infty }}$|⁠ , and the subset |$S_f$| of |$f$| -invariant elements in |$S_{f^{\infty }}$|⁠. When |$X$| is projective, we prove the finiteness of the set |$P_f$| of |$f$| -invariant prime divisors with SDD and give an optimal upper bound $$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$ as |$n\to \infty $|⁠ , where |$d_1(f)$| is the 1st dynamic degree. When |$X$| is an algebraic group (with |$f$| being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound $$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$ as |$n \to \infty $|⁠ , which slightly generalizes a conjecture of S.-W. Zhang for polarized |$f$|⁠. [ABSTRACT FROM AUTHOR]

We further examine the concept of uniform property |$\Gamma $| for |$C^*$| -algebras introduced in our joint work with Winter. In addition to obtaining characterisations in the spirit of Dixmier's work on central sequences in II |$_1$| factors, we establish the equivalence of uniform property |$\Gamma $|⁠ , a suitable uniform version of McDuff's property for |$C^*$| -algebras, and the existence of complemented partitions of unity for separable nuclear |$C^*$| -algebras with no finite dimensional representations and a compact (non-empty) tracial state space. As a consequence, for |$C^*$| -algebras as in the Toms–Winter conjecture, the combination of strict comparison and uniform property |$\Gamma $| is equivalent to Jiang–Su stability. We also show how these ideas can be combined with those of Matui–Sato to streamline Winter's classification by embeddings technique. [ABSTRACT FROM AUTHOR]

In this paper, we introduce properties including groupoid comparison, pure infiniteness, and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid |$C^*$| -algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid |$C^*$| -algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke–Li and Rainone–Sims. We also study the relation among our paradoxical comparison, |$n$| -filling property, and locally contracting property that appeared in the literature for locally compact Hausdorff étale groupoids. [ABSTRACT FROM AUTHOR]

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard–Reifenberg-type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in |$\mathbb{R}^n$| with finite Willmore energy. Especially, we prove the removability of the isolated singularity of multiplicity one surfaces with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption. [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]

We study finite Morse index solutions to the non-local Gelfand–Liouville problem $$\begin{align*}& (-\Delta)^su=e^u\quad\textrm{in}\quad{{\mathbb{R}}^n}, \end{align*}$$ for every |$s\in (0,1)$| and |$n>2s$|⁠. Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$\begin{align*} &u_{n,s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})}\right)\end{align*}$$ is unstable. [ABSTRACT FROM AUTHOR]

We prove effective equidistribution of the horocycle flow in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces. [ABSTRACT FROM AUTHOR]

We construct a filtration of chiral Hodge cohomolgy of a K3 surface |$X$|⁠ , such that its associated graded object is a unitary representation of the |$\mathcal{N}=4$| superconformal vertex algebra with central charge |$c=6$| and its subspace of primitive vectors has the property; its equivariant character for a symplectic automorphism |$g$| of finite order acting on |$X$| agrees with the McKay–Thompson series for |$g$| in Mathieu moonshine. [ABSTRACT FROM AUTHOR]

We generalize Mukai and Shafarevich's definitions of isogenies between K3 surfaces over |${\mathbb{C}}$| to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over |$\bar{{\mathbb{F}}}_p$| by prescribing linear algebraic data when |$p$| is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on |$p$|⁠. [ABSTRACT FROM AUTHOR]

The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank |$d$| perturbations |$A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$|⁠ , |${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$|⁠ , with |${\operatorname{Ran}}{\textbf{B}}$| being cyclic for |$A$|⁠ , parametrized by |$d\times d$| Hermitian matrices |${\boldsymbol{\alpha }}$|⁠ , the singular parts of the spectral measures of |$A$| and |$A_{\boldsymbol{\alpha }}$| are mutually singular for all |${\boldsymbol{\alpha }}$| except for a small exceptional set |$E$|⁠. It was shown earlier by the 1st two authors, see [ 4 ], that |$E$| is a subset of measure zero of the space |$\textbf{H}(d)$| of |$d\times d$| Hermitian matrices. In this paper, we show that the set |$E$| has small Hausdorff dimension, |$\dim E \le \dim \textbf{H}(d)-1 = d^2-1$|⁠. [ABSTRACT FROM AUTHOR]

A Fano problem is an enumerative problem of counting |$r$| -dimensional linear subspaces on a complete intersection in |${\mathbb{P}}^n$| over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes |$F_{r}(X)$| as the complete intersection |$X$| varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups |$W(E_6)$| or |$W(D_k)$|⁠. [ABSTRACT FROM AUTHOR]

Motivated by Lang–Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik's theorem and Prokhorov–Shramov's notion of quasi-minimal model but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai's finiteness result for varieties of general type and thereby generalize Noguchi's theorem (formerly Lang's conjecture). [ABSTRACT FROM AUTHOR]

We prove that, under mild hypothesis, every normal algebraic space that satisfies the |$1$| -resolution property is quasi-affine. More generally, we show that for algebraic stacks satisfying similar hypotheses, the 1-resolution property guarantees the existence of a finite flat cover by a quasi-affine scheme. [ABSTRACT FROM AUTHOR]

We show that the product in the quantum K-ring of a generalized flag manifold |$G/P$| involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the |$J$| -function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators. [ABSTRACT FROM AUTHOR]

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated. [ABSTRACT FROM AUTHOR]

We relate factorizable quantum channels on |$M_n({\mathbb{C}})$|⁠ , for |$n \ge 2$|⁠ , via their Choi matrix, to certain matrices of correlations, which, in turn, are shown to be parametrized by traces on the unital free product |$M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$|⁠. Factorizable maps with a finite dimensional ancilla are parametrized by finite dimensional traces on |$M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$|⁠ , and factorizable maps that approximately factor through finite dimensional |$C^\ast $| -algebras are parametrized by traces in the closure of the finite dimensional ones. The latter set of traces is shown to be equal to the set of hyperlinear traces on |$M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$|⁠. We finally show that each metrizable Choquet simplex is a face of the simplex of tracial states on |$M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$|⁠. [ABSTRACT FROM AUTHOR]

We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples |$(G,\{\mu \},[b])$|⁠ , where |$G$| is a flat affine group scheme over |${\mathbb{Z}}_p$| and |$\mu$| is a cocharacter of |$G$| defined over a finite unramified extension of |${\mathbb{Z}}_p$|⁠. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport–Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data. [ABSTRACT FROM AUTHOR]

We prove that for any prime |$p$| the finite |$p$| -groups of fixed coclass have only finitely many different mod- |$p$| cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of fixed rank. [ABSTRACT FROM AUTHOR]

We prove the finiteness of formal analogs of the spherical function (Spherical Finiteness), the |${\textbf c}$| -function (Gindikin–Karpelevich Finiteness), and obtain a formal analog of Harish-Chandra's limit (Approximation Theorem) relating spherical and |${\textbf c}$| -function in the setting of |$p$| -adic Kac–Moody groups. The finiteness theorems imply that the formal analog of the Gindikin–Karpelevich integral is well defined in local Kac–Moody settings. These results extend Braverman–Garland–Kazhdan–Patnaik's affine Gindikin–Karpelevich finiteness theorems from [ 4 ] and provide an algebraic analog of the combinatorial results of Gaussent–Rousseau [ 10 ] and Hébert [ 14 ]. [ABSTRACT FROM AUTHOR]

We give |$L^1$| -norm estimates for exponential sums of a finite sets |$A$| consisting of integers or lattice points. Under the assumption that |$A$| possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee–Pigno–Smith and Konyagin. These theorems improve upon past work of Petridis. [ABSTRACT FROM AUTHOR]

Let $k$ be a perfect field of characteristic $p> 2$ , and let $K$ be a finite totally ramified extension of $W(k)\big[\frac{1}{p}\big]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions, and let $R = R_0\otimes _{W(k)}\mathcal{O}_K$. Examples of such $R$ include $R = \mathcal{O}_K[\![s_1, \ldots , s_d]\!]$ and $R = \mathcal{O}_K\langle t_1^{\pm 1}, \ldots , t_d^{\pm 1}\rangle $. We show that the generalization of Raynaud's theorem on extending $p$ -divisible groups holds over the base ring $R$ when $e < p-1$ , whereas it does not hold when $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$. As an application, we prove that if $R$ has Krull dimension $2$ and $e < p-1$ , then the locus of Barsotti–Tate representations of $\textrm{Gal}(\overline{R}\big[\frac{1}{p}\big]/R\big[\frac{1}{p}\big])$ cuts out a closed subscheme of the universal deformation scheme. If $R = \mathcal{O}_K[\![s]\!]$ with $e \geq p$ , we prove that such a locus is not $p$ -adically closed. [ABSTRACT FROM AUTHOR]

If $G$ is a group, a virtual retract of $G$ is a subgroup, which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts; and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products, and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns. [ABSTRACT FROM AUTHOR]

The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] defined a notion of wave front set for distributions in the $p$ -adic setting and used it to study some representations of $p$ -adic Lie groups. In this article, we work in the $k\mathopen{(\!(} t \mathopen{)\!)}$ -setting with $k$ a Characteristic 0 field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the framework of [13] and [14] for which we define notions of singular support and $\Lambda$ -wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behavior under natural operations like pullback, tensor product, and products of distributions. [ABSTRACT FROM AUTHOR]

We prove that almost every finite collection of matrices in |$GL_d(\mathbb{R})$| and |$SL_d({\mathbb{R}})$| with positive entries is Diophantine. Next we restrict ourselves to the case |$d=2$|⁠. A finite set of |$SL_2({\mathbb{R}})$| matrices induces a (generalized) iterated function system on the projective line |${\mathbb{RP}}^1$|⁠. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent. [ABSTRACT FROM AUTHOR]

We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve |$X/\mathbb{Q}$| whose Jacobian has Mordell–Weil rank larger than its genus. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of a self-similar action of a group |$G$| on a |$k$| -graph |$\Lambda $| and associate it a universal C |$^\ast $| -algebra |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We prove that |${{\mathcal{O}}}_{G,\Lambda }$| can be realized as the Cuntz–Pimsner algebra of a product system. If |$G$| is amenable and the action is pseudo free, then |${{\mathcal{O}}}_{G,\Lambda }$| is shown to be isomorphic to a "path-like" groupoid C |$^\ast $| -algebra. This facilitates studying the properties of |${{\mathcal{O}}}_{G,\Lambda }$|⁠. We show that |${{\mathcal{O}}}_{G,\Lambda }$| is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of |${{\mathcal{O}}}_{G,\Lambda }$| in terms of the underlying action, and we prove that, whenever |${{\mathcal{O}}}_{G,\Lambda }$| is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether |$\Lambda $| has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs. [ABSTRACT FROM AUTHOR]

We study the metrics of constant |$Q$| -curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$\begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}$$ under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every |$n\ge 3$| also in a supercritical regime. Finally, we state some open problems. [ABSTRACT FROM AUTHOR]

An algebra is said to be |$\tau$| -tilting finite provided it has only a finite number of |$\tau$| -rigid objects up to isomorphism. To each such algebra, we associate a category whose objects are the wide subcategories of its category of finite dimensional modules, and whose morphisms are indexed by support |$\tau$| -tilting pairs. [ABSTRACT FROM AUTHOR]

We prove that every finite symmetric integral tensor category |$\mathcal{C}$| with the Chevalley property over an algebraically closed field |$k$| of characteristic |$p>2$| admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik's conjecture [ 25 , Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme |$\mathcal{G}$| over |$k$| and a grouplike element |$\epsilon \in k\mathcal{G}$| of order |$\le 2$|⁠ , whose action by conjugation on |$\mathcal{G}$| coincides with the parity automorphism of |$\mathcal{G}$|⁠ , such that |$\mathcal{C}$| is symmetric tensor equivalent to |$\textrm{Rep}(\mathcal{G},\epsilon)$|⁠. In particular, when |$\mathcal{C}$| is unipotent, the functor lands in |$\textrm{Vec}$|⁠ , so |$\mathcal{C}$| is symmetric tensor equivalent to |$\textrm{Rep}(U)$| for a unique finite unipotent group scheme |$U$| over |$k$|⁠. We apply our result and the results of [ 17 ] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over |$k$| (e.g. local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field |$k$| of characteristic |$p>0$|⁠ , classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field |$k$| of characteristic |$p>0$|⁠. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [ 4 ], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic |$\ne 2$| is always a Serre subcategory. [ABSTRACT FROM AUTHOR]

We study multiplicative dependence between elements in orbits of algebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott's theorem on boundedness of preperiodic points and Siegel's theorem on finiteness of solutions to |$S$| -unit equations. [ABSTRACT FROM AUTHOR]

We construct a natural semiorthogonal decomposition for the derived category of an arbitrary flat family of sextic del Pezzo surfaces with at worst du Val singularities. This decomposition has three components equivalent to twisted derived categories of finite flat schemes of degrees 1, 3, and 2 over the base of the family. We provide a modular interpretation for these schemes and compute them explicitly in a number of standard families. For two such families the computation is based on a symmetric version of homological projective duality for |${{\mathbb{P}}}^2 \times{{\mathbb{P}}}^2$| and |${{\mathbb{P}}}^1 \times{{\mathbb{P}}}^1 \times{{\mathbb{P}}}^1$|⁠ , which we explain in an appendix. [ABSTRACT FROM AUTHOR]

Let |$K$| be a nontrivial knot in |$S^3$| with the exterior |$E(K)$|⁠ , and |$\gamma \in G(K) = \pi _1(E(K), *)$| a slope element represented by an essential simple closed curve on |$\partial E(K)$| with base point |$* \in \partial E(K)$|⁠. Since the normal closure |$\langle \!\langle \gamma \rangle \!\rangle $| of |$\gamma $| in |$G(K)$| coincides with that of |$\gamma ^{-1}$|⁠ , and |$\gamma $| and |$\gamma ^{-1}$| correspond to a slope |$r \in \mathbb{Q} \cup \{ \infty \}$|⁠ , we write |$\langle \!\langle r \rangle \!\rangle = \langle \!\langle \gamma \rangle \!\rangle $|⁠. The normal closure |$\langle \!\langle r \rangle \!\rangle $| describes elements, which are trivialized by |$r$| -Dehn filling of |$E(K)$|⁠. In this article, we prove that |$\langle \!\langle r_1 \rangle \!\rangle = \langle \!\langle r_2 \rangle \!\rangle $| if and only if |$r_1 = r_2$|⁠ , and for a given finite family of slopes |$\mathcal{S} = \{ r_1, \dots , r_n \}$|⁠ , the intersection |$\langle \!\langle r_1 \rangle \!\rangle \cap \cdots \cap \langle \!\langle r_n \rangle \!\rangle $| contains infinitely many elements except when |$K$| is a |$(p, q)$| -torus knot and |$pq \in \mathcal{S}$|⁠. We also investigate inclusion relation among normal closures of slope elements. [ABSTRACT FROM AUTHOR]

The set of Newton strata in a given Iwahori double coset in the loop group of a reductive group |$G$| is indexed by a finite subset of the set |$B(G)$| of Frobenius-conjugacy classes. For unramified |$G$|⁠ , we show that it has a unique minimal element and determine this element. Under a regularity assumption, we also compute the dimension of the corresponding Newton stratum. We derive corresponding results for affine Deligne–Lusztig varieties. [ABSTRACT FROM AUTHOR]

We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori's program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles. [ABSTRACT FROM AUTHOR]

Let |$k$| be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of |$k$| to that of zero-cycles over |$k$| for Kummer varieties over |$k$|⁠. For example, for any Kummer variety |$X$| over |$k$|⁠ , we show that if the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rational points on |$X$| over all finite extensions of |$k$|⁠ , then the (⁠|$2$| -primary) Brauer–Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on |$X$| over |$k$|⁠. We also obtain similar results for products of Kummer varieties, K3 surfaces, and rationally connected varieties. [ABSTRACT FROM AUTHOR]

We study a family of affine varieties arising from a version of an old problem due to Birkhoff asking for the classification of embeddings of finite abelian |$p$| -groups. We show that all of these varieties are irreducible and have a dense orbit. [ABSTRACT FROM AUTHOR]