We prove a version of Morel’s unstable $\mathbb {A}^1$-connectivity theorem over arbitrary base schemes. In the stable setting, this recovers (and simplifies the proof of) the known connectivity bounds due to Morel, Schmidt–Strunk, Deshmukh–Hogadi–Kulkarni–Yadav, and Druzhinin, and extends them to possibly non-noetherian schemes. Using the recent work of Bachmann–Elmanto–Morrow, this also implies that the slice filtration on homotopy K-theory is convergent for qcqs schemes of finite valuative dimension.

We examine sets $\mathscr A$ of natural numbers having the property that for some real number $p\in (0,2)$, one has the subconvex bound

$$\begin{align*}\int_0^1 \Big| \sum_{n\in \mathscr A\cap [1,N]}e(n\alpha)\Big|^p{\,\mathrm{d}} \alpha \ll N^{-1}|\mathscr A\cap [1,N]|^p. \end{align*}$$

We show that exponential sums over such sets satisfy inequalities analogous to Weyl’s inequality, and in many circumstances of the same strength as classical versions of Weyl’s bound. We also examine equidistribution of polynomials modulo $1$ in which the summands are restricted to these subconvex $L^p$-sets. In addition, we describe applications to problems involving character sums and averages of arithmetic functions.

Building on Beilinson’s work, ‘constructible sheaves are holonomic’, we introduce the notion of holonomicity for étale sheaves, without assuming a priori constructibility. We establish the converse of Beilinson’s result, showing that holonomic sheaves are indeed constructible. This can be seen as an étale analogue of Kashiwara’s theorem on holonomic ${\mathcal D}_X$-modules.

We compute the $K$-theory of the ${C}^{*}$-category generated by order zero, equivariant, properly supported, classical pseudodifferential operators acting on sections of homogeneous bundles over the symmetric space of a real reductive Lie group $G$. Our result uses the Connes-Kasparov isomorphism for $G$, and in fact it is equivalent to the Connes-Kasparov isomorphism. We relate our computation to David Vogan’s well-known parametrization of the tempered irreducible representations of $G$ with real infinitesimal character. When the reductive group $G$ has real rank one, we formulate and prove a Fourier isomorphism theorem for equivariant order zero pseudodifferential operators on the symmetric space, and use it to prove a $K$-theoretic version of Vogan’s theorem.

We compute a Hirzebruch-Riemann-Roch type formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega ,{ \mathcal J})$, allowing $0$ to be a singular value of the moment map ${ \mathcal J}:M\to {\mathbb R}$. Our formula represents an instance of the Guillemin-Sternberg principle, which states that quantization should commute with reduction. The conceptual novelty of our result is that the involved reduced system only depends on the symplectic data of M. To establish this, we derive a complete singular stationary phase expansion of the Witten integral without appealing to any kind of desingularization. As a consequence, our formula expresses the invariant Riemann-Roch number purely in terms of symplectic invariants of the singular symplectic quotient. In particular, it involves a new explicit symplectic invariant of the singularities.

We give a general construction of the motivic path space based at tangential basepoints, extending previous works of P. Deligne, A. B. Goncharov and M. Levine, which were limited to ordinary basepoints or to specific varieties. Given a smooth variety over a field endowed with a simple normal crossings divisor, we encode its tangential basepoints using the language of logarithmic geometry. Building on the recent construction by F. Binda, D. Park and P. A. Østvær of a stable $\infty $-category of $\mathbb {A}^1$-invariant logarithmic motives and its comparison with the usual category of motives, we define in a functorial manner the associated motivic pointed path spaces. In the presence of a motivic t-structure, truncating yields the motivic fundamental groupoid. In general, we construct Betti and de Rham realization functors for logarithmic motives (linearizing the construction of F. Binda, D. Park and P. A. Østvær for the Betti case) and we show that the periods of the motivic fundamental groupoid are given by regularized iterated integration of logarithmic differential $1$-forms, thus yielding a general version of Chen’s theorem with tangential basepoints.

This work is the second in a series, following Part I [Kwa24] (Algebra Number Theory 18.10 (2024)) and preceding Part III [Kwa25] (Math. Ann. 391.1 (2025)). We continue our investigation of spectral moments of $\text {GL}(3)\times \text {GL}(2)\ L$-functions from the perspective of period integrals. Using an identity between two distinct periods for the $\text {GL}(3)$ Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of $\text {GL}(2)\ L$-functions to the shifted fourth moment of $\text {GL}(1) L$-functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (Proc. Lond. Math. Soc. (3) 91 (2005)).

Given a smooth variety X over $\mathbb {C}$, a smooth divisor $i:Y\hookrightarrow X$ and a global function f on X which vanishes on Y and on its critical locus, we compute the map induced on Hochschild homology by the pushforward functor $i_{\ast }:D^b(Y)\to D^{abs}(MF(X,f))$ in terms of the Hochschild-Kostant-Rosenberg isomorphisms.

We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to that of each piece. This result solves a conjecture proposed at a conference in Göttingen in 2003. This result also leads to a higher Cheeger-Müller/Bismut-Zhang theorem. Our approach combines an adiabatic limit along the normal direction of the hypersurface and a Witten-type deformation on the flat vector bundle.

The rational homology of the IA-automorphism group $\operatorname {IA}_n$ of the free group $F_n$ is still mysterious. We study the quotient of the rational homology of $\operatorname {IA}_n$ that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of $\operatorname {IA}_n$. We obtain a representation-stable $\operatorname {GL}(n,\mathbb {Q})$-subquotient of the Albanese homology of $\operatorname {IA}_n$, which conjecturally coincides with the entire Albanese homology of $\operatorname {IA}_n$. In particular, we obtain a lower bound of the dimension of the Albanese homology of $\operatorname {IA}_n$ for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of $\operatorname {IA}_n$ for $n\ge 9$. We also study the Albanese homology of an analogue of $\operatorname {IA}_n$ for the outer automorphism group of $F_n$ and the Albanese homology of the Torelli groups of surfaces. Moreover, we study the relation between the Albanese homology of $\operatorname {IA}_n$ and the cohomology of $\operatorname {Aut}(F_n)$ with twisted coefficients.

In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in [9] via the geometry of orthogonal and hyperbolic Grassmannians over S coincides with the motivic ring spectrum constructed in [4].

We compute the rate of exponential growth of the free inverse monoid of rank r (and hence an upper bound on the corresponding rate for all r-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt {e(2r-1)}$ as $r \to \infty $.

We study analytic torsion and eta-like invariants on contact manifolds admitting a CR structure invariant under a transverse circle action, and equipped with a unitary representation. We show that, when defined using the spectrum of relevant operators arising in this geometry, the spectral series involved can been interpreted in their whole, both from a topological viewpoint, and as purely dynamical functions of the Reeb flow.

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between parabolic induction/restriction and Hermitian duals, Bernstein’s second adjointness and generalizations of the Langlands classification. We check that, in the known cases of equivalences between module categories of affine Hecke algebras and Bernstein blocks for reductive p-adic groups, such equivalences preserve Hermitian duality.

We also initiate the study of generic representations of affine Hecke algebras. Based on an analysis of the Hecke algebras associated to generic Bernstein blocks for quasi-split reductive p-adic groups, we propose a fitting definition of genericity for modules over affine Hecke algebras. With that notion we prove special cases of the generalized injectivity conjecture, about generic subquotients of standard modules for affine Hecke algebras.

We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in $\mathcal {M}_g$ is not Zariski dense when $g\geq 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to $\mathrm {GSp}_{2g}$; these components are defined over $\overline {\mathbb Q}$, and their union is closed under the action of $\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with properties which are similar to the unstable and the stable foliation of a pseudo-Anosov homeomorphism.

Let M be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalising results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $T^1M$, and prove their equidistribution towards the Bowen-Margulis measure. The result holds in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

This is the first part of a series of papers devoted to the study of linear cocycles over chaotic systems. In the present paper, we establish the existence of such cocycles that $\mathcal {C}^\alpha $-stably exhibit fiberwise bounded orbits ($\alpha>0$). The proof is based on a new mechanism which yields stable elliptic-type behavior in $\mathrm {GL}(d,\mathbb {R})$ or $\mathrm {SL}(d,\mathbb {R})$ cocycles. Moreover, we show that this phenomenon is $\mathcal {C}^0$-dense among $\mathrm {SL}(d,\mathbb {R})$ cocycles over a shift of finite type without dominated splitting.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.