In this paper we are going to compute the $ \mathrm {KW} $-Euler classes for rank 2 vector bundles on the classifying stack $ \mathrm {B} \mathrm{N} $, where N is the normaliser of the standard torus in $\mathrm{SL}_2$ and $\mathrm {KW}$ represents Balmer’s derived Witt groups. Using these computations we will recover, through a new and different strategy, the formulas previously obtained by Levine in Witt-sheaf cohomology. In order to obtain our results, we will prove Künneth formulas for products of $\mathrm{GL}_n$’s and $\mathrm{SL}_n$’s classifying spaces and we will develop from scratch the basic theory of twisted symplectic bundles with their associated twisted Borel classes in $\mathrm{SL}$-oriented theories.

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity $1$, threefold singularities, and hypersurface singularities. Given a real value $v>0$, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-$1$ and n-dimensional hypersurface singularities for arbitrary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-$1$ torus action. We exhibit a $3$-dimensional example which shows that this last statement is optimal.

We give a new, short proof of the formula for the first potentially non-zero differential of the Hochschild–Serre spectral sequence for semiabelian varieties over non-closed fields. We show that this differential is non-zero for the Jacobian of a curve when the image of the torsor of theta-characteristics under the Bockstein map is non-zero. An explicit example is a curve of genus 2 whose Albanese torsor is not divisible by 2. When the Albanese torsor is trivial, we show that the Hochschild–Serre spectral sequence for the Jacobian degenerates at the second page. We give a formula for the differential of the Hochschild–Serre spectral sequence for a torus which computes its Brauer group. Finally, we describe the differentials of the Hochschild–Serre spectral sequence for a smooth projective curve, generalising a lemma of Suslin.

Affine Deligne-Lusztig varieties with various level structures show up in the study of Shimura varieties and moduli spaces of shtukas. Among them is the Iwahori level structure which is the most refined one. We study the nonemptiness problem of single affine Deligne-Lusztig varieties at Iwahori level in the basic case. Under a genericity condition (the ‘shrunken Weyl chambers’ condition), an explicit criterion is known. However, no explicit criterion has been available without the condition even conjecturally. We conjecture a new criterion in full generality, and prove it except for finitely many cases. As an application, the nonemptiness problem for special cases and a new conjectural dimension formula are discussed.

We introduce the notion of symplectic determinant laws in analogy to Chenevier’s definition of determinant laws. Symplectic determinant laws are a way to define pseudorepresentations for symplectic representations of algebras with involution over arbitrary $\mathbb Z[\tfrac {1}{2}]$-algebras. We prove that this notion satisfies the properties expected from a good theory of pseudorepresentations, and we compare it to V. Lafforgue’s $\operatorname {Sp}_{2d}$-pseudocharacters. In the process, we compute generators of the invariant algebras $A[M_{2d}^m]^{\operatorname {Sp}_{2d}}$ and $A[G^m]^G$ when $G \in \{\operatorname {Sp}_{2d}, \mathrm O_d, \operatorname {GSp}_{2d} and\ \operatorname {GO}_d\}$ over an arbitrary commutative ring A, generalising results of Zubkov.

In this paper we take the first steps towards the categorification of the Zelevinsky tensor product of finite-dimensional representations of extended affine type A Hecke algebras.

Let $E/F$ be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch–Zagier cycles arising from the associated Hilbert modular varieties can be put in p-adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable p-adic adjoint L-function twisted by the Hecke character of $E/F$, attached to Hida families of Hilbert modular forms over F.

We construct a relative version of the $A_{ {\mathrm {inf}}}$-cohomology theory developed by Bhatt-Morrow-Scholze and relate it to the prismatic theory of Bhatt-Scholze. The construction relies on the fiber product of topoi. As an application we show that there is an étale comparison of the q-crystalline pushforward after inverting $\mu \in A_{ {\mathrm {inf}}}$.

Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity to prove that certain multipliers are bounded on the noncommutative $L_p$-spaces of a free group. Here, we study Cotlar-type identities in full generality, giving a closed characterization for them in terms of m:

$$\begin{align*}\big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G}\setminus \{e\}, h \in \mathrm{G}. \end{align*}$$

$\mathbb{R}$

$\mathrm{G}$

$\widetilde{m}: X \to \mathbb{C}$

$X \setminus \{x_0\}$

$T_m$

$L_p$

$1 < p < \infty$

This result establishes a new connection between group actions on $\mathbb{R}$-trees and Fourier multipliers. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson’s subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers on groups. These include new families such as Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm {PSL}_2(\mathbb{C})$ satisfies Cotlar’s identity when restricted to the Bianchi group $\mathrm {PSL}_2(\mathbb{Z}[\sqrt{-1}])$.

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.

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.

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.