We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

We show that for any commutative Noetherian regular ring $R$ containing $\mathbb{Q}$, the map $K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.

In the framework of Berthelot’s theory of arithmetic ${\mathcal{D}}$-modules, we prove the $p$-adic analogue of Betti number estimates and we give some standard applications.

Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

In this paper we extend the generalized algebraic fundamental group constructed in Esnault and Hogadi, (Trans. Amer. Math. Soc. 364(5) (2012), 2429–2442) to general fibered categories using the language of gerbes. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined in Borne and Vistoli (J. Algebraic Geom. (2014), S1056–3911, 00638-X) for nonsmooth non-pseudo-proper algebraic stacks.

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$

$L(a)\unicode[STIX]{x1D6E5}u$

$u\cdot \unicode[STIX]{x1D6FB}u$

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations:

$$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$

$\unicode[STIX]{x1D70B}$

$d\geqslant 1$

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

We introduce the notion of regularity for a relative holonomic ${\mathcal{D}}$-module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative ${\mathcal{D}}$-modules underlying a regular mixed twistor ${\mathcal{D}}$-module, this functor satisfies the left quasi-inverse property.

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy $K$-theory groups of a Noetherian scheme $X$ of Krull dimension $d$ vanish below $-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy $K$-theory groups vanish below $-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy $K$-theory group.

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble $\text{CLE}_{4}$ and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the $\text{CLE}_{4}$ carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of $\text{CLE}_{4}$) are in fact measurable functions of the GFF.