We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

• – a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

• – a novel definition of double and triple Stanley symmetric functions;

• – a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

• – the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

• – the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

• – equivariant Pieri rules for the homology of the infinite Grassmannian;

• – homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

We consider $C^{r}$-diffeomorphisms ($1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

We prove the analog of the Morel–Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.

We explain how to form a novel dataset of Calabi–Yau threefolds via the Gross–Siebert algorithm. We expect these to degenerate to Calabi–Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities. In particular, we explain how to ‘smooth the boundary’ of a class of four-dimensional reflexive polytopes to obtain polarised tropical manifolds. We compute topological invariants of a compactified torus fibration over each such tropical manifold, expected to be homeomorphic to the general fibre of the Gross–Siebert smoothing. We consider a family of examples related to products of reflexive polygons. Among these we find $14$ topological types with $b_2=1$ that do not appear in existing lists of known rank-one Calabi–Yau threefolds.

We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$. Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$-modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$.

We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

We develop a dimension theory for coadmissible $\widehat {\mathcal {D}}$-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic $\mathcal {D}$-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $\underline {H}^{i}_Z(\mathcal {M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal {M}$ on $X$.

We prove several results showing that the algebraic $K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,

\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]

$\operatorname {Tr}_{L/k}$

$\operatorname {GW}(L) \to \operatorname {GW}(k)$

${\mathbf {C}}$

${\mathbf {R}}$

$\mathbf {A}^1$

A spectral sequence is established whose $E_{2}$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $E_{2}$ page is a characteristic-2 version of $F_{5}$ homology in Khovanov's classification.

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

Nous développons dans cet article des techniques d'aplatissement des faisceaux cohérents en géométrie de Berkovich, en nous inspirant de la stratégie générale que Raynaud et Gruson ont mise en œuvre pour traiter le problème analogue en théorie des schémas. Nous donnons ensuite quelques applications à l’étude des morphismes entre espaces analytiques compacts, et obtenons notamment une description de l'image d'un tel morphisme.

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

In this paper we give an interpretation, in terms of derived de Rham complexes, of Scholze's de Rham period sheaf and Tan and Tong's crystalline period sheaf.

We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{\infty }$-algebra which is freely generated of type ${\mathcal {W}}(2,3,\ldots )$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal ${\mathcal {W}}$-algebras ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ for $n\geq 2$. We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$-algebras.