For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.

Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the canonical ribbon conjecture of Bayer and Eisenbud holds over a field of characteristic $0$ or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.

Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$. Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$. We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.

The trace of the $n$-framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

We prove that a formula for the ‘pluricanonical’ double ramification cycle proposed by Janda, Pandharipande, Pixton, Zvonkine, and the second-named author is in fact the class of a cycle constructed geometrically by the first-named author. Our proof proceeds by a detailed explicit analysis of the deformation theory of the double ramification cycle, both to first and to higher order.

We establish some cohomological bounds in $D$-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $b$-function lemma for non-holonomic $D$-modules.

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 study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

We revisit the paper [Automorphy lifting for residually reducible $l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

The commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

Let $G$ be a connected split reductive group over a finite field $\mathbb{F}_{q}$ and $X$ a smooth projective geometrically connected curve over $\mathbb{F}_{q}$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\unicode[STIX]{x1D700}$ is any positive real number and $s$ is large enough in terms of $\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$ that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.