We prove a theorem that computes, for any augmented operad $\mathcal{O}$, the stable homology of the Lie algebra of derivations of the free algebra $\mathcal{O}(V)$ with twisted bivariant coefficients (here stabilization occurs as $\dim(V)\to\infty$) out of the homology of the wheeled bar construction of $\mathcal{O}$; this can further be used to prove uniform mixed representation stability for the homology of the positive part of that Lie algebra with constant coefficients. This result generalizes both the Loday–Quillen–Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of the Lie algebra of vector fields. We also prove analogous theorems for the Lie algebras of derivations with constant and zero divergence, in which case one has to consider the wheeled bar construction of the wheeled completion of $\mathcal{O}$. Similarly to how cyclic homology of an algebra A may be viewed as an additive version of the algebraic K-theory of A, our results hint at the additive K-theoretic nature of the wheeled bar construction.

We establish a derived geometric Satake equivalence for the quaternionic general linear group ${\textrm{GL}}_{n}({\mathbb H})$. By applying the real–symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$. We show that the stalks are given by the Kostka–Foulkes polynomials for ${\textrm{GL}}_n$ but with all degrees doubled.

We give an explicit quadratic Gröbner basis for generalized Chow rings of supersolvable built lattices, with the help of the operadic structure on geometric lattices introduced in a previous article. This shows that the generalized Chow rings associated to minimal building sets of supersolvable lattices are Koszul. As another consequence, we get that the cohomology algebras of the components of the extended modular operad in genus $0$ are Koszul.

We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax–Schanuel and André–Oort-type theorems in genus zero. As a consequence, one obtains several equivalent characterizations of bi-algebraic varieties. It follows that bi-algebraic varieties in genus zero are foliated by affine-linear varieties. Furthermore, bi-algebraic varieties with constant residues in strata with only simple poles are affine-linear. In addition, we produce infinitely many new linear varieties in strata of genus zero, including infinitely many new examples of meromorphic Teichmüller curves.

In their 1988 paper ‘Gluing of perverse sheaves and discrete series representations’, D. Kazhdan and G. Laumon constructed an abelian category $\mathcal{A}$ associated to a reductive group G over a finite field with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproved in 2001 by R. Bezrukavnikov and A. Polishchuk, who found a counterexample in the case $G = SL_3$. Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension, and suggested that this conjecture could lead toward a proof that Kazhdan and Laumon’s construction is well defined. He proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the present paper, we prove Polishchuk’s conjecture for all types, and prove that Kazhdan and Laumon’s construction is indeed well defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. We go on to show that the concept of Morita equivalence between connected fusion 2-categories corresponds to a notion of Witt equivalence between braided fusion 1-categories. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is $\mathbf{Vect}$ or $\mathbf{SVect}$. We prove that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category.

We prove Chai's conjecture on the additivity of the base change conductor of semiabelian varieties in the case of Jacobians of proper curves. This includes the first infinite family of non-trivial wildly ramified examples. Along the way, we extend Raynaud's construction of the Néron lft-model of a Jacobian in terms of the Picard functor to arbitrary seminormal curves (beyond which Jacobians admit no Néron lft-models, as shown by our more general structural results). Finally, we investigate the structure of Jacobians of (not necessarily geometrically reduced) proper curves over fields of degree of imperfection at most one and prove two conjectures about the existence of Néron models and Néron lft-models due to Bosch–Lütkebohmert–Raynaud for Jacobians of general proper curves in the case of perfect residue fields, thus strengthening the author's previous results in this situation.

Restriction is a natural quasi-order on d-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field; namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property X. For instance, X admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g., homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property.

We show the existence of large $\mathcal C^1$ open sets of area-preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the $\mathcal C^1$ topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class, and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.

For a certain class of real analytic varieties with Lie group actions, we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel’s structure theorem and the endomorphism theorem. We apply these results to give a purely geometric proof of the main result of Bezrukavnikov and Vilonen [Koszul duality for quasi-split real groups, Invent. Math. 226 (2021), 139–193], which proves Soergel’s conjecture [Langlands’ philosophy and Koszul duality, in Algebra – representation theory (Constanta, 2000), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 28 (Kluwer Academic Publishers, Dordrecht, 2001), 379–414] for quasi-split groups.

Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$. We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times }$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm {Br}(L)[p]$ but the mod p Massey product $\langle a,b,c,d\rangle$ is not defined over $L$. Thus, the strong Massey vanishing conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^{* }(\Gamma _L,\mathbb Z/p\mathbb Z)$ of the absolute Galois group $\Gamma _L$ of $L$ is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.

Let J(m) be an $m\times m$ Jordan block with eigenvalue 1. For $\lambda\in\mathbb{C}\setminus\{0,1\}$, we explicitly construct all rank 2 local systems of geometric origin on $\mathbb{P}^1\setminus\{0,1,\lambda,\infty\}$, with local monodromy conjugate to J(2) at $0,1,\lambda$ and conjugate to $-J(2)$ at $\infty$. The construction relies crucially on Katz’s middle convolution operation. We use our construction to prove two conjectures of Sun, Yang and Zuo (one of which was proven earlier by Lin, Sheng and Wang; the other was proven independently of us by Yang and Zuo) coming from the theory of Higgs–de Rham flows, as well as a special case of the periodic Higgs conjecture of Krishnamoorthy and Sheng.

Fix $\alpha >0$. Then by Fejér's theorem $(\alpha (\log n)^{A}\,\mathrm {mod}\,1)_{n\geq 1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided $A>1$. This is the first example of a deterministic sequence modulo $1$ whose gap distribution and all of whose correlations are proven to be Poissonian. The range of $A$ is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of $(\log (n)\, \mathrm {mod}\,1)$, which is necessarily not Poissonian.

We develop a local-to-global formalism for constructing Calabi–Yau structures for global sections of constructible sheaves or cosheaves of differential graded categories. The required data (a morphism between the sheafified Hochschild homology with the topological dualizing sheaf, satisfying a nondegeneracy condition) specializes to the classical notion of orientation when applied to the category of local systems on a manifold. We apply this construction to the cosheaves on arboreal skeleta arising in the microlocal approach to the A-model.

Ramsey’s theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdős conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of $K_k$, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko’s conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common and, in fact, no 2-chromatic common graph unsettled for Sidorenko’s conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known.

We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami et al. [Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742], and a problem listed by Conlon et al. [Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49–118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger et al. [Multiplicities of subgraphs, Combinatorica 16 (1996), 123–131] whether there exists a common graph with chromatic number at least four.

We consider the Bernstein–Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeros of the Bernstein–Sato polynomial. First, we prove each nonvanishing degree of the zeroth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein–Sato polynomial, generalizing a result of M. Saito (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeros of the Bernstein–Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted logarithmic comparison theorem, and more precise statements when f is additionally assumed to be homogeneous. Finally, when f defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeros of the Bernstein–Sato polynomial of f. We show all zeros except the candidate root $-2 + (2 / \deg(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only noncombinatorial candidate root $-2 + (2/ \deg(f))$ is in fact a zero of the Bernstein–Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\deg(f) - 1}$.

Let $k \geqslant 2$ and $b \geqslant 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots , b - 1\}$ are distinct and coprime. Let $\mathcal {S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then every sufficiently large integer is a sum of at most $b^{160 k^2}$ numbers of the form $x^k$, $x \in \mathcal {S}$.

This corrigendum fixes an error in the statement of Lemma 8.3 in the published version of the paper.

We study the Macdonald intersection polynomials $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $\mu ^{(1)},\dots ,\mu ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia–Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia, Haiman, and Tesler. Furthermore, we establish a connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.

The Newell–Littlewood (NL) numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood–Richardson (LR) coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: a description by Extended Horn inequalities (as conjectured in part II of this series), where, using a result of King’s, this description is controlled by the saturated LR-cone and thereby recursive, just like the Horn inequalities; a minimal list of defining linear inequalities; an eigenvalue interpretation; and a factorization of Newell–Littlewood numbers, on the boundary.