Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants.

For a Lie algebra ${\mathfrak g}$ of holomorphic vector fields on a complex manifold M, any holomorphic ${\mathfrak g}$-invariant hypersurface is given in terms of a ${\mathfrak g}$-invariant divisor. This generalizes the classical notion of scalar relative ${\mathfrak g}$-invariant. Any ${\mathfrak g}$-invariant divisor gives rise to a ${\mathfrak g}$-equivariant line bundle, and a large part of this paper is therefore devoted to the investigation of the group $\mathrm {Pic}_{\mathfrak g}(M)$ of ${\mathfrak g}$-equivariant line bundles. We give a cohomological description of $\mathrm {Pic}_{\mathfrak g}(M)$ in terms of a double complex interpolating the Chevalley-Eilenberg complex for ${\mathfrak g}$ with the Čech complex of the sheaf of holomorphic functions on M.

We also obtain results about polynomial divisors on affine bundles and jet bundles. This has applications to the theory of differential invariants. Those were actively studied in relation to invariant differential equations, but the description of multipliers (or weights) of relative differential invariants was an open problem. We derive a characterization of them with our general theory. Examples, including projective geometry of curves and second-order ODEs, not only illustrate the developed machinery but also give another approach and rigorously justify some classical computations. At the end, we briefly discuss generalizations of this theory.

Let M be a von Neumann algebra, let $\varphi $ be a normal faithful state on M and let $L^p(M,\varphi )$ be the associated Haagerup noncommutative $L^p$-spaces, for $1\leq p\leq \infty $. Let $D\in L^1(M,\varphi )$ be the density of $\varphi $. Given a positive map $T\colon M\to M$ such that $\varphi \circ T\leq C_1\varphi $ for some $C_1\geq 0$, we study the boundedness of the $L^p$-extension $T_{p,\theta }\colon D^{\frac {1-\theta }{p}} MD^{\frac {\theta }{p}}\to L^p(M,\varphi )$ which maps $D^{\frac {1-\theta }{p}} x D^{\frac {\theta }{p}}$ to $D^{\frac {1-\theta }{p}} T(x) D^{\frac {\theta }{p}}$ for all $x\in M$. Haagerup–Junge–Xu showed that $T_{p,{\frac {1}{2}}}$ is always bounded and left open the question whether $T_{p,\theta }$ is bounded for $\theta \not ={\frac {1}{2}}$. We show that for any $1\leq p<2$ and any $\theta \in [0, 2^{-1}(1-\sqrt {p-1})]\cup [2^{-1}(1+\sqrt {p-1}), 1]$, there exists a completely positive T such that $T_{p,\theta }$ is unbounded. We also show that if T is $2$-positive, then $T_{p,\theta }$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $\theta \in [1-p/2,p/2]$.

We define a local homomorphism $(Q,k)\to (R,\ell )$ to be Koszul if its derived fiber $R\otimes ^{\mathsf {L}}_Q k$ is formal, and if $\operatorname {Tor}^{Q}(R,k)$ is Koszul in the classical sense. This recovers the classical definition when Q is a field, and more generally includes all flat deformations of Koszul algebras. The non-flat case is significantly more interesting, and there is no need for examples to be quadratic: all complete intersection and all Golod quotients are Koszul homomorphisms. We show that the class of Koszul homomorphisms enjoys excellent homological properties, and we give many more examples, especially various monomial and Gorenstein examples. We then study Koszul homomorphisms from the perspective of $\mathrm {A}_{\infty }$-structures on resolutions. We use this machinery to construct universal free resolutions of R-modules by generalizing a classical construction of Priddy. The resulting (infinite) free resolution of an R-module M is often minimal and can be described by a finite amount of data whenever M and R have finite projective dimension over Q. Our construction simultaneously recovers the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring, and produces analogous resolutions for various other classes of local rings.

We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris.

We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the T-equivariant quantum cohomology $QH^{\bullet }_T(G/P)$ of any flag variety $G/P$ with the structure constants, with respect to the affine Schubert basis, for the T-equivariant Pontryagin homology $H^T_{\bullet }(\mathcal {G}r)$ of the affine Grassmannian $\mathcal {G}r$ of G, where G is any simple simply-connected complex algebraic group.

Our approach is to construct an $H_T^{\bullet }(pt)$-algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson’s map. More precisely, the map is defined via Savelyev’s generalized Seidel representations, which can be interpreted as certain Gromov-Witten invariants with input $H^T_{\bullet }(\mathcal {G}r)\otimes QH_T^{\bullet }(G/P)$. We determine these invariants completely, in a way similar to how Fulton and Woodward did in their proof of the quantum Chevalley formula.

For a smooth affine group scheme G over the ring of p-adic integers and a cocharacter $\mu $ of G, we develop the deformation theory for G-$\mu $-displays over the prismatic site of Bhatt–Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic F-gauges introduced by Drinfeld and Bhatt–Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of p-divisible groups.

We define a new class of enumerative invariants called k-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the k-leaky double Hurwitz numbers introduced in [CMR25]. These numbers are defined as intersection numbers of the logarithmic DR cycle against $\psi $-classes and logarithmic classes coming from piecewise polynomials encoding fixed branch point conditions. We give a tropical graph sum formula for these new invariants, allowing us to show their piecewise polynomiality in any genus. Investigating the piecewise polynomial structure further (and restricting to genus zero for this purpose), we also show a wall-crossing formula. We also prove that in genus zero the invariants are always nonnegative and give a complete classification of the cases where they vanish.

We write arbitrary separable nuclear $\mathrm {C}^*$-algebras as limits of inductive systems of finite-dimensional $\mathrm {C}^*$-algebras with completely positive connecting maps. The characteristic feature of such ${\mathrm {CPC}^*}$-systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a $\mathrm {C}^*$-algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.

Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$. Given an exterior Lipschitz domain $\Omega $, let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform $\nabla \mathcal {L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$, even if $\mathcal {L}=\Delta $ is the Laplace operator and $\Omega $ is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \Omega $ is $C^1$. We prove that for $p>2$ and $p\in [n,\infty )$, it holds that

$$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$

for $f\in \mathring {W}^{1,p}(\Omega )$. Here, $\mathcal {K}_p(\mathcal {L}_D^{1/2})$ is the kernel of $\mathcal {L}_D^{1/2}$ in $\mathring {W}^{1,p}(\Omega )$, which coincides with $\tilde {\mathcal {A}}^p_0(\Omega ):=\{f\in \mathring {W}^{1,p}(\Omega ):\ \mathcal {L}_Df=0\}$ and is a one-dimensional subspace. As an application, we provide a substitution of $L^p$-boundedness of $\sqrt {t}\nabla e^{-t\mathcal {L}_D}$ which is uniform in t for $p\ge n$ and $p>2$.

Anosov representations of hyperbolic groups form a rich class of representations that are closely related to geometric structures on closed manifolds. Any Anosov representation $\rho :\Gamma \to G$ admits cocompact domains of discontinuity in flag varieties $G/Q$ [GW12, KLP18] endowing the compact quotient manifolds $M_\rho $ with a $(G,G/Q)$–structure. In general, the topology of $M_\rho $ can be quite complicated.

In this article, we will focus on the special case when $\Gamma $ is a the fundamental group of a closed (real or complex) hyperbolic manifold N and $\rho $ is a deformation of a (twisted) lattice embedding $\Gamma \to \mathrm {Isom}^\circ (\mathbb {H}_{\mathbb {K}}) \to G$ through Anosov representations. In this case, we prove that $M_\rho $ is a smooth fiber bundle over N, and we describe the structure group of this bundle and compute its invariants. This theorem applies in particular to most representations in higher rank Teichmüller spaces, as well as convex divisible representations, AdS-quasi-Fuchsian representations and $\mathbb {H}_{p,q}$–convex cocompact representations.

Even when $M_\rho \to N$ is a fiber bundle, it is often very difficult to determine the fiber. In the second part of the paper, we focus on the special case when N is a surface, $\rho $ a quasi-Hitchin representation into $\mathrm {Sp}(4,{\mathbb C})$, and $M_\rho $ carries a $(\mathrm {Sp}(4,{{\mathbb C}}),\mathrm {Lag}({{\mathbb C}}^4))$–structure. We show that in this case the fiber is homeomorphic to ${{\mathbb C}}\mathbb {P}^2 \# \overline {{{\mathbb C}}\mathbb {P}}^2$.

This fiber bundle $M_\rho \to N$ is of particular interest in the context of possible generalizations of Bers’ double uniformization theorem in the context of higher rank Teichmüller spaces, since for Hitchin-representations it contains two copies of the locally symmetric space associated to $\rho (\Gamma )$. Our result uses the classification of smooth $4$–manifolds, the study of the $\mathrm {SL}(2, {{\mathbb C}})$–orbits of $\mathrm {Lag}({{\mathbb C}}^4)$ and the identification of $\mathrm {Lag}({{\mathbb C}}^4)$ with the space of (unlabelled) regular ideal hyperbolic tetrahedra and their degenerations.

We compute the open Gromov-Witten disk invariants and the relative quantum cohomology of the Chiang Lagrangian $L_\triangle \subset \mathbb {C}P^3$. Since $L_\triangle $ is not fixed by any anti-symplectic involution, the invariants may augment straightforward J-holomorphic disk counts with correction terms arising from the formalism of Fukaya $A_\infty $-algebras and bounding cochains. These correction terms are shown in fact to be nontrivial for many invariants. Moreover, examples of nonvanishing mixed disk and sphere invariants are obtained.

We characterize a class of open Gromov-Witten invariants, called basic, which coincide with straightforward counts of J-holomorphic disks. Basic invariants for the Chiang Lagrangian are computed using the theory of axial disks developed by Evans-Lekili and Smith in the context of Floer cohomology. The open WDVV equations give recursive relations which determine all invariants from the basic ones. The denominators of all invariants are observed to be powers of $2$ indicating a nontrivial arithmetic structure of the open WDVV equations. The magnitude of invariants is not monotonically increasing with degree. Periodic behavior is observed with periods $8$ and $16.$

An ordered r-matching is an r-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of r-dimensional orders. The theory of ordered 2-matchings is well developed and has connections and applications to extremal and enumerative combinatorics, probability and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered r-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.

In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic 1-space.

We prove that, for any finite set of minimal r-graph patterns, there is a finite family $\mathcal F$ of forbidden r-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum r-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author [30], where the above statement was established for a single pattern.

We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order n and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal {F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.

This paper focuses on dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. More precisely, the paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a gas consisting of hard spheres, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time. We expect that this paper can serve as a guideline for deriving a generalized Boltzmann equation that incorporates higher-order interactions among particles.

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet })$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet })$ can be ‘corrected’ into a complex $({\mathcal {A}}_{\bullet })$ of pseudodifferential operators, where ${\mathcal {A}}_k - A_k$ is a zero-order correction within this class. The induced complex $({\mathcal {A}}_{\bullet })$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

In this paper, we study the sphere packing problem in Euclidean space where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii r, such that no two centers $x_1$ and $x_2$ satisfy $\sqrt {\tfrac {4}{3}} < \frac {1}{2r}|x_1-x_2| <\sqrt {\tfrac {5}{3}}$, has center density less or equal than $(3/2)^{24}$. Equality occurs for periodic packings if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem, and gives evidence towards the conjecture that extremal lattices are optimal unconstrained sphere packings in $48$ dimensions. We also provide results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, showing that even unimodular extremal lattices are again uniquely optimal. Moreover, in the one-dimensional case, where it is not at all clear that periodic packings are among those with largest density, we nevertheless give a condition on the set of constraints that allows this to happen, and we develop an algorithm to find these periodic configurations by relating the problem to a question about dominos.