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We give a new proof of an unpublished result of Dale Peterson, proved by Lam and Shimozono, which identifies explicitly the structure constants, with respect to the quantum Schubert basis, for the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.

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.

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 $\theta \in [1-p/2,p/2]$.

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.

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>2$ by showing that the Kisin–Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.