We study the injectivity of the Kudla–Millson lift of genus-2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree-4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L. We describe their behavior with respect to the split of a hyperbolic plane in L. This generalizes the results of Borcherds to genus higher than 1.

We introduce and explore the Uniform Izumi–Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if R is a normal domain essentially of finite type over a field, there exists a constant C such that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.

We provide a complete classification of when the homeomorphism group of a stable surface, $\Sigma$, has the automatic continuity property: Any homomorphism from $\mathrm{Homeo}({\Sigma})$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable, second-countable Stone space has the automatic continuity property. Under the presence of stability, this answers two questions of Mann.

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) $\mathcal{D}$-modules. We employ Kazhdan–Lusztig theory and Saito’s theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck–Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian; in particular, showing that the weight filtration is controlled by the admissible augmented Dyck patterns of Raicu and Weyman. As an application, upon restriction to the opposite big cell, we recover the composition factors and weight filtration on local cohomology with support in generic determinantal varieties.

We study the geometry of Calabi–Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian Yang–Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov–Hausdorff topology.

We give a categorical formulation of the p-adic local Langlands correspondence for ${\mathrm{GL}}_2({{{\mathbb{Q}_{p}}}})$ as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of$\mathrm{Gal}(\overline{{\mathbb{Q}}}_p/{{{\mathbb{Q}_{p}}}})$. The Montréal functor appears as the‘Whittaker coefficient’ for the universal Galois representation, in the sense of the geometric Langlands program. Moreover, we relate our version of the p-adic local Langlands correspondence for ${\mathrm{GL}}_2({{{\mathbb{Q}_{p}}}})$ to the cohomology of modular curves through a local–global compatibility formula.

Let k be a perfect field of characteristic p and let W(k) be its ring of Witt vectors. We construct an equivalence of categories between the full subcategory of the derived category of quasi-coherent sheaves on the syntomification of W(k) spanned by objects whose Hodge–Tate weights are between $[0,p-2]$ and an appropriate derived category of Fontaine–Laffaille modules.

Kreck proved that two 2q-manifolds are stably diffeomorphic if and only if they admit normally bordant normal $(q{-}1)$-smoothings over the same normal $(q{-}1)$-type $(B,\xi)$. We show that ‘stably diffeomorphic’ can be replaced by ‘diffeomorphic’ if the normal smoothings have isomorphic Q-forms (consisting of the intersection form of the manifold and the induced homomorphism on $H_q$), when the manifolds are simply-connected, $q=2k$ is even and $H_q(B)$ is free. This proves a special case of Crowley’s Q-form conjecture. The basis of the proof is the construction of an extended surgery obstruction associated to a normal bordism. As an application, we identify the inertia group of a $(2k{-}1)$-connected 4k-manifold with the kernel of a certain bordism map. By the calculations of Senger and Zhang and earlier results, these kernels are now known in all cases. For $k=2,4$, the combination of these results determines the inertia groups. We also obtain, for a simply-connected 4k-manifold M with normal $(2k{-}1)$-type $(B,\xi)$ such that $H_{2k}(B)$ is free, an algebraic description of the stable class of M, that is, the set of diffeomorphism classes of manifolds stably diffeomorphic to M. Using this description, we explicitly compute the stable class of manifolds M with rank-2 hyperbolic intersection form.

We prove the vanishing of bounded cohomology with separable dual coefficients for many groups of interest in geometry, dynamics, and algebra. These include compactly supported structure-preserving diffeomorphism groups of certain manifolds; the group of interval exchange transformations of the half line; piecewise linear and piecewise projective groups of the line, giving strong answers to questions of Calegari and Navas; direct limit linear groups of relevance in algebraic K-theory, thereby answering a question by Kastenholz and Sroka and a question of two of the authors and Löh; and certain subgroups of big mapping class groups, such as the stable braid group and the stable mapping class group, proving a conjecture of Bowden. Moreover, we prove that in the recently introduced framework of enumerated groups, the generic group has vanishing bounded cohomology with separable dual coefficients. At the heart of our approach is an elementary algebraic criterion called the commuting cyclic conjugates condition that is readily verifiable for the aforementioned large classes of groups.

We give a proof of Mukai’s theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai’s biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our approach is based on Lazarsfeld’s construction that produces vector bundles on a variety from globally generated line bundles on a divisor, on Mukai’s theory of stable vector bundles on K3 surfaces, and on Brill–Noether properties of curves and (sensu Mukai) of K3 surfaces.

We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a $\mathrm{ku}_\ast$-module using the descent spectral sequence for the map $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which is the motivic spectral sequence for $\mathrm{THH}(\mathrm{ku})$ in the sense of Hahn–Raksit–Wilson. We reduce the computation of homotopy groups to the algebra of the universal formal group law, providing a systematic way to compute THH of quotients of MU. We compute the $E_2$-page of the motivic spectral sequence computing $\operatorname{THH}(\mathrm{ku})$, and we show that it degenerates at the $E_2$-page.

We establish the following family version of Habegger’s bounded height theorem on abelian varieties [Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Invent. Math. 176 (2009a), 405–447]: a locally closed subvariety of an abelian scheme with Gao’s tth degeneracy locus [Gao, Generic rank of Betti map and unlikely intersections, Compositio Math. 156 (2020a), 2469–2509] removed, intersected with all flat group subschemes of relative dimension at most t, gives a set of bounded total height. Our main tools include the Ax–Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan and Zhang [Adelic line bundles on quasi-projective varieties, Annals of Mathematics Studies, vol. 221 (Princeton University Press, Princeton, NJ, 2026)]. As two applications, we generalize Silverman’s specialization theorem [Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211] to a higher-dimensional base, and establish a bounded height result towards Zhang’s ICM conjecture [Zhang, Small points and Arakelov theory, in Proceedings of the international congress of mathematicians, Vol. II (Berlin, 1998), Extra Vol. II (1998), 217–225].

We propose a method for computing generating functions of genus-zero invariants of a gauged linear sigma model (GLSM) $(V, G, \theta, w)$. We show that certain derivatives of I-functions of quasimap invariants of $[V\mathbin{/\mkern-6mu/}_\theta G]$ produce I-functions (appropriately defined) of the GLSM. When G is an algebraic torus, we obtain an explicit formula for an I-function, and check that it agrees with previously computed I-functions in known special cases. Our approach is based on a new construction of these invariants that applies whenever the evaluation maps from the moduli space are proper, and includes insertions from light marked points, which may collide with each other and with basepoints.

For log canonical (lc) algebraically integrable foliations on Kawamata log terminal (klt) varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.

We construct a family of special cycle classes on the regular integral model of an orthogonal Shimura variety, and show that these cycle classes appear as Fourier coefficients of a Siegel modular form. Passing to the generic fiber of the Shimura variety recovers a result of Bruinier and Raum, originally conjectured by Kudla.

We show that every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite. As a consequence, the crossed products of minimal such actions are ${\mathcal{Z}}$-stable and classified by their Elliott invariant.

Consider a quadratic polynomial $Q(\xi_{1},\ldots,\xi_{n})$ of independent Rademacher random variables $\xi_{1},\ldots,\xi_{n}$. To what extent can $Q(\xi_{1},\ldots,\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if $Q(\xi_{1},\ldots,\xi_{n})$ ‘robustly depends on at least m of the $\xi_{i}$’ in the sense that there is no way to pin down the value of $Q(\xi_{1},\ldots,\xi_{n})$ by fixing values for fewer than m of the variables $\xi_{i}$, then we have $\mathrm{Pr}[Q(\xi_{1},\ldots,\xi_{n})=0]\le O(1/\sqrt{m})$. This also implies a similar result in the case where $\xi_{1},\ldots,\xi_{n}$ have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility.

We show that maximal analytic Hardy fields are $\eta_1$ in the sense of Hausdorff. We also prove various embedding theorems about analytic Hardy fields. For example, the ordered differential field $ {\mathbb T} $ of transseries is shown to be isomorphic to an analytic Hardy field.

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.