This paper explores the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and gives unconditional proofs. Namely, we prove the existence of a universal bound for the dimension of de Rham cohomology of flat bundles with bounded rank and irregularity on surfaces. In any dimension, we prove a Lefschetz recognition principle stating the existence of hyperplane sections distinguishing flat bundles with bounded rank and irregularity after restriction. We obtain in any dimension a universal bound for the degrees of the turning loci of flat bundles with bounded rank and irregularity. Along the way, we introduce a new operation on the group of $b$-divisors on a smooth surface (the partial discrepancy) and prove a closed formula for the characteristic cycles of flat bundles on surfaces in terms of the partial discrepancy of the irregularity $b$-divisor attached to any flat bundle by Kedlaya.

Let $F$ be a totally real field in which $p$ is unramified and let $B$ be a quaternion algebra over $F$ which splits at at most one infinite place. Let $\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place $v|p$, $B$ ramifies at $v$ and $F_v$ is isomorphic to $\mathbb {Q}_p$ and $\overline {r}$ is generic at $v$. We prove that the admissible smooth representations of the quaternion algebra over $\mathbb {Q}_p$ coming from mod $p$ cohomology of Shimura varieties associated to $B$ have Gelfand–Kirillov dimension $1$. As an application we prove that the degree-two Scholze's functor (which is defined by Scholze [On the $p$-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of $\mathrm {GL}_2(\mathbb {Q}_p)$. We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

Tachikawa's second conjecture for symmetric algebras is shown to be equivalent to indecomposable symmetric algebras not having any nontrivial stratifying ideals. The conjecture is also shown to be equivalent to the supremum of stratified ratios being less than $1$, when taken over all indecomposable symmetric algebras. An explicit construction provides a series of counterexamples to Tachikawa's second conjecture from each (potentially existing) gendo-symmetric algebra that is a counterexample to Nakayama's conjecture. The results are based on establishing recollements of derived categories and on constructing new series of algebras.

Let $X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$ or, equivalently, the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$. We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional ‘affine $A_1$-case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.

Compared with algebraic varieties the local monodromy of Drinfeld modules appears to be hopelessly complex: the image of the wild inertia subgroup under Tate module representations is infinite save for the case of potential good reduction. Nonetheless, we show that Tate modules of Drinfeld modules are ramified in a limited way: the image of a sufficiently deep ramification subgroup is trivial. This leads to a new invariant, the local conductor of a Drinfeld module. We establish an upper bound on the conductor in terms of the volume of the period lattice. As an intermediate step we develop a theory of normed lattices in function field arithmetic including the notion of volume. We relate normed lattices to vector bundles on projective curves. With the aid of Castelnuovo–Mumford regularity this implies a volume bound on norms of lattice generators, and the conductor inequality follows. Last but not least we describe the image of inertia for Drinfeld modules with period lattices of rank $1$. Just as in the theory of local $\ell$-adic Galois representations this image is commensurable with a commutative unipotent algebraic subgroup. However, in the case of Drinfeld modules such a subgroup can be a product of several copies of $\mathbf {G}_a$.

The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

Several authors have studied homomorphisms from first homology groups of modular curves to $K_2(X)$, with $X$ either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a $1$-cocycle from $\mathrm {GL}_2(\mathbb {Z})$ to the second $K$-group of the function field of a suitable group scheme over $X$, from which the maps of interest arise by specialization.

We introduce $F$-gauges over a prism, construct syntomic cycle classes, and prove the prismatic Poincaré duality for proper smooth schemes.

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$-type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$. We start by proving a general theorem allowing to lift supercuspidal $\overline {\mathbf {F}}_{\ell }$-representations of $\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline {\mathbf {F}}_{\ell }$-representation $\pi$ of $\operatorname {GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi _\pi$ is irreducible and conjugate-selfdual, then $\pi$ is either $\operatorname {GL}_n(F)$-distinguished or $(\operatorname {GL}_{n}(F),\omega _{E/F})$-distinguished (where $\omega _{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\operatorname {GL}_2(F)$-distinguished representations of $\operatorname {GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\operatorname {PGL}_2$. We show that the ‘classical’ Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil–Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\operatorname {SL}_2(F)$-distinguished modular representations of $\operatorname {SL}_2(E)$.

We use sheaves of spectra to quantize a Hamiltonian $\coprod _n BO(n)$-action on $\varinjlim _{N}T^*\mathbf {R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed by Gaitsgory and Rozenblyum [A study in derived algebraic geometry, vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, 2017)] to give an enrichment of stratified Morse theory by the $J$-homomorphism. This provides a key step in the work of Jin [Microlocal sheaf categories and the $J$-homomorphism, Preprint (2020), arXiv:2004.14270v4] on the proof of a claim of Jin and Treumann [Brane structures in microlocal sheaf theory, J. Topol. 17 (2024), e12325]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold $L\subset T^*\mathbf {R}^N$ is given by the composition of the stable Gauss map $L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow B\mathrm {Pic}(\mathbf {S})$. We put special emphasis on the functoriality and (symmetric) monoidal structures of the categories involved and, as a byproduct, we produce several concrete constructions of (commutative) algebra/module objects and (right-lax) morphisms between them in the (symmetric) monoidal $(\infty, 2)$-category of correspondences, generalizing the construction out of Segal objects of Gaitsgory and Rozenblyum, which might be of independent interest.

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.