In this work, we establish a local-smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. As an application, we prove that the Gromov–Hausdorff limit of compact manifolds with bounded curvature concentration, Ahlfors n-regularity, and bounded Sobolev constants is a topological manifold away from finitely many points. In the complete non-compact case, we show that manifolds with Euclidean-type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces. In contrast with all previous works, we remove the Ricci-curvature condition.

David Vogan proved that if G is a real reductive group, and if K is a maximal compact subgroup of G, then every irreducible representation of K is included as a minimal K-type in precisely one tempered, irreducible unitary representation of G with real infinitesimal character, and that moreover it is included there with multiplicity one, and is the unique minimal K-type in that representation. We shall prove that the Connes–Kasparov isomorphism in operator K-theory is equivalent to a K-theoretic version of Vogan’s result.

Let F be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized relative trace formula to prove a refined hybrid subconvex bound for $L(1/2,\pi\times\chi).$ In particular, we obtain the Burgess subconvex bound

\begin{align*}L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon}.\end{align*}

We present a comprehensive study of the geometry of Hilbert p-adic eigenvarieties at classical parallel weight 1 intersection points of their cuspidal and Eisenstein loci. For instance, we determine all such points at which the weight map is étale. The Galois theoretic approach presents genuine difficulties due to the lack of good deformation theory for pseudo-characters irregular at p and reflects the richness of the local geometry. We believe that our geometric results lead to deeper insight into the arithmetic of Hilbert automorphic forms and we produce in support several applications in Iwasawa theory.

We investigate the realisability of the Casson–Sullivan invariant for homeomorphisms of smooth 4-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We prove that for all pairs of orientable, homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$. As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected 4-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on 4-manifolds which are diffeomorphic but not stably pseudo-isotopic.

We prove a Poincaré duality for arithmetic p-adic pro-étale cohomology of smooth dagger curves over finite extensions of $\mathbf{Q}_p$. We deduce it, via the Hochschild–Serre spectral sequence, from geometric comparison theorems combined with Tate and Serre dualities. The compatibility of all the products involved is checked via reduction to the ghost circle, for which we also prove a Poincaré duality (showing that it behaves like a proper smooth analytic variety of dimension $1/2$). Along the way we study functional analytic properties of arithmetic p-adic pro-étale cohomology and prove that the usual cohomology is nuclear Fréchet and the compactly supported one: of compact type.