We give a construction of integral local Shimura varieties which are formal schemes that generalise the well-known integral models of the Drinfeld p-adic upper half spaces. The construction applies to all classical groups, at least for odd p. These formal schemes also generalise the formal schemes defined by Rapoport-Zink via moduli of p-divisible groups, and are characterised purely in group-theoretic terms.

More precisely, for a local p-adic Shimura datum $(G, b, \mu)$ and a quasi-parahoric group scheme ${\mathcal {G}} $ for G, Scholze has defined a functor on perfectoid spaces which parametrises p-adic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over $O_{\breve E}$. Scholze-Weinstein proved this conjecture when $(G, b, \mu)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G, \mu)$ of abelian type when $p\neq 2$, and when $p=2$ and G is of type A or C. We also relate the generic fibre of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu , {\mathcal {G}})$.

Let $W_{\mathrm {aff}}$ be an extended affine Weyl group, $\mathbf {H}$ be the corresponding affine Hecke algebra over the ring $\mathbb {C}[\mathbf {q}^{\frac {1}{2}}, \mathbf {q}^{-\frac {1}{2}}]$, and J be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing J as a subalgebra of the $(\mathbf {q}^{-\frac {1}{2}})$-adic completion of $\mathbf {H}$ via Lusztig’s map $\phi $, we use Harish-Chandra’s Plancherel formula for p-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf {q}$, with denominator depending only on the two-sided cell containing w, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

Along the way, we show that upon specializing $\mathbf {q}=q>1$, the map from J to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter.

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair $({\mathcal {O}},V)$ on any del Pezzo surface of degree $4$, such that V is a bundle of rank r and degree d. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be included into a $5$-dimensional linear space of compatible Poisson brackets. We also construct new examples of linear spaces of compatible Feigin-Odesskii Poisson brackets of dimension $>5$, coming from del Pezzo surfaces of degree $>4$.

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek–Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an ‘inflating’ sphere through a point p in a Damek–Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma $ starting from p, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by ‘overinflating the horospheres’ by pushing the center of the sphere beyond the point at infinity of $\gamma $ along a virtual prolongation of $\gamma $. They give a new family of isoparametric hypersurfaces in Damek–Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. Díaz-Ramos and M. Domínguez-Vázquez [17] in Damek–Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.

In this paper, we prove the following result advocating the importance of monomial quadratic relations between holomorphic CM periods. For any simple CM abelian variety A, we can construct a CM abelian variety B such that all non-trivial Hodge relations between the holomorphic periods of the product $A\times B$ are generated by monomial quadratic ones which are also explicit. Moreover, B splits over the Galois closure of the CM field associated with A.

Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification.

We prove that a schön subvariety of the torus is cohomologically tropical if and only if it is wunderschön and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincaré duality.

We study other properties of cohomologically tropical and wunderschön varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin and Zharkov.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

The influence of certain arithmetic conditions on the sizes of conjugacy classes of a finite group on the group structure has been extensively studied in recent years. In this paper, we explore analogous properties for fusion categories. In particular, we establish an Ito-Michler-type result for modular fusion categories.

For every , we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures, and thus, the ergodic measure space is not connected, while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti [11, Conjecture 2] in $C^r$-topology for Lorenz attractors.

For an even positive integer n, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm {O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands L-functions associated to the group $\mathrm {GL}_1 \times \mathrm {O}(2n)$ over F. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin–Selberg L-functions attached to a pair of Hilbert modular forms.

We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

Let S and T be smooth projective varieties over an algebraically closed field k. Suppose that S is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of k, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational and motivic functor. This generalizes Kahn’s result on the torsion order of S. We also exhibit an example of S over $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.

We consider families of special cycles, as introduced by Kudla, on Shimura varieties attached to anisotropic quadratic spaces over totally real fields. By augmenting these cycles with Green currents, we obtain classes in the arithmetic Chow groups of the canonical models of these Shimura varieties (viewed as arithmetic varieties over their reflex fields). The main result of this paper asserts that generating series built from these cycles can be identified with the Fourier expansions of non-holomorphic Hilbert-Jacobi modular forms. This result provides evidence for an arithmetic analogue of Kudla’s conjecture relating these cycles to Siegel modular forms.

In the first part of the paper, we prove a mirror symmetry isomorphism between integral tropical homology groups of a pair of mirror tropical Calabi-Yau hypersurfaces. We then apply this isomorphism to prove that a primitive patchworking of a central triangulation of a reflexive polytope gives a connected real Calabi-Yau hypersurface if and only if the corresponding divisor class on the mirror is not zero.

We construct an explicit algebraic example of a subshift of finite type over a group $\Gamma $ with an invariant Markov measure which has completely positive sofic entropy (with respect to ‘most’ sofic approximations) and yet does not have a direct Bernoulli factor because its model spaces shatter into exponentially many clusters of sub-exponential size. The example and its analysis are related to random low-density parity-check (LDPC) codes.

We introduce the notion of integrable modules over $\imath $quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath $quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath $quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\imath $quantum group.

We define a notion of tracial $\mathcal {Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple nonelementary C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal {Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal {Z}$-absorbing C*-algebras which are not $\mathcal {Z}$-absorbing. We use techniques from reduced free products of von Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal {Z}$-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.