We extend the construction of the p-adic L-function interpolating unitary Friedberg–Jacquet periods in previous work of the author to include the p-adic variation of Maass–Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this p-adic L-function can be viewed as a higher-dimensional generalisation of the work of Bertolini–Darmon–Prasanna and Castella–Hsieh, and the inclusion of this extra variable arising from the p-adic iteration of differential operators will play a key role in relating values of this p-adic L-function to p-adic regulators of special cycles on unitary Shimura varieties.

One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical i.i.d. transmission time $L_{xy}$ is multiplied by $(W_xW_y)^\mu $, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: finite and infinite Geometric Inhomogeneous Random Graphs, including Hyperbolic Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees.

We show that as the penalty function, that is, $\mu $ increases, the transmission time between two far away vertices sweeps through four universal phases: explosive (with tight transmission times), polylogarithmic, polynomial but strictly sublinear, and linear in the Euclidean distance. The strictly polynomial growth phase is a new phenomenon that so far was extremely rare in spatial graph models. All four growth phases are robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend nontrivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution L near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes.

We study fibrations of the projective model for the symmetric space associated with $\operatorname {\mathrm {SL}}(2n,\mathbb {R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only nondegenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\operatorname {\mathrm {Sp}}(2n,\mathbb {R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.

Given a continuous linear cocycle $\mathcal {A}$ over a homeomorphism f of a compact metric space X, we investigate its set $\mathcal {R}$ of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set $\mathcal {R}$ is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb {R}^n$ and a positive constant $\alpha $, the goal is to find the minimizers of $\sum _{x \in L} e^{-\alpha \|x - z\|^2}$ over all $z \in \mathbb {R}^n$.

By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions—when $\alpha $ tends to infinity—the minimizers in the limit are found at deep holes of the lattice. In this paper, we consider minimizers which already stabilize for all $\alpha \geq \alpha _0$ for some finite $\alpha _0$; we call these minimizers stable cold spots.

Generic lattices do not have stable cold spots. For several important lattices, like the root lattices, the Coxeter-Todd lattice, and the Barnes-Wall lattice, we show how to apply the linear programming bound for spherical designs to prove that the deep holes are stable cold spots. We also show, somewhat unexpectedly, that the Leech lattice does not have stable cold spots.

The depth of a subgroup H of a finite group G is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb {C}H \subseteq \mathbb {C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f \leftrightarrow \pi (f)$, called geometric and spectral hypotheses, under which one obtains “nice” PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $\sigma $ of $\operatorname {\mathrm {PGL}}_2(\mathbb Q_p)$, we study extensively the case that $\pi (f)$ is a projection onto the line of the newform if $\pi $ is isomorphic to $\sigma $ or its unramified quadratic twist, and $\pi (f) = 0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.

We investigate the Stark operator restricted to a bounded domain $\Omega \subset \mathbb {R}^2$ with Dirichlet boundary conditions. In the semiclassical limit, a three-term asymptotic expansion for its individual eigenvalues has been established, with coefficients dependent on the curvature of $\Omega $. We analyse the accumulation of eigenvalues beneath the leading-order terms in these expansions, establishing Weyl-type asymptotics. Furthermore, we derive weak asymptotics for the density of the spectral projector onto these low-lying states.

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations and we show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. As corollaries of this result, we obtain that if a countable Borel equivalence relation is either:

1. 1. measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $\sigma $-compact Polish space,

2. 2. treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space,

3. 3. treeable, amenable and Borel bounded,

then it is hyperfinite.

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by $h_q(n)$. Morier-Genoud and Ovsienko defined the q-analogue of a rational number $[r/s]_q$ in various ways, most of which depend directly or indirectly on the continued fraction expansion of $r/s$. As our first application we show that $[r/s]_q=q\,h_q(n-1)/h_q(n)$ where $r/s$ occurs as the nth entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset ${\cal F}(n)$ whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that $[r/s]_q$ can be computed by taking products of certain matrices which are q-analogues of the standard generators for the special linear group $\operatorname {\mathrm {SL}}(2,{\mathbb Z})$. We express the entries of these products in terms of the polynomials $h_q(n)$.

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\boldsymbol {\Sigma }^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices.

Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$, for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours.

The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C, introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$. In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

We show sharp well-posedness with analytic data-to-solution mapping in the semilinear regime for dispersion-generalized KP-I equations on $\mathbb {R}^2$ and $\mathbb {R} \times \mathbb {T}$. On $\mathbb {R}^2$ we cover the full subcritical range, whereas on $\mathbb {R} \times \mathbb {T}$ the sharp well-posedness is strictly subcritical. We rely on linear and bilinear Strichartz estimates which are proved using decoupling techniques and square function estimates. Nonlinear Loomis-Whitney inequalities are a further ingredient. These are presently proved for Borel measures with growth condition reflecting the different geometries of the plane $\mathbb {R}^2$, the cylinder $\mathbb {R} \times \mathbb {T}$, and the torus $\mathbb {T}^2$. Finally, we point out that on tori $\mathbb {T}^2_\gamma $, KP-I equations are never semilinear.

We show that the moduli spaces of Scholze’s p-adic shtukas with framing satisfy a p-adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$, even for exceptional groups G, and extends work of Oswal-Shankar-Zhu-Patel who proved a p-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p-adic rigid analytic version of Brody hyperbolicity.

We prove a version of Sylvester’s law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the R-matrix associated to the standard q-deformation of $GL(N,\mathbb {C})$. For q positive, this particular REA comes equipped with a natural $*$-structure, by which it can be viewed as a q-deformation of the $*$-algebra of polynomial functions on the space of self-adjoint N-by-N-matrices. We will show that this REA satisfies a type I-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb {C})$, any irreducible representation of the REA is determined by its extended signature, which is a classical signature vector extended by a parameter in $\mathbb {R}/\mathbb {Z}$. It is this latter result that we see as a quantized version of Sylvester’s law of inertia.

Let G be a simple algebraic group over an algebraically closed field $\Bbbk $ of positive characteristic. We consider the questions of when the tensor product of two simple G-modules is multiplicity free or completely reducible. We develop tools for answering these questions in general, and we use them to provide complete answers for the groups $G = \mathrm {SL}_3(\Bbbk )$ and $G = \mathrm {Sp}_4(\Bbbk )$.

We consider a toy model of rate-independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump “as late and as little as possible”, a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$-spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.

For Lebesgue generic $({x}_1,x_2)\in \mathbb {R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in \mathbb {N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q< T$ such that

$$ \begin{align*}a <q\cdot \|qx_1\| \cdot \|qx_2\|\le b\quad \text{and} \quad \|qx_1\|, \|qx_2\|\le c \end{align*} $$

for given parameters $a,b, c$ satisfying certain growth conditions.

In this paper, we give a complete list of strongly tempered anomaly-free hyperspherical Hamiltonian spaces-those that are dual to symplectic vector spaces under the relative Langlands duality. We show that the period integrals attached to the list contain many previously studied Rankin-Selberg integrals and period integrals, thus giving a new conceptual understanding of these integrals. The list also proposes many new interesting period integrals to study.