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 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.

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 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.

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 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.

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 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.

Two central problems in extremal combinatorics are concerned with estimating the number $\mathrm {ex}(n,\mathcal {H})$, the size of the largest $\mathcal {H}$-free hypergraph on n vertices, and the number $\mathrm {forb}(n,\mathcal {H})$ of $\mathcal {H}$-free hypergraph on n vertices. It is well known that ${\mathrm { forb}}(n,\mathcal {H})=2^{(1+o(1))\mathrm {ex}(n,\mathcal {H})}$ for k-uniform hypergraphs that are not k-partite. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many k-partite (or degenerate) hypergraphs $\mathcal {H}$, ${\mathrm { forb}}(n, \mathcal {H}) = 2^{O(\mathrm {ex}(n, \mathcal {H}))}$. However, there are few known instances of degenerate hypergraphs $\mathcal {H}$ for which ${\mathrm { forb}}(n,\mathcal {H})=2^{(1+o(1))\mathrm {ex}(n,\mathcal {H})}$ holds.

In this paper, we show that ${\mathrm { forb}}(n,\mathcal {H})=2^{(1+o(1))\mathrm {ex}(n,\mathcal {H})}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $\mathcal {H}$ for which ${\mathrm { forb}}(n,\mathcal {H})=2^{(1+o(1))\mathrm {ex}(n,\mathcal {H})}$ holds. As a corollary of our main results, we obtain a sharp estimate of ${\mathrm { forb}}(n,C^{(k)}_\ell )=2^{(\left \lfloor \frac {\ell -1}{2} \right \rfloor +o(1))\binom {n}{k-1}}$ for the k-uniform linear $\ell $-cycle, for all pairs $k\geq 5, \ell \geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\geq 5, \ell \geq 3$. Our methods also lead to some sharp results on the related random Turán problem.

As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.

Let X be a smooth proper rigid analytic space over a complete algebraically closed field extension K of $\mathbb {Q}_p$. We establish a Hodge–Tate decomposition for X with G-coefficients, where G is any commutative locally p-divisible rigid group. This generalizes the Hodge–Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb {G}_a$. For this, we introduce geometric analogs of the Hodge–Tate spectral sequence with general locally p-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over X and in the relative setting of smooth proper morphisms $X\rightarrow S$ of seminormal rigid spaces. We deduce applications to analytic Brauer groups and the geometric p-adic Simpson correspondence.

We study the homological stability of spin mapping class groups of surfaces and of quadratic symplectic groups using cellular $E_2$-algebras. We get improvements in their stability results, which for the spin mapping class groups we show to be optimal away from the prime $2$. We also prove that in both cases the $\mathbb {F}_2$-homology satisfies secondary homological stability. Finally, we give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups.

We consider the Maxwell–Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from nonrelativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell–Schrödinger equations for extended charges toward the nonrelativistic Vlasov–Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell–Schrödinger equations and for the Vlasov–Maxwell system for extended charges.

Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies.

We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan.

Several different characterizations of Lorentzian polynomials on cones are provided.

We investigate positivity and probabilistic properties arising from the Young–Fibonacci lattice $\mathbb {YF}$, a 1-differential poset on words composed of 1’s and 2’s (Fibonacci words) and graded by the sum of the digits. Building on Okada’s theory of clone Schur functions, we introduce clone coherent measures on $\mathbb {YF}$ which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, clone coherent measures are generally not extremal on $\mathbb {YF}$. Our first main result is a complete characterization of Fibonacci positive specializations – parameter sequences which yield positive clone Schur functions on $\mathbb {YF}$. Second, we establish a broad array of correspondences that connect Fibonacci positivity with: (i) the total positivity of tridiagonal matrices; (ii) Stieltjes moment sequences; (iii) the combinatorics of set partitions; and (iv) families of univariate orthogonal polynomials from the (q-)Askey scheme. We further link the moment sequences of broad classes of orthogonal polynomials to combinatorial structures on Fibonacci words, a connection that may be of independent interest. Our third family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or limits supported on the discrete component of the Martin boundary of the Young–Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin–Kerov on asymptotics of the Plancherel measure on $\mathbb {YF}$. We also establish Cauchy-like identities for clone Schur functions whose right-hand side is presented as a quadridiagonal determinant rather than a product, as in the case of classical Schur functions. We construct and analyze models of random permutations and involutions based on Fibonacci positive specializations along with a version of the Robinson–Schensted correspondence for $\mathbb {YF}$.

Under the assumption that the adjusted Brill-Noether number $\widetilde {\rho }$ is at least $-g$, we prove that the Brill-Noether loci in ${\mathcal M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to ${\mathcal M}_g$ is dominant if $n+\widetilde {\rho } \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski.

In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with nongeneral behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first-named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves.

We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and description often rely on specific knowledge of the parent object and its automorphisms. In many cases, questions of reproducibility and comparison arise. Here we present a categorical framework that addresses these questions. We prove that every characteristic structure is the image of a functor equipped with a natural transformation. This shifts the local description in the parent object to a global one in the ambient category. Through constructions in representation theory, such as tensor products, we can combine characteristic structure across multiple categories. Our results are constructive and stated in the language of a constructive type theory which facilitates their implementation in proof checkers.

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of $\mathsf {D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take $G=\mathcal {O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the q-Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.

We introduce the index ${\mathcal N}(\omega _1,\omega _2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega $ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal {N}(\omega ,\omega _D)$, where $\omega _D$ is obtained from $\omega $ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal {N}$.

The compactification $\overline {\mathfrak M}_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $\chi (X)=3$ in the moduli space of stable surfaces parametrises the so-called stable I-surfaces.

We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.