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.

We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$, there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1 \leq b_2 \}$, for infinite $B\subset {\mathbb N}$ and prove that any set $A\subset {\mathbb N}$ with lower density $\underline {\!\mathrm {d}}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

In this paper we study degree-penalized contact processes on Galton-Watson (GW) trees and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex u with degree $d_u$ infects its neighboring vertex v with degree $d_v$ with rate $\lambda / f(d_u, d_v)$ for some positive function f. In the case $f(d_u, d_v)=\max (d_u, d_v)^\mu $ for some $\mu \ge 0$, the infection is slowed down to and from high-degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts.

We show that new phase transitions occur in terms of the parameter $\mu $ (at $1/2$) and the degree distribution D of the GW tree.

• • When $\mu \ge 1$, the process goes extinct for all distributions D for all sufficiently small $\lambda>0$;

• • When $\mu \in [1/2, 1)$, and the tail of D weakly follows a power law with tail-exponent less than $1-\mu $, the process survives globally but not locally for all $\lambda $ small enough;

• • When $\mu \in [1/2, 1)$, and $\mathbb {E}[D^{1-\mu }]<\infty $, the process goes extinct almost surely, for all $\lambda $ small enough;

• • When $\mu <1/2$, and D is heavier than stretched exponential with stretch-exponent $1-2\mu $, the process survives (locally) with positive probability for all $\lambda>0$.

We also study the product case, where $f(d_u,d_v)=(d_u d_v)^\mu $. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu \ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

We prove a local-global principle for parametrized $\infty $-categories: we show that any functor $\mathcal {B} \to \mathcal {C}$ is determined by the following data: the collection of fibers $\mathcal {B}_X$ for X running through the set of equivalence classes of objects of $\mathcal {C}$ endowed with the action of the space of automorphisms $\mathrm {Aut}_X(\mathcal {B})$ on the fiber, the local data, together with a locally cartesian fibration ${\mathcal D} \to \mathcal {C}$ and $\mathrm {Aut}_X(\mathcal {B})$-linear equivalences ${\mathcal D}_X \simeq {\mathcal P}(\mathcal {B}_X)$ to the $\infty $-category of presheaves on $\mathcal {B}_X$, the gluing data. As applications we compute the mapping spaces of the conditionally existing internal hom of $\infty \mathrm {Cat}_{/\mathcal {C}}$ and extend the $\infty $-categorical Grothendieck-construction by proving that $\infty $-categories over any $\infty $-category $\mathcal {C}$ are classified by normal lax 2-functors to a double $\infty $-category of correspondences.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [33, 28] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb {R}^n$ of k-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma $ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec {q}$ of length r, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D such that each connected component of $\mathbb {R}^n \setminus Z(P)$ intersects at most $\sim \frac {|\Gamma |}{D^{n - k - r}}$ elements of $\Gamma $. Also, under some mild conditions on $\vec {q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most D defined with respect to $\vec {q}$, such that each connected component of $\mathbb {R}^n \setminus Z(P')$ intersects at most $\sim \frac {|\Gamma |}{D^{n-k}}$ elements of $\Gamma $. To do so, given a k-dimensional semi-Pfaffian set $\mathcal {X} \subseteq \mathbb {R}^n$, and a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D, we establish a uniform bound on the number of connected components of $\mathbb {R}^n \setminus Z(P)$ that $\mathcal {X}$ intersects; that is, we prove that the number of connected components of $(\mathbb {R}^n \setminus Z(P)) \cap \mathcal {X}$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemerédi-Trotter-type theorems, and also prove bounds on the number of joints between Pfaffian curves.

We establish, as $\rho \to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of particles of size $\rho $ with fixed centers $x_j\in {\mathbb R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form

$$ \begin{align*} E_\rho = \sum_j \mu_j -4\pi\rho \sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(\rho)\,, \end{align*} $$

where $\mu _j$ is the minimal energy after zooming in at scale $\rho $ around each particle, and $v_j\in {\mathbb R}^3$ is determined by the far-field behavior of the corresponding single-particle minimizer. The Coulomb-like interaction in this expansion agrees with the electrostatic analogy: a linearized approximation commonly used in the physics literature for colloid interactions in nematic liquid crystal. We obtain here for the first time a precise estimate of the energy error introduced by that linearization, by developing new tools that address the lack of convergence rate when zooming in at scale $\rho $.

The study of Koszul binomial edge ideals was initiated by V. Ene, J. Herzog, and T. Hibi in 2014, who found necessary conditions for Koszulness. The binomial edge ideal $J_G$ associated to a finite simple graph G is always generated by quadrics. It has a quadratic Gröbner basis if and only if the graph G is closed. However, there are many known nonclosed graphs G where $J_G$ is Koszul. We characterize the Koszul binomial edge ideals by a simple combinatorial property of the graph G.

In [7], Hjorth, assuming $\mathsf {{AD+ZF+DC}}$, showed that there is no sequence of length $\omega _2$ consisting of distinct $\boldsymbol {\Sigma }^1_2$-sets. We show that the same theory implies that for $n\geq 0$, there is no sequence of length $\boldsymbol {\delta }^1_{2n+2}$ consisting of distinct $\boldsymbol {\Sigma }^1_{2n+2}$ sets. The theorem settles Question 30.21 of [16], which was also conjectured by Kechris in [17] (see Conjecture in Chapter 4 of [17] and the last paragraph of Chapter 4 of [17]).

We consider the countably many families $\mathcal {L}_d$, $d\in \mathbb {N}_{\geq 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell–Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell–Weil rank jump property for $d\neq 2,3$ and moreover the Mordell–Weil rank jump property for $d\equiv 3\ \mod 4$, $d\neq 3$. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell–Weil rank jump property and the presence of special multisections of the fibration.