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 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 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.