We obtain a generalisation of the Stroock–Varadhan support theorem for a large class of systems of subcritical singular stochastic partial differential equations driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup $\mathcal {H}$ of the renormalisation group such that any renormalisation procedure determines a unique coset $g\circ \mathcal {H}$ . The support of the solution then depends only on this coset and is obtained by taking the closure of all solutions obtained by replacing the driving noises by smooth functions in the equation that is renormalised by some element of $g\circ \mathcal {H}$ .

One immediate corollary of our results is that the $\Phi ^4_3$ measure in finite volume has full support, and the associated Langevin dynamic is exponentially ergodic.

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.

In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures.

Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and log-height of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$. We prove effective bounds on the geometry of the intersection ${\mathcal B}\cap V$. In particular, when $\operatorname {codim} V=\dim {\mathcal L}$ we prove that $\#({\mathcal B}\cap V)$ is bounded by a polynomial in $\delta _{V}$ and $\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${\mathcal B}\cap V$ by an algebraic map $\Phi $. For instance, under suitable conditions we show that $\Phi ({\mathcal B}\cap V)$ contains at most $\operatorname {poly}(g,h)$ algebraic points of log-height h and degree g.

We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak {gl}_2(\mathbb {Q}_p)$ . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight k, we show its corresponding Galois representation has Hodge–Tate–Sen weights $0,k-1$ and prove a converse result.

A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.

In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ universality class. The Tracy–Widom distributions appear as special self-similar solutions of the KP and Korteweg–de Vries equations. In addition, it is noted that several known exact solutions of the KPZ equation also solve the KP equation.

We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin’s parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

Define the Collatz map ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ on the positive integers $\mathbb {N}+1 = \{1,2,3,\dots \}$ by setting ${\operatorname {Col}}(N)$ equal to $3N+1$ when N is odd and $N/2$ when N is even, and let ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ denote the minimal element of the Collatz orbit $N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $ . The infamous Collatz conjecture asserts that ${\operatorname {Col}}_{\min }(N)=1$ for all $N \in \mathbb {N}+1$ . Previously, it was shown by Korec that for any $\theta> \frac {\log 3}{\log 4} \approx 0.7924$ , one has ${\operatorname {Col}}_{\min }(N) \leq N^\theta $ for almost all $N \in \mathbb {N}+1$ (in the sense of natural density). In this paper, we show that for any function $f \colon \mathbb {N}+1 \to \mathbb {R}$ with $\lim _{N \to \infty } f(N)=+\infty $ , one has ${\operatorname {Col}}_{\min }(N) \leq f(N)$ for almost all $N \in \mathbb {N}+1$ (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a $3$ -adic cyclic group $\mathbb {Z}/3^n\mathbb {Z}$ at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.

Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant. Moreover, this constant can be taken equal to the operator norm of the local operator that generates the Hamiltonian (the local interaction strength). The result still holds true if one restricts to arbitrarily small quantum perturbations of classical Hamiltonians. The proof combines a robustness analysis of Robinson’s aperiodic tiling, together with tools from quantum information theory: the quantum phase estimation algorithm and the history state technique mapping Quantum Turing Machines to Hamiltonians.

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.

The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions.

More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions.

These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.

The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé and our previous results on the BCOV invariant, we establish this conjecture for Calabi–Yau hypersurfaces in projective spaces. Our contribution takes place on the B-side, and together with the work of Zinger on the A-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang–Lu–Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla–Selberg type theorem expressing it in terms of special $\Gamma $ -values for certain Calabi–Yau manifolds with complex multiplication.

Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ( $\mathrm {GW}/{\mathrm {PT}}$ ) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.