An error is noted in the averaging argument in “Convergence of exclusion processes and the KPZ equation to the KPZ fixed point”, J. Amer. Math. Soc. 36 (2023), 251-289. Consequently, the method proves tightness for non-nearest neighbor exclusion processes under the 1 : 2 : 3 1:2:3 scaling, but convergence to the KPZ fixed point only for perturbations of TASEP vanishing in the limit.
Transitivity properties of the group of morphisms generated by Vieta involutions on the solutions to the Markoff equation modulo primes are established, yielding forms of strong approximation for the Markoff surface. These are applied to show that almost all Markoff numbers are highly divisible.
A long-standing conjecture of De Giorgi asserts that every monotone solution of the Allen–Cahn equation in R n + 1 \mathbb {R}^{n+1} is one-dimensional if n ≤ 7 n \leq 7 . A stronger version of the conjecture, also widely studied and often called “the stable De Giorgi conjecture”, proposes that every stable solution in R n \mathbb {R}^n must be one-dimensional for n ≤ 7 n \leq 7 . To this date, both conjectures remain open for 3 ≤ n ≤ 7 3 \leq n \leq 7 . An elegant variant of this problem, advocated by Caffarelli, Córdoba, and Jerison since the 1990s, considers a free boundary version of the Allen–Cahn equation. This variant features a step-like double-well potential, leading to multiple free boundaries. Locally, near each free boundary, the solution satisfies the Bernoulli free boundary problem. However, the interaction of the free boundaries causes the global behavior of the solution to resemble that of the Allen–Cahn equation. In this paper, we establish the validity of the stable De Giorgi conjecture in dimension 3 3 for the free boundary Allen–Cahn equation and, as a corollary, we prove the corresponding De Giorgi conjecture for monotone solutions in dimension 4 4 . To obtain these results, a key aspect of our work is to address a classical open problem in free boundary theory of independent interest: the classification of global stable solutions to the one-phase Bernoulli problem in three dimensions. This result, which is the core of our paper, implies universal curvature estimates for local stable solutions to Bernoulli, and serves as a foundation for adapting some classical ideas from minimal surface theory—after significant refinements—to the free boundary Allen–Cahn equation.
In this paper and its sequel, we develop a technique for controlling the distribution of l∞-Selmer groups in degree l twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. It also can be used to prove distributional results about the l∞-class groups in the family of degree l cyclic extensions of a given number field. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the l∞-Selmer group. This paper gives a technique for finding the distribution of l∞-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.
We construct a theory of (étale) Berkovich motives. This is closely related to Ayoub’s theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. Applying the theory to discrete fields, one still recovers the étale version of Voevodsky’s theory. Two notable features of our setting which do not hold in other settings are that over any base, the cancellation theorem holds true, and under only minor assumptions on the base, the stable ∞ \infty -category of motivic sheaves is rigid dualizable.
The r-colour Ramsey number R_r(k) is the minimum n ∈ℕ such that every r-colouring of the edges of the complete graph K_n on n vertices contains a monochromatic copy of K_k. We prove, for each fixed r ⩾ 2, that R_r(k) ⩽ e^-δk r^rk for some constant δ= δ(r) > 0 and all sufficiently large k ∈ℕ. For each r ⩾ 3, this is the first exponential improvement over the upper bound of Erdős and Szekeres from 1935. In the case r = 2, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.
We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley–Stembridge conjecture.
For every finite group H and every finite H-module A, we determine the subgroup of negligible classes in H^2(H,A), in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime p, every integer n≥ 3, and every field F containing a primitive p-th root of unity, there exists a continuous n-dimensional mod p representation of the absolute Galois group of F(x_1,…,x_p) which does not lift modulo p^2. This answers a question of Khare and Serre, and disproves a conjecture of Florence.
We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on SL_2(ℝ)/SL_2(ℤ), a result of independent interest.
We prove that the small quantum t-connection on a closed monotone symplectic manifold is of exponential type and has quasi-unipotent regularized monodromies at t=0. This answers a conjecture of Katzarkov-Kontsevich-Pantev and Galkin-Golyshev-Iritani for those classes of symplectic manifolds. The proof follows a reduction to positive characteristics argument, and the main tools of the proof are Katz's local monodromy theorem in differential equations and quantum Steenrod operations in equivariant Gromov-Witten theory with mod p coefficients.
Sen’s theorem on the ramification of a p p -adic analytic Galois extension of p p -adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing p p -adic Hodge theory for general valuation rings, we establish a geometric analogue of Sen’s criterion for any p p -adic analytic Galois extension of p p -adic varieties: its (Riemann-Zariski) stalkwise perfectoidness is necessary for the non-vanishing of the geometric Sen operators. As the latter is verified for general Shimura varieties by Pan and Rodríguez Camargo, we obtain the perfectoidness of every completed stalk of general Shimura varieties at infinite level at p p . As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for general Shimura varieties.
Kim, Kresch and Oh defined unramified Gromov-Witten invariants. For a threefold, Pandharipande conjectured that they are equal to Gopakumar-Vafa invariants (BPS invariants) in the case of Fano classes and primitive Calabi-Yau classes. We prove the conjecture using a wall-crossing technique. This provides an algebro-geometric construction of Gopakumar-Vafa invariants in these cases.
We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained in , our result establishes an analogue of the Gell-Mann–Brueckner formula c_1ρlog(ρ)+c_2ρ for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.
For a smooth projective curve X X over C p \mathbb C_p and any reductive group G G , we show that the moduli stack of G G -Higgs bundles on X X is a twist of the moduli stack of v-topological G G -bundles on X v X_v in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings’ p p -adic Simpson correspondence for X X , which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of p p -adic representations of π 1 ( X ) \pi _1(X) to an open substack of the stack of semi-stable Higgs bundles of degree 0 0 .
We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses.
Given a d-dimensional vector space V ⊂ℂ[u] of polynomials, its Wronskian is the polynomial (u + z_1) ⋯ (u + z_n) whose zeros -z_i are the points of ℂ such that V contains a nonzero polynomial with a zero of order at least d at -z_i. Equivalently, V is a solution to the Schubert problem defined by osculating planes to the moment curve at z_1, …, z_n. The inverse Wronski problem involves finding all V with a given Wronskian (u + z_1) ⋯ (u + z_n). We solve this problem by providing explicit formulas for the Grassmann-Plücker coordinates of the general solution V, as commuting operators in the group algebra ℂ[𝔖_n] of the symmetric group. The Plücker coordinates of individual solutions over ℂ are obtained by restricting to an eigenspace and replacing each operator by its eigenvalue. This generalizes work of Mukhin, Tarasov, and Varchenko (2013) and of Purbhoo (2022), which give formulas in ℂ[𝔖_n] for the differential equation satisfied by V. Moreover, if z_1, …, z_n are real and nonnegative, then our operators are positive semidefinite, implying that the Plücker coordinates of V are all real and nonnegative. This verifies several outstanding conjectures in real Schubert calculus, including the positivity conjectures of Mukhin and Tarasov (2017) and of Karp (2021), the disconjugacy conjecture of Eremenko (2015), and the divisor form of the secant conjecture of Sottile (2003). The proofs involve the representation theory of 𝔖_n, symmetric functions, and τ-functions of the KP hierarchy.
We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.
We prove that the geodesic flow on a geometrically finite locally symmetric space of negative curvature is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The approach is based on constructing a suitable anisotropic Banach space on which the infinitesimal generator of the flow admits an essential spectral gap. A key step in the proof involves estimating certain oscillatory integrals against the Patterson-Sullivan measure. For this purpose, we prove a general result of independent interest asserting that measures on ℝ^d that do not concentrate near proper affine subspaces enjoy polynomial Fourier decay outside of a sparse set of frequencies. As an intermediate step, we show that the L^q-dimension (1<q≤∞) of iterated self-convolutions of such measures tend towards that of the ambient space. Our analysis also yields that the Laplace transform of the correlation function of smooth observables extends meromorphically to the entire complex plane in the convex cocompact case and to a strip of explicit size beyond the imaginary axis in the case the manifold admits cusps.
Building on To\"en's work on affine stacks, we develop a certain homotopy theory for schemes, which we call "unipotent homotopy theory." Over a field of characteristic $p>0$, we prove that the unipotent homotopy group schemes $\pi_i^{\mathrm{U}}(\,\cdot\,)$ introduced in our paper recover the unipotent Nori fundamental group scheme, the $p$-adic \'etale homotopy groups, as well as certain formal groups introduced by Artin and Mazur. We prove a version of the classical Freudenthal suspension theorem as well as a profiniteness theorem for unipotent homotopy group schemes. We also introduce the notion of a formal sphere and use it to show that for Calabi-Yau varieties of dimension $n$, the group schemes $\pi_i^{\mathrm{U}}(\,\cdot\,)$ are derived invariants for all $i \ge 0$; the case $i=n$ is related to recent work of Antieau and Bragg involving topological Hochschild homology. Using the unipotent homotopy group schemes, we establish a correspondence between formal Lie groups and certain higher algebraic structures.
We make progress on several interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, exceptional estimates for orthogonal projections, and the dimension of Furstenberg sets. We give a new proof of the following asymmetric sum-product theorem: Let $A,B,C \subset \mathbb{R}$ be Borel sets with $0 < {\dim_{\mathrm{H}}} B \leq {\dim_{\mathrm{H}}} A < 1$ and ${\dim_{\mathrm{H}}} B + {\dim_{\mathrm{H}}} C > {\dim_{\mathrm{H}}} A$. Then, there exists $c \in C$ such that $${\dim_{\mathrm{H}}} (A + cB) > {\dim_{\mathrm{H}}} A. $$ Here we only mention special cases of our results on projections and Furstenberg sets. We prove that every $s$-Furstenberg set $F \subset \mathbb{R}^{2}$ has Hausdorff dimension $$ {\dim_{\mathrm{H}}} F \geq \max\{ 2s + (1 - s)^{2}/(2 - s), 1+s\}.$$ We prove that every $(s,t)$-Furstenberg set $F \subset \mathbb{R}^{2}$ associated with a $t$-Ahlfors-regular line set has $${\dim_{\mathrm{H}}} F \geq \min\left\{s + t,\tfrac{3s + t}{2},s + 1\right\}.$$ Let $\pi_{\theta}$ denote projection onto the line spanned by $\theta\in S^1$. We prove that if $K \subset \mathbb{R}^{2}$ is a Borel set with ${\dim_{\mathrm{H}}}(K)\le 1$, then $$ {\dim_{\mathrm{H}}} \{\theta \in S^{1} : {\dim_{\mathrm{H}}} \pi_{\theta}(K) < u\} \leq \max\{ 2(2u - {\dim_{\mathrm{H}}} K),0\}, $$ whenever $u \leq {\dim_{\mathrm{H}}} K$, and the factor "$2$" on the right-hand side can be omitted if $K$ is Ahlfors-regular.
