Papers

We establish an a priori $C^2$-estimate for complex Hessian equations on compact Hermitian manifolds. As an application, we obtain a regularity result for solutions to complex Hessian equation with densities of the right hand side depending on the solution. This generalizes Székelyhidi–Tosatti's result to the setting of complex Hessian equations on Hermitian manifolds.

We answer, in the principal setting, a question of Matui on the almost finite–purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.

We construct an odd smooth initial density for the unforced incompressible porous media (IPM) equation on $\mathbb T^2$ whose solution develops a finite-time blowup. The components $\partial_{x_2}\rho(t,0)$ and $\partial_{x_1}u_1(t,0)$ tend to $+\infty$ as the maximal smooth existence time is approached, while the density and velocity remain bounded in $L^2$. Starting from a stationary solution, we iterate the angular amplification of localized oscillations with rapidly growing frequencies, adapting the mechanism of Córdoba and Martínez-Zoroa [7] for forced IPM. Each perturbation is constructed from time zero and creates the geometry configuration for the next iteration stage. A one-sided upper bound for the pressure Hessian is applied to control backward preparation and initial perturbations, following an idea from the Euler construction of OpenAI [14].

Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control–fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases–that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli $X/Z$ interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.

We spread out geometrically irreducible perverse sheaves. Over an arithmetic base scheme, such sheaves extend from the generic fiber to a relatively perverse, universally locally acyclic sheaf over a dense open of the base scheme. As an application, we prove the arithmetic Kashiwara conjecture by Esnault and Kerz for geometric traits in equicharacteristic and generalize the decomposition theorem for arithmetic complexes to fields finitely generated over a separably closed field. We also recover the Hard Lefschetz theorem for arithmetic perverse sheaves.

We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.

In this note, we prove a stronger version of the Kővári-Sós-Turán theorem for partition-wise $k$-stable $r$-hypergraphs. More precisely, we show that for every $k,r\in\mathbb{N}_{\geq 2}$ there is $\eta=\eta(r,k)>0$ such that if $H=(V;E)$ is a partition-wise $k$-stable $r$-uniform $K^{(r)}_{t,\ldots,t}$-free hypergraph with $|V|=n$, then $|E| = O_{r,k,t}(n^{r-\eta})$. Crucially, $\eta$ is independent of $t$. The proof follows the pseudofinite regime used by Chernikov and Starchenko to prove an analogous version of Erdős-Hajnal for stable hypergraphs.

We prove that the resonance of any hyperplane arrangement is separable. Using the main result from arXiv:2512.10160 this proves Suciu's conjecture on the fundamental group of every hyperplane arrangement, in its most effective form.

We consider the defocusing quintic wave equation $\partial_t^2u-\Delta u+u^5=0$ for real-valued functions on $\mathbb{R}^{1+10}$, which is energy-supercritical. We construct a discretely self-similar solution in a backward light cone, of the form $$u(t,x)=(T-t)^{-1/2}W^*\big(\omega^*\log\frac1{T-t},\frac{|x|}{T-t}\big),$$ where the nonzero profile $W^*$ is $2\pi$-periodic in its first variable and real-analytic up to and across the light cone. Cutting off its initial data gives smooth, compactly supported, radial data whose solution coincides with the discretely self-similar solution in the backward light cone, is smooth on $[0,T]\times\mathbb{R}^{10}$ except at the vertex $(T,0)$ of the cone, and blows up there at the self-similar rate. The solutions we construct here appear to be the first real-valued blow-up solutions of defocusing energy-supercritical NLW. Existence of the real profile is proved by a computer-assisted Newton–Kantorovich argument in a Banach algebra of Fourier–Chebyshev coefficients, with rigorous error control. We find the approximate profile by numerically continuing rotating self-similar profiles of the complex-valued equation. This step is not essential for the blow-up proof, so it is left as non-rigorous numerics. The main analytic ingredients are an exact description of the mode operators for the linear part, which are hypergeometric and upper triangular in the Chebyshev basis, and a bound for their inverses that is uniform at high frequencies. All computer-assisted proofs, including the requisite codes and the approximate objects, are accessible on GitHub at \cite{code}.

We extract rank and regularized exponents from cyclic generic factors of the quantum connection and show that these data persist under the birational comparisons considered here. Applications include showing non-rationality of cubic threefolds, ordinary Gushel–Mukai threefolds, very general quartic fourfolds, and products of the threefolds with $\PP^1$.

We give a fast algorithm for solving min-cost $k$-commodity flow. The basic idea is to construct an auxiliary linear program that has low rank and whose minimum value is at most $1/k$ times the minimum value of the original problem (thus, solving this auxiliary linear program will make at least $1/k$ fraction of progress in the original problem). Low-rank linear programs can be solved quickly using black-box techniques. Thus, our algorithm runs in time $\tilde{O}(\operatorname{poly}(k)(n^{2.5}+m\sqrt{n}))$ on a directed graph with $n$ vertices and $m$ edges. We do not rely on any fast matrix multiplication.

We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie algebra of every maximal nested unipotent subgroup has the special property of being what we call "integral"; we also show that this fails for X = affine space of dimension > 3. As an application of our results, we show that Nagata's automorphism is contained in a unique Borel subgroup.

We establish an $L^1$-theory for a class of elliptic operators with unbounded coefficients, extending our earlier semigroup generation results and those of Canale–Rhandi–Tacelli to the endpoint $p=1$. The $L^1$-realization generates a positive analytic $C_0$-semigroup that is compact for every positive time. We characterize the generator domain through weighted estimates and prove consistency with the corresponding $L^p$-semigroups for $1<p<\infty$.

Hyper-connections widen the residual stream of a Transformer to $n$ parallel streams. Their manifold-constrained version (mHC) mixes the streams at each layer with a doubly stochastic matrix, which it computes by Sinkhorn normalization of exponentiated logits. We give a geometric theory of this design on the Birkhoff polytope. First, a doubly stochastic mixer splits the stream into a mean channel, on which mHC is exactly a residual network, and a difference channel, which each layer contracts by its second singular value $\sigma_2 \le 1 - n \min_{ij} H_{ij}$. Thus the extra width is a fading memory with a horizon of $1/(1-\sigma_2)$ layers, and among nonnegative mixers only the permutations do not collapse. Second, the Sinkhorn-logit map is a global chart, and its logit gradient is exactly the Fisher-Rao gradient. Thus logit gradient flow follows a squared Fisher-Rao metric, and the straight-through update is exactly entropic mirror descent. Third, under logit gradient flow the logarithm of each entry moves at a rate of at most $4n^3\|\nabla f\|_\infty \varepsilon$, where $\varepsilon$ is the distance to the nearest permutation. Thus gradient flow approaches and leaves the vertices only at rate $1/t$, but mirror descent moves at an exponential rate. Fourth, the local convergence factor of Sinkhorn is $\sigma_2^2$, so a fixed iteration budget limits the horizon. Experiments confirm the predicted rates.

This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the $\Gamma$-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational $K$-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their $\Gamma$-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin–Osher–Fatemi (ROF) and Chan–Esedoglu models. Our main result in this setting is a contribution to the inverse regularity problem associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational $K$-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and $\Gamma$-versions of the Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational $K$-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.

We prove counting formulae for the number of directed Hamiltonian paths and directed Hamiltonian cycles in any complete multipartite graph. The method is based on a collapsing technique relating general path graphs with coloured vertices to legally coloured path graphs.

In this work, we study transparent boundary conditions for the time-fractional heat equation on a metric star graph. This equation is the standard continuum model for subdiffusive heat and mass transport in disordered and porous branched media. Using the Laplace transform to solve the exterior branch problems, we demonstrate that the non-local vertex operator observed by the incoming bond is an Abel-type convolution kernel of order $\beta/2$, which generalizes the classical half-derivative kernel recovered at $\beta=1$. Our main result is that the diffusion-coefficient sum rule $\kappa_1=\kappa_2+\kappa_3$, known to eliminate thermal backflow for the classical heat equation, remains unchanged for every fractional order $\beta$: the transparency of the junction depends only on the network's diffusivities and not on the memory of the transport law. We rigorously prove this and uniqueness for the coupled vertex problem. We confirm our results numerically at the level of the underlying coupled partial differential equation system with an implicit L1 finite-difference scheme. We compare the star-graph solution to a reference solution on an unbranched line with matching diffusivity.

The information bottleneck (IB) seeks a representation $U$ of a source $X$ that retains as much information as possible about a target $Y$, subject to a constraint on $I(U;X)$. A classical argument shows that it suffices to consider representations with at most $|\mathcal{X}|+1$ symbols, and this bound is known to be tight whenever $|\mathcal{X}| \geq 3$. We show that the binary case behaves differently: if $X$ is binary and $Y$ is finite, then for every joint distribution of $(X,Y)$ and every rate constraint, the IB optimum is attained by a binary $U$. Hence the bound $|\mathcal{U}| \leq |\mathcal{X}|+1$ sharpens to $|\mathcal{U}| \leq |\mathcal{X}|$ for binary sources. The proof combines a separating hyperplane argument with the observation that, for a binary source, the ratio of the second derivatives of the two entropy functions involved is concave.

We prove stability of nonrelativistic fermionic matter in the presence of fixed nuclei \(\{R_j\}\) on an asymptotically Euclidean Riemannian manifold \((M,g)\). Upper and lower bounds on sectional curvatures are assumed, but no specific sign conditions on the curvature are imposed. The proof follows the route of Lieb-Thirring and electrostatic inequalities, the latter being proved by means of a Voronoi decomposition type argument which is however based on the Green functions \(\{G(\,\cdot\,,R_j)\}\) rather than the distance function.

We count maximal cylinders on zero holonomy $\mathbb{Z}$-covers of genus $2$ square-tiled surfaces, up to $\mathbb{Z}$-action, obtaining quadratic asymptotics. We also show that the leading term of the asymptotic, called the Siegel-Veech constant, can be recovered via a large-genus approximation by intermediate finite covers. Our work applies to the infinite staircases introduced by P. Hubert and G. Weitze-Schmithüsen. For many members of this family, we explicitly compute the associated Siegel-Veech constants. In particular, we exhibit the first infinite family of examples of zero holonomy $\mathbb{Z}$-cover in which the number of cylinders grows sub-quadratically.

A classic result of Poly and Raby \cite{Poly-Raby:1984} states that, in finite-dimensional Euclidean spaces, $\mathcal{C}^k$-smoothness (with $k\geq2$) of the squared distance function near a point of a closed set characterizes smoothness of the set as a submanifold, with the same order of differentiability. In this work, we show that this characterization splits in infinite-dimensional Hilbert spaces: $\mathcal{C}^k$-smoothness of the squared distance function characterizes weakly $\mathcal{C}^k$-submanifolds, while $\mathcal{C}^k$-submanifolds are characterized by this smoothness together with an additional pointwise equicontinuity condition on the highest-order derivative of the squared distance. The diffeomorphism of Poly and Raby, its associated graph representation, and the description of its inverse remain the common geometric basis of both characterizations. The development of this work was assisted by GPT-6 Astra.

We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.

Motivated by the Merino–Welsh conjecture, we consider the smallest $c\ge0$, denoted by $c_*$, for which the inequality $T_M(c,0)T_M(0,c)\ge T_M(1,1)^2$ holds for every loopless and coloopless finite matroid $M$. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [Adv. Math. 446 (2024), 109674] give the lower bound $x_0$, where $x_0\approx2.22668$ is the largest real root of the polynomial $x^3-9(x-1)$. Later, Csikvári [European J. Combin. 137 (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at $c=x_0$. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that $c_*=x_0$, without computer-assisted finite verification.

This survey studies C*-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality, and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang-Su algebra $\mathcal{Z}$. We present the main dynamical tools known to obtain $\mathcal{Z}$-stability, both in the settings of amenable and nonamenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.

On the set $\mathbb{YF}_n$ of words in $1$ and $2$ with digit sum $n$, we consider the probability distribution $\mu_P$, corresponding to the Plancherel measure on paths in the Young–Fibonacci graph. We study the asymptotics as $n$ grows of the entropy of the distribution $\mu_P$ and of the minimum number of words on which $\mu_P$ is almost entirely concentrated.

We introduce a novel Bregman proximal algorithm for convex composite optimization by applying the standard Bregman proximal gradient (BPG) method to a mirror-coordinate reparameterization of the objective. In primal variables, the resulting algorithm alternates between a preconditioned gradient step followed by a right Bregman proximal update. Our analysis relies on relative smoothness holding in the mirror coordinates instead of primal coordinates. Under this condition, we prove monotonic decrease of the objective in the general convex setting. We then specialize the method to Poisson inverse problems using weighted negative entropy as the potential. The resulting scheme recovers the classical Richardson-Lucy multiplicative updates and extends them to general convex regularizers. Building on a recent convergence analysis of multiplicative updates, we establish a sublinear convergence rate in function values for the Poisson setting. Finally, we demonstrate the performance of the regularized algorithm on several imaging inverse problems with Poisson-distributed observations. The code is publicly available at https://github.com/Tmodrzyk/MU-Bregman.

Let $(M^n,g_0)$ be a closed connected Einstein manifold with positive scalar curvature, where $n\ge6$. Using an Obata-type identity, we prove that every smooth metric conformal to $g_0$ with constant sixth-order $Q$-curvature is Einstein.

In this paper, we rigorously justify the low Mach number limit of the full compressible Navier-Stokes equations in a three-dimensional smooth bounded domain, subject to the no-slip boundary condition for the velocity and well-prepared initial data. In the presence of heat conduction and large temperature variations, we establish the uniform estimates for strong solutions on a time interval independent of the Mach number by constructing an energy functional in a Sobolev framework. The no-slip boundary condition cannot provide the vorticity boundary relation used in the slip-boundary setting. To overcome this difficulty, a divergence-free Stokes lifting of the boundary trace for the time derivative of the modified velocity is employed to define a test function vanishing on the boundary, which enables the transfer of a priori estimates from the velocity to the temperature.

A perfect isometry $I$ (introduced by Broué) between two blocks $b$ and $c$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $\psi$ of $c$ to $\pm$ an irreducible character of $b$. Broué proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $\psi$. This element is called the Broué invariant of $I$ by Boltje. The goal of this paper is to show that if $I$ comes from a Morita equivalence with an endopermutation source $V$, then, up to a sign, the Broué invariant of $I$ is determined by local data of $b$ and $c$. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of $V$ modulo $p$. As a corollary, we obtain that the Isaacs–Navarro refinement of the Alperin–McKay conjecture holds for inertial blocks.