Papers

The Bárány-Larman conjecture states that for any $d+1$ sets of $r$ points each in $\mathbb{R}^d$, considered as color classes, we can partition their union into $r$ rainbow $(d+1)$-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when $r$ is a prime number. We also show that the optimal colorful Tverberg theorem of Blagojević, Matschke, and Ziegler cannot be extended to prime powers.

Adsorption and desorption redistribute soluble surfactant between a liquid film and its free surface, but cannot change the total inventory in a closed periodic domain. We audit this requirement in nonlinear weighted-residual models for soluble-surfactant-laden falling films. Although a previous correction restores conservation of the nonlinear surface Marangoni flux, a separate inconsistency remains in the depth-integrated bulk diffusion. We derive the resulting mass imbalance and construct explicit positive periodic concentration fields that exhibit it. The missing contribution arises from the streamwise concentration gradient appearing in the normal diffusive flux at a sloping interface. Including this contribution restores local conservation of the coupled bulk-surface transport system while leaving the hydrodynamic equations, the leading-order adsorption-desorption closure, and the linear stability properties of the uniform equilibrium unchanged. In an independent no-slip calculation using the parameters of an earlier travelling-wave study, the reference model loses $0.0228832\%$ of its initial surfactant mass by $t=4500$, whereas the corrected run has an absolute relative mass change of $1.2\times10^{-15}$. The main hump and capillary ripples remain closely comparable at $t\approx4500$. This benchmark therefore shows that agreement with travelling-wave calculations does not, by itself, guarantee that a reduced model respects the conservation laws inherited from the underlying transport problem. Although the drift is small in this case, the corrected equations conserve the reduced total inventory for every smooth periodic solution. We also distinguish this exact conservation property from higher-order compatibility with the full curved-interface boundary conditions.

The interval $[1,c]_T$ between the identity element and a Coxeter element $c$ in the absolute order on a Coxeter group $W$ is a generalization of the poset of noncrossing partitions arising when $W$ is the symmetric group. When $W$ is finite, this poset is always a lattice, and it is natural to associate to every element $w\in [1,c]_T$ its moved space $\mathsf{Mov}(w)=\mathrm{Im}(w - \mathrm{Id}_V)$ in the geometric representation $V$ of $W$. It has dimension equal to the reflection length $\ell_T(w)$ of $w$, and gives a realization of $[1,c]_T$ inside the lattice of subspaces of $V$. It is an important tool in the study of $[1,c]_T$. When $W$ is infinite, the moved space of an element $w\in [1,c]_T$ no longer has dimension $\ell_T(w)$ in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element $w\in [1,c]_T$ in an arbitrary Coxeter group, that we call absolute moved space of $w$. This subspace $\mathsf{AM}(w)$ of $V$ always contains $\mathsf{Mov}(w)$ and has dimension equal to $\ell_T(w)$, and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from $[1,c]_T$ to reflection subgroups of $W$, which to $w\in [1,c]_T$ associates the subgroup $P(w)$ generated by reflections lying below $w$ in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of $[1,c]_T$ when $W$ has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which $[1,c]_T$ fails to be a lattice.

We study the optimization landscape of low-tubal-rank tensor sensing through a balanced factorization. Under a tubal restricted isometry condition, we establish a quantitative strict-saddle landscape with no spurious local minima for arbitrary Fourier multi-rank profiles. We further show that the local geometry depends on the Fourier-slice ranks rather than the tubal rank alone. Uniform ranks yield quadratic growth transverse to the solution orbit, whereas nonuniform ranks produce quartically flat directions through hidden frequency-wise overparameterization, even when the factor width equals the exact tubal rank. Numerical experiments illustrate the global optimization behavior and the contrasting local geometries.

In this paper, we define the triangulated category of initial motives, which contains, for example, all motives of projective homogeneous varieties for semisimple algebraic groups of inner type. By analyzing its Chow weight structure, we prove that the family of isotropic localizations indexed by finitely generated extensions of the base field is conservative and Pic-injective on initial motives. This result provides an extension, in the mixed setting, of the work of Quéguiner-Mathieu and the first author on higher Tate traces of Chow motives.

In this paper, we study the two- and three-dimensional barotropic compressible Navier-Stokes equations with degenerate density-dependent viscosity coefficients in the whole space or in a ball for arbitrarily large spherically symmetric initial data. For initial data allowing vacuum, we establish the global existence of weak solutions and derive uniform-in-time a priori estimates. As a consequence, we prove that the vacuum state of weak solutions will vanish in finite time. The key ingredients include a treatment of the pressure terms based on separate estimates in the low- and high-density regions, the Bresch-Desjardins entropy estimates, weighted radial estimates, and a coupled control of the velocity and effective velocity. In the three-dimensional case, this approach also allows us to treat the endpoint case of the adiabatic exponent. For sufficiently regular initial data with strictly positive density, we establish the global existence and large-time behavior of classical solutions.

If $T$ is an optimal transport map between bounded convex domains $\Omega$ and $\Omega'$, we characterize the regularity/singularity dichotomy for $DT$ at $(x,T(x)) \in \partial \Omega \times \partial \Omega'$. This is obtained through a new approach to the regularity theory based on affine degenerations of the tangent cones to $(\Omega, \Omega')$ at $(x,T(x))$. We formulate a general principle relating the optimal boundary regularity to a stability criterion and an $\mathrm{SL}(n)$ moduli space of pairs of convex cones. Among other results, we establish geometric criteria for existence and non-existence of homogeneous optimal transport maps between convex cones.

A sequence of non-linear iterations with non-identically distributed random perturbations is studied. The asymptotic behaviour of the sequence is described via its visitations measures, which are deterministic measures that allow one to estimate the time spent by the sequence in the neighbourhoods of its partial limits. The results obtained are compared with statements regarding the statistical convergence of the sequence of iterations under study

Rounding errors in finite element computations can lead to a complete loss of accuracy, stalled convergence, and incorrect results. Moreover, the effects of rounding errors accumulated within automatically generated and compiled kernels are difficult to analyze a priori. We present the first software framework for automated rounding error estimation within finite element kernels. The proposed methodology is based on an a posteriori technique called Running Error Analysis (REA), where a forward error estimate is automatically computed concurrently with the value. An open-source implementation is provided for the FEniCS Form Compiler (FFCx), based on a C++ backend for generating type-generic templated kernels over a custom arithmetic type that tracks both the value and its error estimate. We demonstrate REA on two examples. First, we use it to detect catastrophic cancellation in the assembly of a Neo-Hooke hyperelastic model in the small deformation regime. A series expansion circumvents the cancellation problem and the computed error estimates show this. Second, we study the assembly of the Laplace operator on a near-degenerate mesh. We demonstrate that our REA implementation typically incurs only 2-4x performance overhead. Applications of this work include robust reduced-precision computations in embedded systems, numerical debugging of new, possibly ill-conditioned or unstable PDE formulations, and guiding the design of mixed-precision kernels.

The nerve theorem provides a combinatorial model for the homotopy type of a topological space from a suitable open cover. We extend this local-to-global approach by replacing the intersections of the cover with compatible categorical models. These local models are assembled through the Grothendieck construction, yielding a small category whose classifying space has the weak homotopy type of the original space. Under suitable hypotheses, the resulting category is finite and acyclic. We develop two versions of this construction. The first is indexed by the usual poset of nonempty finite intersections of the cover. The second uses the membership poset, inspired by the finite-space construction of Sancho de Salas, which retains only the membership patterns realized by points of the space and is defined for point-finite covers. The classical nerve theorem and its componentwise variant are recovered as particular cases. The construction is recursive: categorical models of local pieces, together with functors representing their inclusions, can be combined to produce categorical models of more complex spaces.

This paper addresses the stabilization of switched affine systems under dwell-time constraints around a desired operating point that is not a common equilibrium of the subsystems. Since asymptotic stabilization is generally unattainable in this setting, the objective is to ensure that trajectories converge to and remain within a prescribed region containing the desired operating point. The main contribution is the development of Lyapunov-based tools that accommodate target regions more general than those induced by quadratic Lyapunov functions. In particular, we establish a constructive Lyapunov characterization for a broad family of weighted $p$-norm functions, which generalizes the classical quadratic Lyapunov equivalence and underpins the proposed hybrid-control and dwell-time developments. Focusing on the case in which the desired region is a polytope, and building on existing Lyapunov-based methods, we develop a constructive two-step method that starts from a polytopic Lyapunov certificate for the average system and yields a smooth weighted $p$-norm Lyapunov function for all sufficiently large $p$. We prove that this extension guarantees practical asymptotic stability of the desired operating point and yields a Lyapunov-certified invariant region under minimum dwell-time switching. Numerical examples illustrate the effectiveness of the proposed method.

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.

Thin fluid-filled channels in porous media are prevalent in a wide range of environmental settings including fractures in geological formations. Transport of fluid in such multi-domain systems is commonly described by Darcy's law and formulated either in a fully dimensional framework or through a hybrid-dimensional approach, where fractures are represented as inclusions of co-dimension one enclosed in the porous matrix. These mixed-dimensional models enable accurate representation of the corresponding full-dimensional systems while significantly reducing computational complexity. Models available in literature usually do not properly account for channel-wall roughness while applying coupling conditions on the fracture-matrix interface and are usually developed for flow directions parallel to the fracture. The goal of this work is to advance and study the hybrid-dimensional Stokes-Darcy model developed by the authors in their previous work. We consider a thin open channel embedded within a porous matrix, where the channel walls are in contact with porous regions of distinct permeabilities. We incorporate channel-wall roughness effects into the model and improve it to accommodate arbitrary flow directions. We prove existence and uniqueness of the weak solution for the new model and demonstrate its advantages over existing hybrid-dimensional Stokes-Darcy models using pore-scale resolved simulations.

We study the $q$-matroids associated with nondegenerate $\F_{q^m}$-linear near-MRD codes, which we call near-uniform $q$-matroids. Using the correspondence between rank-metric codes and their associated $q$-systems, we determine explicitly their rank functions and characterize their cyclic flats. We show that near-uniform $q$-matroids form a class of representable paving $q$-matroids. and prove an upper bound on the number of their cyclic flats, which is shown to be sharp for certain parameters. Finally, when $n>m$, exploiting the rank distribution of near-MRD codes, we obtain an exact count of the nontrivial cyclic flats of the associated $q$-matroids.

Concurrent games and strategies based on event structures are an established and expressive model of interactive computation. In this paper we revisit the problem of equipping this model with symmetry. Symmetry is essential for many applications, but it is also a source of mathematical complexity, not least because strategies with symmetry can be compared up to several distinct notions of equivalence. Our first contribution is a complete characterization of two classes of strategies for which composition admits an identity. For these two classes the identity laws hold, respectively, up to a "weak" and a "strong" notion of equivalence. The main result is that the weak class of strategies corresponds precisely to those inducing Street fibrations over the game. We also show how the established "thin" approach to symmetry also fits in this general framework. A second contribution is a formal connection between games and setoid-valued profunctors. This follows a long tradition of relating games and relational models, but here additionally includes a very general model of symmetries. Our characterization of strategies as Street fibrations makes the connection to profunctors significantly easier to establish.

We study stochastic systems driven by a Brownian sheet and formulate a Markov property adapted to their two-parameter structure. For time-space homogeneous Itô sheets, the future evolution from a point $(t,x)$ is not determined by the single value $Y(t,x)$ alone, but also by the boundary data along the horizontal and vertical edges issuing from $(t,x)$. We therefore show that the enlarged state consisting of the triple $(Y,D_1Y,D_2Y)$ satisfies a natural Markov property with respect to the past sigma-algebra generated by the two coordinate histories of the Brownian sheet. The result is illustrated by examples showing the strong martingale property for the Brownian sheet and for simple Brownian sheet systems. We also recall a Brownian sheet Girsanov theorem and use it to treat drifted sheets.

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion from a Kähler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in\Delta$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigenera for Kähler families.

In a continuous coordination game, under myopic best response, players change strategies, and thus strategic communities change their shapes in time. We show that the boundaries of these strategic communities evolve according to Free Boundary Mean Curvature Flow in the continuous time limit. The convergence relies on an approximation scheme for the Mean Curvature Flow, which has previously been used to describe other threshold dynamics. With this equivalence, we can characterize Nash equilibria of the continuous coordination game as minimal surfaces and explore related consequences in biased versions of the same game.

A majority coloring of a graph $G = (V,E)$ is a coloring of $V(G)$ such that, for each vertex $v$, the number of neighbors of $v$ with the same color as $v$ is at most $deg(v)/2$. A strong majority coloring is a coloring of $V(G)$ such that, for each vertex $v$, every monochromatic subset of $N(v)$ has size at most $deg(v)/2$. The (strong) majority coloring game is a two-player Maker-Breaker-type game, in which two players Alice and Bob color the vertices of a graph $G$ alternately, maintaining the (strong) majority condition. The least number of colors such that Alice has a winning strategy in such a game is called the (strong) majority game chromatic number of the graph $G$, denoted $\mu_g(G)$ (or $\mathrm{Maj}_g(G)$ for the strong version). For the majority coloring game, we prove that $\mu_g(G) \le 3$ under the following cases: $G$ is a $2$-caterpillar, $G$ is a rooted tree with all leaves at depth $k \le 4$, and $G$ is a subdivision of some graph. The latter resolves a problem posed by Bosek–Grytczuk–Jakóbczak in 2019, who also asked whether $\mu_g(T) \le 3$ for every tree $T$. For the latter question, we discuss various difficulties that arise when natural strategies are attempted by Alice to win the majority coloring game on trees. We include a comparison with the marking game and relaxed coloring game on trees, and with the majority coloring game on locally finite acyclic graphs $G$ with $\delta(G) > 1$. For the strong majority coloring game, we compute $\mathrm{Maj}_g(C_n)$ exactly for each cycle $C_n$, $n \ge 3$. We also initiate the study of the computational complexity of the strong majority coloring game; specifically, we prove that the decision version of the Strong Majority Game Chromatic Number problem is PSPACE-complete. We also show that the Strong Majority 2-Coloring problem is NP-complete on Eulerian graphs.

We find explicit formulas for the EHZ and cylindrical capacities of rotated $L_{p}$-ellipsoids. In addition, we provide an algorithm to estimate the EHZ capacity.

We prove that if $f_1,\ldots,f_N$ is an uniformly bounded orthonormal system, then for all $p\in[1,2)$ there is an estimate for its Kolmogorov widths: $d_n(\{f_1,\dots,f_N\},L_p) \gtrsim \min\{1,n^{-1/p}N^{1/2}\}$, $n\le N/2$. In the regime $n\asymp N$ this provides the lower bound $N^{-\alpha_p}$ with a sharp exponent $\alpha_p:=1/p-1/2$. Besides that, it follows that a good approximation of such ONS requires the dimension $n\gtrsim N^{p/2}$. Our second result is an approximation theorem for Fourier matrices of finite abelian groups (this includes the usual DFT matrices). For every $\eta>0$ there is $a=a(\eta)>0$ such that any Fourier matrix $F$ of sufficiently large order has an approximation of rank $N^{1-a}$ with the row-wise $\ell_1$-error at most $N^{1/2+\eta}$; the exponent $1/2$ is sharp. Consequences include the approximation of the same kind for circulant matrices; the approximation of the trigonometric functions $\exp(2\pi i\langle \lambda,x\rangle)$, $\lambda\in \Lambda=K\cap\mathbb{Z}^d$, $K$ is symmetric convex, with dimension $|\Lambda|^{1-a}$ and an optimal error $|\Lambda|^{-\alpha_p+\eta}$; bounds for widths of weighted Wiener classes.

In this paper, we study the final data problem for the defocusing nonlinear Schrödinger equation (NLS) $$ i\partial_t u + \frac12\Delta u = |u|^p u $$ in weighted spaces $\dot\Sigma^s(\mathbb R^d):=L_x^2(\mathbb R^d;|x|^{2s}\mathrm{d} x)$. Let $s_c=\frac d2-\frac2p$. Under the scaling of the NLS, $\dot{\Sigma}^{-s_c}(\mathbb R^d)$ is the scaling-critical weighted space. In three dimensions, within the range of exponents considered in this paper, we show that the final data problem is locally well-posed for $s\geq -s_c$ and ill-posed for $s<-s_c$. This is opposite to the corresponding initial data problem, which is locally well-posed for $s\leq -s_c$ and ill-posed for $s>-s_c$. For the three-dimensional quadratic equation, we further obtain a unique global solution for every radial final datum in the critical space $\dot\Sigma^{\frac12}(\mathbb R^3)$.

Hilbert C*-modules whose coefficient C*-algebra is equipped with a finite faithful trace generate Banach Gelfand triples. This provides a natural setting for the interpretation of Hilbert C*-modules as function spaces. In particular, when the modules are finitely generated and projective, their external tensor products interact well with projective tensor products. This allowed us to derive abstract kernel theorems for Hilbert C*-modules. As an application, we apply our results to Heisenberg modules and obtain new kernel theorems for time-frequency analysis.

With a view toward applications in Riemannian geometry, we explore coloop splitting properties of regular matroids. Nienhaus showed by classification in rank four that a regular matroid has a cocircuit whose deletion yields two coloops unless the matroid takes a particular form. In the latter case, one can split off any element of the ground set as a coloop. We reprove this using Seymour's structure theorem for regular matroids and prove an extension to matroids of ranks five and six. As an application to Riemannian geometry, we prove that the torus symmetry assumption in a recent result of Mouillé, Nienhaus, and the second author can be relaxed from rank ten to rank nine.

Let B be a plane curve which has simple normal crossings. We prove that, if B = B_1 \cup B_2 has two irreducible components with deg B_1, deg B_2 \geq 5 or deg B_1 = 4, deg B_2 \geq 7, and if B is general, then the associated algebraic exceptional set is empty. Based on this, we show that for a general plane curve B satisfying \bar{c}_1^2 - \bar{c}_2 > 0, the open surface \mathbb{P}^2 \setminus B is Kobayashi hyperbolic.

Let $G$ be a connected graph, and let $\lambda_1(G) > \lambda_2(G)$ denote its two largest adjacency eigenvalues. The spectral gap of $G$ is defined as the difference $\lambda_1(G) - \lambda_2(G)$. For integers $r\geq 2$ and $s\geq 0$, the double kite $DK(r,s)$ is formed by taking two vertex-disjoint copies of the complete graph $K_r$ and joining one specified vertex of each clique to a path with $s$ internal vertices. Stanić (2013) conjectured that every connected $n$-vertex graph with minimum adjacency spectral gap is a double kite. In this paper, we confirm this conjecture for sufficiently large $n$.

Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of $\phi_0,\ldots,\phi_p$ is asymptotically optimal for $\omega_p(M,g)$ as $p\to\infty$. We show that this conjecture is false in every dimension, even for positively curved real-analytic metrics. Nevertheless, we show that a less rigid notion of nodal sweepouts recovers the volume spectrum. Let \[ E_N(M,g):=\operatorname{span}\{\phi_0,\ldots,\phi_N\}, \qquad a_N\in H^1\bigl(\mathbb P(E_N(g));\mathbb Z_2\bigr) \] be the generator, and for $p\geq1$ define \[ \mathcal N_p^N(g) := \left\{ \Psi \mid X\text{ is a finite complex},\ \Psi\in C\bigl(X,\mathbb P(E_N(g))\bigr),\ \Psi^*(a_N^p)\neq0 \right\}. \] We define the nodal $(p,N)$-width by \[ \nu_p^N(M,g) := \inf_{\Psi\in\mathcal N_p^N(g)} \sup_{x\in\operatorname{dmn}(\Psi)} \mathcal H_g^{n}\bigl(\{\Psi_x=0\}\bigr). \] We prove the following filtration formula \[ \omega_p(M,g) = \lim_{N\to\infty}\nu_p^N(M,g) \] for every $p\geq1$. We state several questions on nodal geometry and min-max theory.

Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretization of the infinite-dimensional operator and the underlying surface can lead to inaccurate and misleading results. This paper introduces the SurfSpec suite of algorithms to compute eigenvalues and spectral measures of operators on surfaces with error control and convergence guarantees. We combine methods based on complex contour integration and residuals to compute discrete spectra and high-frequency eigenmodes, including of nonlinear eigenvalue problems. We then compute convolutions of spectral measures with rational kernels to deal with continuous spectra. In both cases, the key tool is the operator resolvent applied to functions, which is computed using high-order methods that solve surface PDEs. We illustrate our algorithms on a variety of differential operators and surfaces, including a triceratops.

Basis-restricted priors in Bayesian inverse problems can lose coverage when the truth has components outside the basis. We show that estimating the observation-noise variance can hide this loss. Under a linear forward model, when the in-span prior variance dominates the noise, the maximum-likelihood noise estimate absorbs the out-of-basis energy in the complement of the model range. Residual-magnitude and observation-coverage checks then stay near nominal while field coverage falls. We study repeated problems sharing one forward operator and one basis, fixed independently of the tested data. After projection onto the complement, and conditionally on the fitted noise scale, every exact test is a test of the scale-free direction of the residuals. We test the shape of their sample spectrum with John's sphericity statistic. Under Gaussian noise its null model is exact at finite sample size, and we derive its null mean and its power at proportional dimension. On synthetic problems and in a preregistered GEBCO topography study, the test detects structured out-of-basis variation that cross-validation and observation-coverage checks largely miss. It cannot detect Gaussian out-of-basis variation that is isotropic in the complement, since that is indistinguishable from a change of noise scale.

We prove that the relative commutant of a finite unitary tuple in a tracial matrix ultraproduct is internal whenever its conjugation operator has a spectral gap off the commutant. This implies the existence of a non-hyperlinear group.