Papers

For a domain $\Omega$ that is a deformation of a unit ball in $\mathbb{R}^n$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy having $n+1$ wells. This sequence converges in the $L^1$ topology to a partition of $\Omega$ whose skeleton is given by a simplex cone that contains an $(n+1)$-junction point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated $\Gamma$-limit of the sequence of Allen-Cahn functionals. The results established in this article generalize those in the author's earlier article with Peter Sternberg (MR5033050), which dealt with the case $n=3$. We also weaken the one crucial assumption from the author's earlier article with Peter Sternberg (MR5033050).

We compute the relative smooth surgery structure sets of the thickenings $\mathbb{OP}^{2}\times\mathbb{D}^{k}$ of the Cayley projective plane $\mathbb{OP}^{2}$ for every $k\geq 1$ with $k\equiv 0\pmod 4$, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the $2$-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension $16+k$, homotopy equivalent to $\mathbb{OP}^{2}\times\mathbb{S}^{k}$ and distinguished by their Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group $\widetilde{\operatorname{Diff}}(\mathbb{OP}^{2})$ in every degree congruent to $3$ modulo $4$; and we construct smooth $\mathbb{OP}^{2}$-bundles over $\mathbb{S}^{4}$, $\mathbb{S}^{8}$, and $\mathbb{S}^{12}$ whose total spaces have non-vanishing $\widehat{\mathfrak{A}}$-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on $\mathbb{OP}^{2}$ in degrees $3$, $7$, and $11$.

We develop a general theory of selflessness for C*-correspondences, with several applications. Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness. Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of “phantom” traces on ultrapowers.

We improve the best known sphere packings in every dimension from $38$ to $43$. Our packings in dimensions $38$ to $42$ come from one chain of cross-sections of the extremal even unimodular lattice $P_{48p}$, $\sqrt3E_6 \subset \sqrt3E_7 \subset \sqrt3E_8 \subset K_9 \subset K_{10}$, whose first three members are cut from the fixed lattice $\sqrt3(E_8 \perp E_8)$ of an order-three automorphism; their orthogonal complements are lattice packings and set the records in dimensions $42$ down to $38$. Each successive section is a determinant-minimal extension of its predecessor. Our $43$-dimensional packing is an antipode packing: six translates of the complement of a $5$-dimensional section. We also improve some kissing numbers. Conway and Sloane's twelve $1982$ cross-section packings appear never to have had theirs computed; we compute them and find that, in dimensions $42$ to $47$, they exceed the previously tabulated lower bounds. Our chain does better in dimensions $40$, $41$ and $42$, and a further antipode packing beats the record in dimension $45$.

In this paper, we show that the modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold exists for all time and converges smoothly to an equidistant surface of constant mean curvature as $t\to \infty$. This result generalizes earlier work to the modified mean curvature flow setting and removes the restrictive global gradient bound initially required for the standard mean curvature flow by Huang, Zhou, and the second author \cite{HLZ2020}.

We propose a signature-induced, optimal transport framework for path-space model risk, in which ambiguity between stochastic path laws is factorized through optimal transport costs on signature coordinates under a common coupling. Via an ambient feature-space relaxation, we derive for affine signature scores and affine half-space events explicit robust expectation and probability bounds that depend only on the baseline model. In both cases, the correction is governed by the same effective budget, which depends only on the affine score and the budget vector of the ambiguity set. Moreover, the same quantity leads to a budget-aware sparse signature surrogate method for more general, possibly implicit, or data-driven, path functionals. We illustrate the resulting methodology through controlled stress tests and benchmark model-misspecification problems in finance and insurance.

We develop a dynamical framework for non-commutative de Finetti theory. We first establish the non-commutative Hewitt-Savage 0-1 law for quantum stochastic processes. We identify the factorization condition of the distribution of a spreadable process which characterizes tail-triviality, which is also characterized dynamically in terms of a certain attractivity property of the distribution. These three equivalent ergodic conditions identify a distinguished level in the non-commutative hierarchy of ergodic properties, which all collapse to ergodicity in the classical probability setting. To pass from the ergodic to the general case, we construct the conditional expectation onto the tail algebra for a spreadable process, in the GNS representation of the distribution of the canonical bilateral extension of the process. In this representation we establish the non-commutative Olshen Theorem identifying the tail algebra with the stationary algebra, and with the exchangeable algebra when the process is exchangeable. The resulting conditional expectation in particular inherits the same type of factorization property as distributions satisfying the Hewitt-Savage 0-1 law. This factorization strengthens conditional independence conditions arising in previous literature, while collapsing to the same notion in the classical case. This dynamical viewpoint leads to the identification of the minimal distributional symmetry underlying non-commutative de Finetti Theory, which we call weak spreadability, and which in the classical setting is equivalent to exchangeability. We prove that a stationary process is weakly spreadable if and only if its tail algebra admits a unique normal conditional expectation satisfying Hewitt-Savage type factorization, thereby establishing a general non-commutative de Finetti Theorem.

Over the past decade, the study of the asymptotic growth of Weil-Petersson volumes of the moduli space of hyperbolic surfaces has yielded numerous results on the length spectrum and on the spectrum of the Laplacian of typical large genus surfaces. We compute the exact asymptotic value of the volume polynomials $V_{g,n}(x_1,\ldots x_n)$ for $\mathbf{x}=(x_1,\ldots x_n)$ the lengths of the boundary components such that $x_i=\mathcal{O}(\sqrt{g})$: $$\prod_{j=1}^{n}\frac{x_j}{2}\cdot\frac{V_{g,n}(x_1,\ldots x_n)}{V_{g,n}}\!=\!\frac{1}{2^n}\exp\left({\frac{|\mathbf{x}|}{2}\!-\!\frac{1}{8\pi^{2}g}\left(\frac{|\mathbf{x}|}{2}\right)^{2}}\right)\!\left(1\!+\!\mathcal{O}_{n}\left(\frac{1}{\min x_j}\right)\right).$$ This result relies on the analysis of the expansion of Witten-Kontsevitch intersection numbers, for which we obtain an analogous explicit result. We also refine the bound over the coefficients of the expansion in terms of $s$ the degree of the expansion. From the expansion of the volumes, we deduce an exact estimate of the average number of non-separating simple geodesics of length of order $\sqrt{g}$. Our result therefore explains the behavior of counting functions at the cutoff $\sqrt{g}$, at which simple geodesics become negligible with respect to non-simple ones. The existence of this cut-off was conjectured by Lipnowski and Wright and proven by Wu and Xue.

We study families of quiver Grassmannians, parametrizing subrepresentations of representations of Dyn\-kin quivers, which are naturally associated to a projective and an injective representation. Our main result is an explicit description of the flat irreducible locus of such a family. More precisely, we prove that a member of this family is irreducible and of the expected dimension if and only if the ambient quiver representation degenerates to the direct sum of the defining projective and injective representations.

We compare the closed $2$-form $\omega_f$ constructed by Rungi-Tamburelli and Goldman's symplectic form $\omega_G$ on the $\mathrm{SL}(3,\mathbb R)$-Hitchin component. In particular, we establish that the semi-pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component defined by Rungi-Tamburelli is non-degenerate everywhere and, after aligning the normalization of the Killing form on $\mathfrak{sl}(3,\mathbb{R})$, coincides with the one recently found by Collier-Toulisse-Wentworth.

We consider the numerical solution of shape optimization problems for thermoelasticity with temperature-dependent material parameters. We show the existence of the shape derivative and derive an expression for generic functionals of domain integral type. Numerical results are presented for two settings: minimization of the compliance under a volume constraint, and minimization of the volume under a constraint on the $L^2$-norm of the von Mises stress. We use the finite element method for solving the underlying boundary value problems and the level set method for the representation of the actual domain.

This paper develops a wavenumber-explicit stability and preasymptotic error analysis for finite element approximation of the two-dimensional Helmholtz scattering problem with a uniaxial perfectly matched layer (UPML) truncation. The analysis is based on direct estimates of the stretched Green kernel associated with the Cartesian complex coordinate transformation. We establish explicit stability for the truncated UPML problem. In particular, we prove that the inf-sup constant of the truncated UPML formulation is $\mu_L = O(k^{-1})$ in a natural $k$-weighted $H^1$ norm. As a consequence, we also obtain an exponential decay estimate for the PML truncation error. Based on this stability estimate, we formulate a linear continuous interior penalty finite element method (CIP-FEM) on the truncated UPML domain. A key ingredient of the analysis is to establish a piecewise $H^{1+s}$-regularity result (for any $0<s<1$), which accounts for coefficient jumps across PML interfaces and singularities induced by Cartesian corner geometries. This fractional regularity is sufficient to derive wavenumber-explicit preasymptotic error estimates. Numerical experiments confirm the theoretical predictions and illustrate the effectiveness of the UPML CIP-FEM in the high-frequency regime.

Let $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, put $\beta=\dim_{\mathbb{R}}\mathbb{K}$, and let $G_n^{\mathbb{K}}$ be an $n\times n$ matrix with i.i.d. standard $\mathbb{K}$-Gaussian entries, namely a standard $\mathbb{K}$-Ginibre matrix. We prove that the normalized row-ordered permanent $W_n^{\mathbb{K}}=\operatorname{per}_{\mathbb{K}}G_n^{\mathbb{K}}/\sqrt{n!}$ has a radial density $p_n^{\mathbb{K}}$ satisfying $\|p_n^{\mathbb{K}}\|_\infty=p_n^{\mathbb{K}}(0)\lesssim_\beta n^{(\beta+2)/4}$ and $\sup_{z\in\mathbb{K}}\mathbb{P}(|W_n^{\mathbb{K}}-z|\leq\varepsilon)\lesssim_\beta n^{(\beta+2)/4}\varepsilon^\beta$. In particular, for $\mathbb{K}=\mathbb{C}$, this resolves the Permanent Anticoncentration Conjecture of Aaronson and Arkhipov. The proof compares the squared Gaussian permanent with the squared (Study) determinant in Laplace-transform order.

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.

We exhibit a finite-dimensional central division algebra, of degree four over its center, together with an explicit element of reduced trace zero that is not a single commutator. This answers negatively the single-commutator question for finite-dimensional central division algebras, the case left open by Amitsur and Rowen. The algebra is a skew Laurent series ring, the element has only two terms, and the proof uses the $x$-adic valuation, a first-obstruction lemma based on two commuting involutions of the associated graded ring, and a parity argument for quadratic forms over iterated Laurent series fields.

It is an old idea that higher-dimensional super-gravity (SuGra) is put on-shell just by imposing Bianchi identities on super-field strengths over super-spacetimes subject to super-torsion constraints. We give a modernized, rigorous account and review recent developments, pointing out how this perspective lends itself to the construction of infrared completions of SuGra by electromagnetic flux quantization in differential nonabelian cohomology theories $\mathcal{A}_{\mathrm{dff}}$ lifting the coefficient $L_\infty$-algebra $\mathfrak{l}\mathcal{A}$ of the super-Bianchi identities. After surveying necessary background, we highlight: (1.) our recent proof that solutions of 11D SuGra are equivalent to the $\mathfrak{l}S^4$-Bianchi identities on super $C$-field flux over super-torsion-free 11D super-spacetimes, (2.) how from this the nonlinearly self-dual gauge sector on 6D M5-brane worldvolumes is equivalent to the $\mathfrak{l}_{S^4}S^7$-Bianchi identity on super $B$-field flux, (3.) and the recent complete discussion of superspace dimensional reduction of this situation to 10D IIA SuGra via $\mathfrak{l}\mathrm{Cyc}(S^4)$-Bianchi identities on super NS/RR-flux, together with its extension to further reduction to 9 SuGra via $\mathfrak{l}\mathrm{Tor}(S^4)$-Bianchis on the corresponding super-fluxes. We close by indicating how this implies consistent infrared completions of 11D SuGra by $C$-field flux quantization in 4-Cohomotopy, of 10D IIA SuGra by NS/RR-field flux-quantization in a form of twisted unstable K-theory, and of M5-brane worldvolumes by $B$-field flux-quantization in twisted (twistorial) relative Cohomotopy. The latter admits geometric engineering of experimentally relevant topological quantum orders.

Gabidulin codes are a rank metric analog of Reed-Solomon codes. They are of great importance in rank-metric cryptography, as they are employed in highly efficient rank-based schemes, such as RQC and the Loidreau cryptosystem. However, until now, there has been no constant-time implementation of these codes, even though such an implementation is crucial for real-life development of cryptosystems. In this paper, we propose the first constant-time decoding algorithm of augmented Gabidulin (AG) codes, a simple variation of Gabidulin codes, in which columns of zeros are added to the end of a Gabidulin code. AG codes are used in the most efficient variants of the RQC cryptosystem. We prove that AG codes decoding can be achieved with quadratic complexity by reducing the decoding of an AG code to the decoding of a shorter standard Gabidulin code. In addition to the description of the complete AG codes decoding procedure, we present a constant-time algorithm for the left division of $q$-polynomials that can be used in every constant-time implementation of Gabidulin code variants. These algorithms are integrated into the RQC-Block-MS-AG scheme, and we evaluate the performance of our implementation through benchmarks. Our results show that our constant-time implementation outperforms the original RQC, though it remains approximately four to five times slower than HQC. However, it achieves ciphertext and key sizes about four times smaller, highlighting an appealing trade-off between performance and compactness.

Sequence A339987 of the OEIS counts (3,1)-regular graphs having one more vertex than edges by half the number of vertices. A recurrence relation satisfied by this sequence was guessed by Kauers and Koutschan in 2023. We confirm it in three ways: first, by a representation as the diagonal of a triple sum and an elaborate variant of traditional creative telescoping that makes an a posteriori validation possible; second, by a residue representation and a direct calculation by reduction-based creative telescoping; third, by a combinatorial recurrence on graph families and a calculation by differential elimination. Each of those three approaches leads to a formally complete proof and involves a computer calculation in one way or another.

We obtain effective characterizations of complex semi-abelian varieties in arbitrary dimension in terms of logarithmic irregularity and the first two logarithmic plurigenera, among varieties of maximal Albanese dimension, and among varieties whose compactification has large irregularity (these hypotheses are sharp). To the authors' knowledge, this is the first result in this direction for quasi-projective varieties in arbitrary dimension.

For a finite group acting on a finite set, a statistic called relational complexity can be calculated for the action. This notion was defined by Gregory Cherlin and motivated by considerations in model theory. Another related statistic is the height of the action, which provides an upper bound for relational complexity. In this paper, both concepts are defined and some basic results proved. The main focus later on is examining the primitive actions of $ PSL_{2} (q) $ and $ PGL_{2} (q) $ and computing both the height and relational complexity for each one.

In this paper, we mainly investigate some basic properties of strong $R$-spaces. It is shown that the property of being a strong $R$-space is closed-hereditary, saturated-hereditary and retractive, but not finite productive. Hence the category $\mathbf{S}$-$\mathbf{Top}_r$ of strong $R$-spaces and continuous mappings is not reflective in the category $\mathbf{Top}_0$ of $T_0$-spaces and continuous mappings. It is proved that a $T_0$-space $(X, \tau)$ is a strong $R$-space iff every nonempty $\tau$-closed subset of $X$ is compact in $(X, \tau^{d})$, where $\tau^d$ is the de Groot dual of $\tau$; consequently, if $(X, \tau)$ is a strong $R$-space (especially, if $(X, \tau)$ is a coherent well-filtered space), then $\tau \subseteq \tau^{dd}$. Therefore, for any locally compact strong $R$-space $(X, \tau)$, we have $\tau=\tau^{dd}$. Finally, we investigate conditions under which the Smyth power space and Scott power space of a $T_0$-space is a strong $R$-space. Several such conditions are given.

We establish an upper bound on the lifespan of bounded weak solutions to the three-dimensional isentropic compressible Euler equations on $[0,T)\times\mathbb R^3$, under a localized positivity condition on the initial data and the assumption that the essential support of the disturbance propagates into an undisturbed background state no faster than predicted by classical local well-posedness. We further show that any entropy-admissible bounded weak solution with bounded inverse density that persists beyond this upper bound must propagate into the undisturbed region at a strictly super-classical speed and that this accelerated propagation is necessarily accompanied by a jump discontinuity in an associated one-sided $L^\infty$-profile of the solution.

This paper investigates the principal spectral theory of a nonlocal dispersal operator with coupled diffusion and aims to establish a variational characterization of the spectral bound for the case where the system is not strongly coupled. In this setting, a key difficulty arises since the principal eigenfunction may have components that are identically zero, rendering existing generalized eigenvalue methods inapplicable. To overcome this, we reorder the components of the operator using a permutation matrix, thereby decomposing it into suitable suboperators, and characterize the spectral bound of the original operator in terms of the spectral bounds of these suboperators. Building on this principal spectral theory, we provide a variational characterization of the basic reproduction ratio for nonlocal dispersal systems and analyze the dynamical behavior of a class of multi-genotype stem cell regeneration models with epigenetic transitions, both in the presence and absence of gene mutations, without assuming the existence of a principal eigenvalue. Furthermore, we investigate the threshold dynamics when the system is not strongly coupled.

The atypical behaviour of random walks in time-dependent random environment was recently related to Kardar-Parisi-Zhang (KPZ) growth. While this is now well-understood in spatial dimension $d=1$, further efforts are necessary to better understand these connections in dimensions $d>1$. In this paper, we study this problem numerically for $d=1, 2$ and $3$, focusing on a discrete model with Dirichlet distributed transition probabilities. This model is a generalization of an integrable model in $d=1$, and it has the advantage of admitting an explicit, product-form, stationary measure. We verify that the growth of the variance of the logarithm of point-to-point probabilities, namely from the origin to position $x$ in time $t$, is compatible with KPZ growth in dimension $d=1$ and $d=2$. In spatial dimension $d=3$, we confirm the existence of a phase transition as the angle $\vert x\vert /t$ increases and we obtain a lower bound based on an exact second moment calculation. We find that in the weak disorder phase the point-to-point probability acquires a heavy tailed distribution, and that in the strong disorder phase the cumulants of its logarithm grow with time. Further, we show that for this model, we can compute exactly the sample to sample variance of the thermal average $\overline{ \langle x \rangle^2}$ and that it is related to the extreme diffusion coefficient introduced recently.

In this paper we prove that whenever a projective plane of odd order n contains an oval, it may be used to construct a triple array with n + 1 rows, n.n columns and n(n + 1) symbols. In particular, for any odd prime power s, we use the projective plane over the field of order s to construct an explicitly given (s + 1) times s.s triple array on s(s + 1) symbols. This is only the third infinite series of triple arrays (up to transposition) to be found. Our construction proves a recent conjecture of Gordeev and Ohman (2025) in the case of odd prime powers.

We prove a flexible criterion for fixed price one, applying in particular to higher-rank lattices over local fields and automorphism groups of affine buildings, including lattices in exotic buildings. It also recovers fixed price one for amenable groups. We then establish a locally compact version, and deduce that every product of two noncompact, compactly generated, unimodular locally compact groups, as well as every lattice in such a product, has fixed price one.

We solve certain questions related to the geometry and singularities of quasi-$F$-split varieties with trivial canonical bundle. First, we prove that regular quasi-$F^{\infty}$-split varieties are not geometrically uniruled (this generalizes and significantly simplifies the earlier results of Patakfalvi and Zdanowicz) and have geometrically canonical singularities. Second, we show that there exist quasi-$F$-split surfaces with trivial canonical bundle which are not quasi-$F^{\infty}$-split, answering negatively a question raised by Kawakami, Takamatsu, Tanaka, Witaszek, Yobuko and Yoshikawa. Third, we show that normal quasi-$F$-split varieties with trivial canonical bundle are geometrically normal (this extends a result of Kawakami, Takamatsu and Yoshikawa), and finally we prove that quasi-$F^e$-pure normal varieties $X$ such that $mp^eK_X$ is Cartier for $m$ coprime to $p$ are log canonical, under a resolution of singularities hypothesis.

Let $G=(V,E)$ be a finite connected graph with boundary $B$. We prove that for a generic positive edge weight function $w \in \mathbb{R}^{|E|}$, the Steklov eigenvalues of $(G,B,w)$ are simple and every Steklov eigenfunction does not vanish on the boundary. More precisely, the exceptional weights are contained in a zero set of a non-identically zero polynomial and hence form a set of Lebesgue measure zero and Hausdorff dimension at most $|E|-1$. Our results provide a discrete extension of the genericity theorem for the Steklov problem on compact manifolds.

The Morse complexity of a manifold is the minimal number of handles required to build it. We explore the Morse complexity of manifolds, bordisms, and homology classes, proving nontrivial upper bounds using surgery theory and lower bounds using index theory. Our most involved result shows that for Lie groups which admit discrete series representations, the Morse complexity of their locally symmetric spaces grows linearly with volume. This implies that such locally symmetric spaces do not admit open book decompositions.

We show that for any two knots $K_1$, $K_2$ in $\mathbb{R}^3$, there exist diagrams $D_1$ and $D_2$ that represent $K_1$ and $K_2$, respectively, such that the Khovanov–Rozansky triply graded homologies of $D_1$ and $D_2$ are isomorphic. The methods expand on Abel's paper [Abe17].