Papers

The universal R-matrix of the quantum affine algebra associated to a finite-dimensional simple complex Lie algebra admits a Gauss decomposition into an uper unitriangular part, an abelian part, and a lower unitriangular part. In this paper, we provide a simple conjugation formula for the unitriangular R-matrices with one tensor factor evaluated at an arbitrary finite-dimensional representation of the quantum affine algebra. Our formula involves the T-series of Frenkel–Hernandez and the Theta series introduced in a previous work. We also extend our conjugation formula to the Yangian case, making use of associators for triple tensor product representations of shifted Yangians.

This paper concerns the three-dimensional forward and inverse acoustic obstacle scattering problem in the time domain. For the forward problem, a retarded potential formulation discretized by convolution quadrature and Galerkin methods is introduced. By introducing the retarded boundary integral defined on a homothetic surface, we propose a novel time-domain convolution quadrature based iterative method to reconstruct both the shape and location of a rigid obstacle. The retarded integral in the time domain is reformulated into a system of integrals in the s-domain. The resulting s-domain integrals are very fast to compute, as they only involve non-singular integrals over the homothetic surfaces. Moreover, the Fréchet derivative with respect to the boundary can be derived straightforwardly. We also prove that the scattered field generated by the homothetic surface converges to the exact field in the time domain. To improve the stability of the inversion algorithm, an incremental truncation technique is proposed, and numerical experiments confirm the effectiveness and robustness of our method.

This article introduces a notion of Bridgeland-Enriques general K3 surfaces motivated by the study of Enriques categories over K3 surfaces and the invariant Bridgeland stability conditions. The family of Bridgeland-Enriques general K3 surfaces of degree 10 detects a categorical degeneration of special Gushel-Mukai threefolds. Also, the families of Bridgeland-Enriques general K3 surfaces with higher degrees are closely related to Hodge-special Gushel-Mukai fourfolds and double EPW sextics.

Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between $p$- and $q$-dimensional excitations occurs in $d=p+q+2$ spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension $d=p+q+1$, applicable to excitations obeying $\mathbb Z_N$ fusion for arbitrary $N$ and all excitation dimensions $p$ and $q$. The corresponding invariant is the Berry phase accumulated in a simple $4N$-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the $N$-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry $\prod X$ and the nearest-neighbor controlled-$Z$ symmetry $\prod CZ$. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.

We answer negatively two natural general forms of Kawan's questions on invariance entropy for control systems, open for more than fifteen years, by a single construction. We show that finite strict invariance entropy need not coincide with ordinary invariance entropy, and that strict invariance entropy need not be lower semicontinuous under Hausdorff perturbations of the initial set. The construction is a continuous-time control system in which a Cantor coordinate stores an infinite symbolic instruction, an exponentially contracting coordinate makes late mismatches geometrically invisible, and a compact matching graph forces exact symbolic agreement. It identifies a source of information complexity not generated by dynamical expansion, but by the persistence of exact viability constraints under thin invariant geometry and by the order of limits in invariance entropy.

This paper establishes a unified Minkowski theory for exterior p-capacitary volumes and resolves the classical Pólya-Szegö conjecture on the electrostatic capacity of convex bodies.

We characterize the pairs of increasing events $A,B$ for which there is equality in the BK inequality. Namely, we show that $P(A\circ B)=P(A)P(B)$ if and only if all the configurations in $A\times B$ admit disjoint witnesses for $A$ and $B$. We discuss the strengthened BK inequality, and we provide a new simplified proof of this inequality.

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H\subseteq G$ is $\Delta(G)$-overfull if $|E(H)|>\Delta(G)\left\lfloor |V(H)|/2\right\rfloor$. In any edge coloring of $G$, each color class restricted to $H$ is a matching of size at most $\left\lfloor |V(H)|/2\right\rfloor$. Thus, if $G$ contains a $\Delta(G)$-overfull subgraph, then $G$ cannot be edge-colored with only $\Delta(G)$ colors. By Vizing's Theorem, $\chi'(G)\le \Delta(G)+1$, and hence $G$ is class $2$. In 1986, Chetwynd and Hilton conjectured that whenever $\Delta(G)>|V(G)|/3$, the converse also holds: every class $2$ graph $G$ contains a $\Delta(G)$-overfull subgraph. This statement, commonly known as the Overfull Conjecture, is one of the most influential conjectures in graph edge coloring. It would imply a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\Delta(G)>|V(G)|/3$, and would also imply several other longstanding conjectures in the area, including the Just-overfull Conjecture and the Vertex-splitting Conjecture. In previous work, the third author verified the conjecture for large graphs $G$ with maximum degree at least $13|V(G)|/14$. In this paper, we confirm the conjecture for robust expanders satisfying certain density constraints. As a consequence, for every $0<\varepsilon<1$, the conjecture holds for all sufficiently large graphs $G$ with maximum degree at least $(1+\varepsilon)|V(G)|/2$.

This paper studies the imperfect Bose gas after the Kac density law and the mean-field Euler equations have selected a condensed density with positive zero-mode excess. In this BEC regime the selected chemical potential cancels the mean-field shift, so the selected one-particle Hamiltonian is exactly the free one. The resulting zero-mode covariance defines a mean-field BEC ideal in the resolvent algebra, while the nonregular quotient and the direct-integral center record distinct representation-theoretic data. Occupation-number and Brownian-loop formulations recover the same density selection, excess density, ODLRO data, local tests, and the separation between finite-density BEC and Buchholz's stricter infinite-occupation proper-condensate criterion.

We study the existence and characterization of optimal tensor representations of elements in the space $L_1(\mu,Y)$ of Bochner integrable functions. We completely describe the set of norm-attaining elements in two settings. First, when the Banach space $Y$ is strictly convex, and second, when $Y=L_1(\nu)$ and $\mathbb K=\mathbb R$. In both situations, our analysis yields the existence of non-norm-attaining tensors whenever the underlying measures are not purely atomic. Finally, we introduce a geometric property over $Y$ ensuring that every element in $L_1(\mu, Y)$ admits an optimal representation. In particular, this holds for Lipschitz-free spaces over complete scattered metric spaces, for $C(K)$ spaces when $K$ is a compact Hausdorff totally disconnected space, and for $c_0(\Gamma)$ where $\Gamma$ is any index set. As a byproduct, we settle two open questions regarding projective norm-attainment.

We prove the existence of globally stable quasistatic evolutions for a cohesive fracture model with unprescribed crack path and without any topological restriction, in arbitrary dimension. The surface energy density is assumed to be concave and to exhibit an activation threshold, modeling depinning effects and fracture process zones in quasi-brittle materials. We devise a new notion of convergence for memory variables supported on evolving crack sets, inspired by $\sigma$-convergence in brittle fracture, guaranteeing compactness and lower semicontinuity properties. In contrast to the brittle case, global stability is not preserved under passage to the limit because of oscillation and branching phenomena in the approximating cracks. To overcome this difficulty, we deviate from the classical scheme for proving energetic solutions by first proving the energy balance and convergence of the surface energies, and only afterwards recovering the global stability condition.

We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegative constraint. For every admissible class $\mathcal{C}$, we prove two transfer principles. First, equality cases regularize optimally: if a non-matching graph $H$ is $\mathcal{C}$-Sidorenko, then every equality case $t(H,W)=p(W)^{e(H)}$ with $W\in\mathcal{C}$ is regular. Consequently, relative forcing is equivalent to relative regular-forcing for every non-matching $\mathcal{C}$-Sidorenko graph. Second, in the range $v(H)\le e(H)$, ordinary $\mathcal{C}$-Sidorenko is equivalent, as a universal property over $\mathcal{C}$, to the spectral inequality $t(H,W)\ge \rho(W)^{2e(H)-v(H)}p(W)^{v(H)-e(H)}$ for every non-zero $W\in\mathcal{C}$. The spectral transfer is obtained from a Perron-biased tensor regularization theorem detecting the Perron spectral radius on the exponential scale. We also prove quantitative near-equality variants and apply the framework to doubly nonnegative graphons and bounded doubly nonnegative kernels. This yields spectral equivalences for Sidorenko-good graphs in the range $v(F)\le e(F)$, and identifies Sidorenko-good forcing with regular-KNRS forcing for non-matching Sidorenko-good graphs.

We prove a global well–posedness result for the quintic NLS on $\mathbb{T}$ for initial data in $H^{s}(\mathbb{T})$, $s > 1/3$. This improves the previous best bound of $s > 2/5$.

Let $C$ be a closed, convex, pointed and generating cone in a finite-dimensional real vector space $V$, and let \( D_C=(-C)\cup\{\bot\}\) be the negative cone with a new least element, ordered by the cone order. Keimel proved that these cone domains are FS-domains and asked whether they are always retracts of bifinite domains. We give a sharp answer: \[D_C\text{ is an RB-domain}\quad\Longleftrightarrow\quad C\text{ is simplicial}. \] Thus every non-simplicial proper cone gives an FS-domain which is not an RB-domain. The proof converts the RB approximation property into finite-valued $C$-isotone approximations of the identity. The analytic obstruction is elementary and finite-dimensional: first in Euclidean space, cone-upper sets are represented, up to null sets, as Lipschitz epigraphs; Rademacher's theorem, Fubini's theorem and integration by parts then force the matrix tested against any finite-valued isotone map to lie in the cone generated by the positive rank-one operators $v\otimes\ell$, $v\in C$, $\ell\in C^*$. If such maps approximate the identity, the identity operator lies in this rank-one cone, which is possible exactly when the cone is simplicial. This answers Keimel's question in the negative for the Lorentz cone and other non-simplicial cones.

We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author. Unlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai's original reconstruction theorem via ultracategories.

We show that the tautological ring of $\mathcal{M}_{g,n}^{\mathrm{ct}}$ is not Gorenstein for $g\geq 2$ and $2g+n\geq 12$. We prove new cases of Pixton's conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including $\mathcal{M}_{6}^{\mathrm{ct}}$, $\mathcal{M}_{5,2}^{\mathrm{ct}}$, and $\mathcal{M}_7^{\mathrm{ct}}$. These are the first known cases where Pixton's conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.

The aggregation with exponential weights (AEW) estimator is not fully understood in the basic setting of model selection aggregation with squared loss. In particular, whether it is minimax-rate optimal in expectation for large enough fixed temperatures and under random design has been an open problem since its introduction, which was explicitly posed by Lecué and Mendelson (2013). In this paper, we settle this problem by showing that without requiring a Bernstein-type assumption, the AEW indeed achieves the excess risk $T \log (M) / (n+1)$ in expectation, whenever the temperature $T$ satisfies $(L^2/T)\exp(B/T)\leq \mu /2$. Here, the number of dictionary elements is $M$, the estimator has observed $n$ i.i.d. samples from any distribution, and the loss is assumed to be bounded by $B$, $L$-Lipschitz continuous and $\mu$-strongly convex. For squared loss, we show that $T\geq 4 b^2$ suffices when the predictions and labels are $[0,b]$-valued. Because AEW is known to be suboptimal in expectation for temperatures below some constant, this shows that AEW has a sharp phase transition when the temperature is large enough but constant, as conjectured by Lecué and Mendelson.

We construct an exhaustive family of constant spacetime mean curvature (STCMC) surfaces for initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid. In particular, we obtain such a foliation as the long time limit of the volume preserving spacetime mean curvature flow starting from the constant mean curvature foliation constructed by Neves-Tian (Geom. Funct. Anal., 2009). As an application, inspired by the definition of STCMC center of mass for initial data sets proposed in the asymptotically Euclidean setting by Cederbaum-Sakovich (Calc. Var. PDE, 2021), we study the center of mass of an asymptotically hyperboloidal initial data set.

Let $\mathcal{H}_{g \to b,d; \mathbf{e}}$, with $\mathbf{e}=(e_1,\ldots, e_n)$, be the Hurwitz space, parametrizing all morphisms $\pi: C\to B$ of degree $d$, with $n$ points $x_1,\ldots, x_n\in C$ of ramification order $e_1,\ldots, e_n$ respectively, and where $C$ and $B$ are smooth, irreducible, projective curves of genera $g$ and $b$ respectively. In this paper we study the question of when there exist components of $\mathcal{H}_{g \to b,d; \mathbf{e}}$ whose members $\pi: C \to B$ all factor through an intermediate curve, in which case we say that these components are of non–simple type. We give necessary and sufficient conditions for the existence of components of non–simple type. Then we prove that for $b\geq 2$ there are always components of simple type, and for $b\in \{0,1\}$ there are such components under suitable sufficient conditions. However there are easy examples for $b\in \{0,1\}$ in which there are never components of simple type.

Let \(q\) be a power of a prime \(p\), and let \(n\) be a positive integer. A subspace \(U\subseteq \mathbb F_q^n\) is called cyclically covering if the union of all its cyclic shifts covers \(\mathbb F_q^n\), and \(h_q(n)\) denotes the maximum possible codimension of such a subspace. This paper studies cyclically covering subspaces via cyclic codes. We first prove that \(h_q(n)=0\) if and only if every nonzero cyclic code in \(\mathbb F_q^n\) contains a full-weight codeword. We also relate \(h_q(n)\) to the maximum weights of cyclic codes. In particular, when \(h_q(n)>0\), we obtain sharp bounds for the maximum weight of cyclic codes without full-weight codewords and provide explicit examples attaining these bounds. Moreover, we study the number of cyclic codes containing no full-weight codeword. We determine this number completely over \(\mathbb F_2\), and give lower bounds over \(\mathbb F_3\). From this, we prove that if \(q\ge 3\) is an odd prime and \(m\ge 4\) is an integer, then \(h_q\left(\frac{q^m+1}{2}\right)>0\).

We formulate an avatar of the Fyodorov-Hiary-Keating conjecture for black hole microstate counts in quantum gravity. By holography, this implies sharp bounds on interval counts of high-dimension primary operators in conformal field theory. The extremal fluctuations of these counts are characterized by a random variable, with a prescribed tail distribution. At large $N$, these order-one erratic fluctuations set a quantitative limit on the resolution of the semiclassical AdS gravitational path integral. Gaussian random models for state counts arise naturally in this context; we express the phenomenon of erratic $N$-dependence in AdS/CFT as a decorrelation property of these models. Our broader point is to suggest that AdS black hole microstate spectra and their field theory duals should exhibit the extreme value statistics of random matrices, lying in the universality class of Gaussian log-correlated fields.

We classify the finite posets whose probabilistic powerdomain is an RB-domain. For a finite nonempty poset \(P\), let \(\Vone(P)\) be the probability powerdomain of $P$, which is the probability simplex ordered by the stochastic order. We prove that \(\Vone(P)\) is an RB-domain if and only if \(P\) has a least element and the undirected Hasse graph of \(P\) is a tree. Consequently, the probabilistic powerdomain does not preserve RB-domains; the four-point diamond gives a finite counterexample. The proof separates two obstructions. First, if \(P\) has no least element, then the face of probability measures supported on the minimal points must be fixed pointwise by every deflation below the identity. Secondly, once a least element exists, the Hasse graph is connected, and a cycle in it makes the local stochastic cone non-simplicial. A Euclidean finite-step cone argument then rules out the finite-valued monotone approximations supplied by the RB property.

We consider an optimal insulation problem for a bounded domain in $\mathbb{R}^N$ driven by the $p$-Laplace operator ($p>1$). We model the convective heat transfer between the body and the environment, which corresponds, before insulation, to a nonlinear Robin boundary value problem. Assuming the body is surrounded by a thin layer of insulating material of size $\varepsilon^{\frac{1}{p-1}}$, we compute the $\Gamma$-limit of the governing energy functional as $\varepsilon \to 0^+$. Furthermore, we study the optimization of the heat content among all possible distributions of the insulating material with a fixed total mass. Finally, we highlight a concentration breaking phenomenon. Under a suitable non-degeneracy condition, if the boundary of the domain is connected or the external temperature profile is constant, the optimal insulating layer fails to cover the entire boundary whenever the total mass is sufficiently small. This is shown to be optimal: an explicit example provides that a disconnected boundary can trigger an anomalous double-phase transition, causing the insulation to fracture again even at intermediate mass regimes.

We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies \[ T(X\oplus_1 Y)=\min\{T(X),T(Y)\}. \] We show that this equality does hold whenever one of the two summands has the Daugavet property. On the other hand, if $D$ is a Banach space with the Daugavet property and $N$ is a suitable absolute norm, then for $X=D\oplus_N D$, one has $T(X\oplus_1 X)<T(X)$.

A coloring of the Hales–Jewett cube $[t]^n$ is symmetric if it is invariant under all coordinate permutations, and one-weight if it reads only an integer-weighted count of the letters. We prove that the two classes coincide – a radix weight realizes every symmetric coloring – so the symmetric lower-bound problem for the Hales–Jewett numbers is exactly a one-dimensional coloring problem about homothetic copies of a $t$-point set, the case $d=1$ of Gallai's theorem. Optimizing the weight yields $\mathrm{HJ}(3,3)\ge22$ and $\mathrm{HJ}(4,2)\ge14$, the latter in closed form from the new Gallai homothety numbers $G_2(\{0,2,3,5\})=67$ and $G_2(\{0,1,5,6\})=80$; new values at three colors – $G_3(\{0,1,3\})=42$, $G_3(\{0,1,4\})=57$ and $G_3(\{0,2,5\})\ge77$ – give $\mathrm{HJ}(3,3)\ge16$ from a one-line certificate. An anatomy of the $(4,2)$ palette locates the source of its compression: it is an extremal object of the bracket regime plus a single boundary scale. An exhaustive census shows how thin the class is: of the $1644$ line-free $2$-colorings of $[3]^3$, exactly $36$ are symmetric. For lines with at most $K$ active coordinates the same machinery gives infinite bracket numbers, $\mathrm{HJ}^{[12]}(3,3)=\mathrm{HJ}^{[12]}(4,2)=\infty$, strictly beyond the sum-type ceilings $\kappa_{\mathrm{sum}}(3,3)=11$ and $\kappa_{\mathrm{sum}}(4,2)=10$; for lines whose active set is an interval the machinery is provably blind, the interval ceiling $\lambda(3,r)$ is settled for every $r$ by assembling the known bounds, and a SAT computation gives the exact value $\mathrm{HJ}^{(1)}(3)=5>4=\mathrm{HJ}(3)$. We close with the Collapse, diagonal-only, and symmetric-extremality conjectures and with open problems on optimal weights. Every certificate displayed in this note has been re-verified by direct enumeration, independently of any solver.

In this paper, we investigate the relationship between quasi-$F$-splitting and log canonicity. We show that if a numerically $\mathbb{Q}$-Gorenstein normal singularity is quasi-$F^e$-split for every $e\geq 1$, then it is numerically log canonical. In dimension two, we prove the converse under the condition that the Gorenstein index is not divisible by the characteristic $p$. We also classify two-dimensional quasi-$F$-split normal singularities.

We study the trianguline variety for split connected reductive groups. We generalize a theorem of Breuil, Hellmann, and Schraen about its local structure, establishing smoothness over the loci determined by various regularity conditions on the triangulation parameter, and normality at certain points outside of these smooth loci. Along the way, we prove a crystallinity criterion for $(\varphi,\Gamma_K)$-modules with $\mathsf G$-structure.

We give counterexamples to two well-known conjectures about matroids: White's conjecture on the generation of the toric ideal by symmetric exchange binomials, and a conjecture of Mason on the log-concavity of the counts of flats of a given rank.

Predictions are increasingly used to guide high-stakes decisions, from treatment selection to policy making. To ensure reliability with imperfect predictions, uncertainty quantification methods such as conformal prediction build prediction sets with coverage guarantees. However, statistical validity alone does not immediately determine the decisions to take, nor the optimality thereof. This gap is especially delicate in counterfactual settings where the outcome that materializes depends on the action taken, so uncertainty cannot be specified independently of the decision rule. We develop a decision-theoretic framework for uncertainty-informed counterfactual decisions. We identify a novel notion of policy-coupled coverage – namely, coverage of the realized outcome under the action induced by the prediction sets themselves – as the optimal and lossless interface between uncertainty and action. It plays three roles. First, it justifies acting via a natural max-min rule as minimax-optimal under distributional ambiguity. Second, optimizing prediction sets under policy-coupled coverage is equivalent both to a stronger universal-coverage formulation and to the direct risk-averse optimization over policies and utility certificates; this equivalence yields the explicit form of the population-optimal prediction sets. Third, it admits a two-stage procedure, Policy-Coupled Risk-Averse Conformal Prediction (PC-RACP), that approximates these optimal sets with rigorous finite-sample coverage. Simulations and a real email-marketing experiment confirm that PC-RACP delivers higher utility than existing approaches while maintaining valid coverage, and that ignoring the counterfactual structure of the decision problem is suboptimal for both validity and utility.

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms. By using the relative inner automorphism group, we define a notion of connectedness for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal connected–covering factorization for arbitrary surjections. We also introduce strongly connected homomorphisms via the relative transvection group, and classify them in terms of perfect normal subgroups of inner automorphism groups. A later part of this paper studies surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.