Papers

We propose boundary theories, CFT$_{p}$ with $p\in \beta\mathbb{N}/\mathbb{N}$, which can be interpreted as an ensemble of theories in the context of AdS/CFT duality, when the boundary spacetime dimension is more than one. These theories are only emergent in the infinite $N$ limit of the boundary CFT and depend on $p$, which corresponds to the different ways one can take a generalized form of the infinite $N$ limit, in the presence of a chaotic or an oscillatory operator. This generalized infinite $N$ limit is called an ultralimit, and $p$ is a free ultrafilter on $\mathbb{N}$. We propose that the gravitational path integral computes an average of CFT$_{p}$ in each of its sectors; and in the appropriate cases, it produces baby universes and spacetime wormholes. We apply this proposal to the Antonini-Sasieta-Swingle (AS$^{2}$) like states.

We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamiltonian with real-analytic dispersion, a Fermi sea confined to a finite momentum range, and nonzero Fermi velocity at every Fermi point, we analyze a particular Abel-regularized transport response (obtained by tracing over the Fermi sea) and prove that its large-time limit equals the Euler characteristic. We derive an explicit finite-time formula, show that quantization requires the large-time limit, and obtain a convergence rate under additional nonstationarity assumptions. We also prove the lattice analog, where the sharp commutator is trace-class and a completely filled band contributes zero. Finite-volume calculations illustrate the prescribed order of the thermodynamic and large-time limits.

Let $s\geq 2$ and let $v_1,\ldots,v_{2^{s-1}}\in F_2^s\setminus\{0\}$ have sum zero. The prescribed-difference matching problem asks whether $F_2^s$ can be partitioned into pairs whose differences, counted with multiplicity, are exactly these vectors. Fix a hyperplane $H\leq F_2^s$. An $h$-hole instance is one in which exactly $h$ prescribed differences lie in $H$, counted with multiplicity. We prove that every four-hole instance has a solution for $s\geq 3$ and every eight-hole instance has a solution for $s\geq 4$. Thus no restriction on the multiplicities or on the sum of the holes is needed beyond the global zero-sum condition. After the internal pairs are placed, their endpoints are deleted from the two affine halves determined by $H$, and the remaining vectors must be paired with the prescribed crossing differences. A locally valid placement of the internal pairs need not admit such a completion. We encode the possible completions by coefficients of signed determinants and use the Walsh transform to sum over all disjoint legal placements while keeping the crossing profile fixed. Nonvanishing of this sum guarantees that at least one placement extends to a full matching. The four-hole and zero-sum eight-hole theorems are proved theoretically. The general eight-hole proof uses three finite exact verifications, independent of the crossing profile: a local filter calculation, a classification of the remaining eight-hole configurations, and integer certificates for those configurations. The verification software, exact inputs, and recorded outputs are archived in a versioned Zenodo record.

Motivated by a conjecture of Burq and Gérard, we investigate quantitative uniqueness principles for functions on $\mathbb T^2$ whose Fourier spectra lie in fixed-width annuli. We obtain observability estimates uniform in the radius, under a mild Sobolev regularity condition, and the generalized geometric control condition (GGCC) on the damping function. This leads to exponential decay for the damped wave equation with rough damping. We also establish an analogous uncertainty principle for spectra near dilates of convex polygonal boundaries, where no Sobolev regularity of the damping is required beyond $L^\infty$.

Hitomezashi is a particular style of sashiko stitching characterized by crossing lines of stitches. Certain hitomezashi designs can be encoded with two binary strings and have been the subject of a wide-array of mathematical research. Dubbed generalized hitomezashi patterns (GHPs) by Sen & Martinez, these designs are reversible and can be 'self-dual.' We investigate the possible 'dual-' or 'flip-symmetries' in GHPs; the inclusion of these symmetries is equivalent to considering two-colour symmetry groups. Following the work of Sen & Martinez, who found the wallpaper symmetry groups that are compatible with GHPs, this paper investigates the possible two-colour symmetries compatible with GHPs. We prove that there are four possible rosette two-colour symmetry types, completely describe the properties on binary strings that create these symmetries, and prove that exactly 17 of the 46 two-colour wallpaper groups are compatible with GHPs.

The multiple $T$-values (MTVs), first studied by Kaneko and Tsumura, are a variation of the multiple zeta values (MZVs) with restricted product structure. Motivated by a deep conjecture of Kaneko and Zagier relating finite MZVs and symmetric MZVs, which was extended to Euler sums by Zhao, we study finite and symmetric multiple $T$-values. In particular, we show that finite MTVs satisfy Hoffman-type duality relations at low height, confirming several conjectures of the second author and discovering new families of identities. In proving relations among symmetric MTVs, our work builds on Xu and Zhao's theory of multiple mixed values, the relations of double zeta values discovered by Gangl, Kaneko and Zagier, and the (weighted) sum formulas of double Euler sums discovered by Berger et al. We then use generating functions derived from the integral structure of MTVs to establish the corresponding relations for finite MTVs. These results aid in the computation of dimensions of the $\Q$-vector space spanned by finite and symmetric MTVs (modulo $\zeta(2)$ products), providing strong evidence for an isomorphism between the two spaces.

We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \(x\) of active features is encoded through an overcomplete dictionary \(W\), and feature recovery is performed by applying \(\operatorname{ReLU}(W^\top W x+b)\) with an appropriate bias vector \(b\). We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order \(d/\log n\). In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with \(n>d+1\), we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization—which should be of independent interest to frame theorists—of the distribution of signs in the Gram matrix.

We consider the following problem: let $n>k$ — positive integers, $G$ — a graph on $n$ vertices (undirected, without loops or multiple edges). Let $h_k(G)$ denote the number of unordered pairs of vertices of the graph $G$ whose degrees differ by less than $k$. We seek to determine the smallest possible value $f(n,k)$ of $h_k(G)$. The interest in this question is motivated by the fact that the bipartite analogue of the problem allowed S. Cichomski and F. Petrov \cite{CP} to prove the Burdzy–Pitman conjecture on the spread of independent identically distributed random variables.

A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction.

We prove that the normalized $k$-th moment of the Laguerre unitary ensemble, as a function of the dimension, is decreasing for fixed $k\in(0,1)\cup (2,\infty)$ and increasing for fixed $k\in(1,2).$ As a corollary, this implies that the average singular value of complex Gaussian random matrix decreases with dimension. Conjectured by Bandeira, Kennedy, and Singer (Math. Program., 2016), this property was recently established by Hutnik, for non-negative integer shape parameter $\alpha$. Our result extends this monotonicity to real $\alpha\geq 0$.

Let $U$ be a Sylow $p$-subgroup in a classical orthogonal group over a finite field with $q$ elements of characteristic $p$ large enough. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. In the paper, we provide a classification of such orbits of pre-maximal dimension. As a corollary, we compute the number of all orbits mentioned above. It turned out that each of these numbers is a polynomial in $q - 1$ with integer non-negative coefficients, which agrees with the Isaacs' conjecture.

Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field with $q$ elements of characteristic $p$ large enough. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. In the paper, we provide a classification of such orbits of pre-maximal dimension for symplectic groups and orbits of maximal dimension for orthogonal groups. As a corollary, we compute the number of all orbits mentioned above. It turned out that each of these numbers is a polynomial in $q - 1$ with integer non-negative coefficients, which agrees with the Isaacs' conjecture.

We study low-rank quantum state tomography from finite-bit Pauli batch responses. To avoid bias introduced by generic quantization, we propose HyperQuant, a mean-preserving hyperbolic quantizer adapted to the second-moment scale of Pauli responses. We establish minimax distortion guarantees and show that exact mean preservation enables direct rank-constrained least-squares recovery without altering the population target. We derive nonasymptotic recovery guarantees and an explicit bit–shot tradeoff under which finite-bit responses retain the error order of unquantized batch averages using fewer response bits. For efficient computation, we develop QuantRGD, a Riemannian gradient method with provable linear convergence to the corresponding statistical neighborhood under explicit resource conditions. Numerical experiments validate the predicted quantization, recovery, and convergence behavior.

We introduce the Quantum Information Space at Infinity (Quinfinity) $\mathcal{Q}_\infty$ as the inverse limit of the symbolic Quantum $N$-Spaces $\mathcal{Q}_N$, identified with the complete local real algebra of formal power series under the adic topology. The pure state pro-variety is formalized as the inverse limit of $N$-level pure state varieties, driven by the algebraic colimit of a direct system of real radical ideals. This real architecture internalizes the imaginary unit, collapsing the metric landscapes onto an invariant Riemannian isometry, which identifies the pro-variety directly with the classical Fisher-Rao statistical manifold. Within this framework, the Heisenberg uncertainty principle undergoes a structural regularization at infinity being intrinsically manifested as a localized truncation obstruction governed by adic algebraic derivations. Concurrently, the continuous Liouville-von Neumann equation operates as an internal derivation tangent to the pro-variety, while the open Gorini-Kossakowski-Sudarshan-Lindblad asymptotic master equation is intrinsically obtained via a non-associative symmetric Jordan product. This dissipative flow acts as a contracting radial vector field that dampens higher-order jet configurations, driving the state trajectories down the hierarchical tree toward the absolute zero element as a non-singular universal attractor.

We prove an $(\infty,2)$-categorical version of Koszul duality for operads and cooperads in spectra by showing that there is an equivalence between a Morita category of operads and bimodules and a dual Morita category of cooperads and bicomodules. This result subsumes various forms of operadic Koszul duality present in the literature.

We give an all-level proof of the four-point $SU(2)$ AGT correspondence with four fundamental hypermultiplets at generic $\beta=-\epsilon_1/\epsilon_2$, by proving an all-level factorization formula for Selberg averages of generalized Jack polynomials. Taking a coefficientwise Jack limit of the generalized Macdonald Pieri rule, we obtain the required one-box matrix elements in the strict Cauchy dual basis. A rational corner-function identity then evaluates the sum over all parent double partitions, while an explicit total derivative of the Selberg kernel yields a triangular recursion on the Dotsenko-Fateev charge balance hyperplane. The unique solution of this recursion is the generalized Kadell formula previously verified only through finite level. Combining this result with the generalized Cauchy identity identifies each double partition term with the corresponding Nekrasov fixed-point contribution.

In this paper, we prove finiteness results for K-trivial varieties in all dimensions. In dimension 2, these results are corollaries of the Global Torelli Theorem for polarized K3 surfaces. In particular, we establish a sufficient condition when two fibers of a projective family of K-trivial varieties are isomorphic via the Hodge theory. As applications, we prove that if a projective family of K-trivial varieties has a dense subset of birationally equivalent fibers, then all fibers are isomorphic.

We prove that the $q$-Ehrhart series of a hyperplane slice of a cube is a rational function with an explicit denominator that satisfies $q$-reciprocity, confirming a conjecture of Reiner and Rhoades for these polytopes. To do this, we find a generating set for the orbit harmonics ideal, which also yields the Hilbert series and graded Frobenius characteristic of the associated quotient. We further show that the harmonic algebra of a hypersimplex $\Delta$ is generated as an algebra by the harmonic space of $\Delta$ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.

Let $A$ be a commutative-associative algebra with an invertible derivation $D$, and put $R=D^{-1}$. We study the operations \begin{equation*} x\succ y=R(x)y,\qquad x\prec y=xD(y), \end{equation*} which define a Novikov-Zinbiel algebra. Using a simple graded model, we represent multilinear $\prec,\succ$-monomials by rational functions and construct an explicit basis for the resulting space. This yields a description of the multilinear components of the free Novikov-Zinbiel algebra. As a consequence, we prove that every multilinear identity satisfied by this construction follows from the defining identities of Novikov-Zinbiel algebras.

This paper addresses centroids, which are fundamental invariants of tensors. Our main results are as follows: (i) The construction of explicit tensors with very large centroids, whereas previously it had been conjectured that none such exist. (ii) An upper bound on the dimension of the centroid that is essentially attained by our examples. (iii) The development of a geometric technique to write down border rank decomposition of tensors using centroids and "extended centroids". (iv) The technique is applied to tensors of this paper to prove they are of minimal border rank. The technique is versatile and enables us to geometrically derive and improve upon previous ad hoc decompositions. (v) The construction of symmetric tensors with large centroids and proof that they are wild in the sense of Buczyńska-Buczyński. Our results also pave the way for new upper bounds on the exponent of matrix multiplication. The geometric technique also constructs new "better" tensors for Strassen's laser method from old, and we apply this to the tensors of Strassen and Schönhage to get better tensors in the sense that they give better upper bounds on the exponent than the original tensors.

In this paper, we develop the weighted level set method for a Kähler manifold $(X^n,\omega)$ admitting an almost holomorphic map to a possibly singular base $Z$, which is not uniruled. As a key intermediate result, we prove that any blowup $\operatorname{Bl}_SX$ of $X$ along smooth submanifolds $S$ of $ \operatorname{codim} S\ge2$ admits a sequence of Kähler metrics with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $\omega$. As a consequence, we establish the sharp \(2\)-systole estimate for every positive scalar curvature Kähler manifold $(X,\omega)$ and prove $\min_XS(\omega) \cdot\operatorname{sys}_2(\omega) \le 2\pi r(r+1)$, where \(r\) is the rational dimension of $X$, with equality if and only if the universal cover splits as $(\widetilde X,\widetilde \omega) \cong (\mathbb P^r,\omega_{\mathrm{FS}}) \times(Y^{n-r},\omega_{\mathrm{RF}})$ up to normalization where $\omega_{\mathrm{FS}}$ is the Fubini-Study metric and $\omega_{\mathrm{RF}}$ is Ricci-flat. We also show a sharp even-systolic inequality in the same setting when the general fibre is the projective space.

Let $q(n)$ denote the largest number of queens that can be placed on an $n\times n$ chessboard so that no queen attacks more than one other queen. We prove that $q(n)=\lfloor4n/3\rfloor$ for every $n\geqslant6$, and that $q(n)=n$ for $n\leqslant5$, which settles a previously conjectural value. As a corollary, we also settle that, in the version of the problem where each queen attacks exactly one other queen, the answer is $2\lfloor2n/3\rfloor$, again as previously conjectured.

In this paper, we study the dependence of the propagation speed on the diffusion ratio in a Lotka-volterra competition-diffusion system under strong competition. The model is known to admits a unique monotone travelling front connecting two exclusion equilibria and the sign of its speed determines which species invades the territory occupied by the other. We establish smoothness results for the wave speed and the wave profile. Our main result shows that the wave speed is a strictly decreasing function of the diffusion ratio on explicit unbounded regions of parameter space. This monotonicity had been observed numerically but, had not previously been proved. The proof combines the smooth dependence of both the travelling front and its speed on the diffusion ratio, an adjoint identity for the linearized operator and a maximum principle argument. At zero speed, we also prove that there exists a smooth threshold, we obtain an exact formula for its derivative and deduce its monotonicity. These threshold results also provide a complete characterization of the sign of the wave speed.

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, so the boundary data of such a stranding is recorded by a row-strict tableau. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis. This non recursive construction recovers Fontaine's bases and produces others. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we ask whether any two strandings of a web are connected by a sequence of local moves called strand reversals. We answer this question affirmatively for $\mathfrak{sl}_3$, extending a known result for closed webs to webs with boundary; in particular, a leading term stranding of any $\mathfrak{sl}_3$ web is always reachable from an arbitrary stranding.

We present a finite element framework to simulate rotating mixing devices using the Mortar Element Method as a domain decomposition strategy. The model is implemented within a matrix-free Navier-Stokes framework which uses a high-order Continuous Galerkin method. The discretized domain is subdivided into rotor and stator parts. An Arbitrary Lagrangian-Eulerian approach accounts for the relative rotor-stator motion, and stabilization is ensured through the Streamline-Upwind/Petrov-Galerkin and Pressure-Stabilizing Petrov-Galerkin methods. The rotor-stator domains are connected by an interface composed of mortar cells, and continuity is weakly enforced in a Discontinuous Galerkin fashion by accounting for boundary integrals at the rotor-stator interface. Verifications of the convergence order in two-dimensional steady and transient examples report optimal rates. The geometric non-conformity created at the mortar interface due to rotor rotation does not introduce significant error in the solution. A three-dimensional example is used to investigate the model's scalability, which yields ideal strong scaling for large problems. A two-dimensional Rushton impeller example uses a torque analysis to showcase the mesh convergence, and the corresponding velocity profile is in agreement with existing numerical results. In a three-dimensional pitched blade turbine case, the power number curve (Np vs Re) shows good agreement with experimental data for Reynolds number values from 1 to 2000. An energy balance analysis reports a numerical dissipation of 1% for Re=200 and of 10% for Re=2000. By exploiting modern hardware capabilities through matrix-free methods, the proposed model is a robust, accurate, and efficient framework suitable for simulating flows with rotating geometries.

In this paper, we consider Robbins' problem, which is a full information variant of the well-known secretary selection problem. In this version of the problem, the goal is to minimize the expected rank of the selected candidate among $n$ that are interviewed sequentially, and a decision to select or not the $m^{th}$ candidate needs to be taken right after the interview (so without seeing the last $n-m$ candidates and without recall). We first show how to model instances of Robbins' problem as infinite Markov Decision Processes (MDPs). Then we propose several finite-state abstractions of these MDPs that allow us to approximate the value of the problem for fixed $n$. While it is known that the full memory of past candidates' values is necessary for optimal expected rank minimization, making the analysis of the problem challenging, we highlight simple memory structures that are sufficient for obtaining near-optimal selection strategies. Additionally, we provide approximate values for Robbins' problem for numbers of candidates $n$ up to 100 for which no good approximations were previously known (the exact value is only known for instances where $n \leq 4$ and numerical approximations were for small values of $n$ not exceeding one digit), for all $n : 5 \leq n \leq 100$, we give better approximation than what was previously known.

For $a,c\in\mathbb{N}$ and $b,d\in\mathbb{Z}$ such that the (non-empty) set \[ R_{a,b,c,d} :=\left\{\frac{an+b}{\,cn+d\,}: n\in\mathbb{N}\right\} \cap\bigl(\mathbb{Q}_{>0}\setminus\{1\}\bigr) \] is multiplicatively recurrent, we give a complete characterization of the set of limit points of every unimodular multiplicative function $f\in\mathcal{M}$ along $R_{a,b,c,d}.$ We show that the possible limit sets are either the finite subgroups of the unit circle or the entire circle, thereby extending the dichotomy of Klurman–Mangerel.

The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős–Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.

Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $\rho$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $\rho$ induces a very good $G$-grading on the structural matrix ring $M_n(\rho,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $\rho$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $\rho$ to the ring, but the converses fail in general. We also give a concrete criterion for epsilon-strongness and show that a very good $G$-grading on $M_n(\rho,R)$ that is strong satisfies $|G|\leq n$. When $\rho$ is an equivalence relation and the neutral component is diagonal, very good gradings correspond bijectively to free partial actions of $G$ on $\{1,\ldots,n\}$ with orbit relation $\rho$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.

We construct Casimir operators of $4D, \,\mathcal{N}=2$ supersymmetry algebra in the harmonic superspace approach, and provide the explicit expressions for them in terms of covariant derivatives in the analytic basis. Specifically, the superisospin operator $C_I$ is expressed in terms of new 'long' harmonic derivatives $\nabla^{\pm\pm}, \nabla^0$ as $C_I = \frac{1}{4}\left(\nabla^0\right)^2 + \frac{1}{2}\left\{\nabla^{++},\nabla^{--}\right\}$. These derivatives involve non-local terms with the inverse Box operator $\Box^{-1}$ and satisfy modified $SU(2)$ algebra relations in which the eigenvalues of $\nabla^0$ on analytic superfields are shifted by -2 relative to the standard $U(1)$ harmonic charges. We also construct $\mathcal{N}=2$ superhelicity and super-isohelicity operators relevant to the massless case (with vanishing $\Box$) and find their eigenvalues for few instructive examples of $4D, \,\mathcal{N}=2$ superfield theories.