Papers

We study the dynamics of Krylov state complexity in finite-dimensional quantum systems. Orthogonal polynomials built on a discrete energy spectrum crowd away from regions where the density of states is low at large Krylov index. The physical implication of this result is that, even using a minimalist Krylov state complexity which is defined over the entire spectrum, the Krylov complexity provably stops growing a little past the Heisenberg length in every low-energy state. This has interesting implications for the proposal that Krylov complexity is related to the wormhole length in 2D quantum gravity.

For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\operatorname{SO}(3)$ connections on the bundle $\Lambda^+$ of self-dual $2$-forms. On the open locus where the self-dual curvature is an orientation-preserving frame, pointwise polar decomposition removes the gauge freedom and replaces the connection by a positive self-adjoint endomorphism field $h$ of $\Lambda^+$. Using the classical reconstruction of the compatible connection $A(h)$, we obtain a global formulation of the Yang–Mills equation as the determined second order system $$ \Phi_g(h):=F_{A(h)}^+h^{-1}-\operatorname{Id}=0. $$ Here, the tensor $F_{A(h)}^+$ is the self-dual curvature of $A(h)$, regarded as an endomorphism of $\Lambda^+$, and $\operatorname{Id}$ is the identity of $\Lambda^+$. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator $D_h\Phi_g$. For fields of the form $h=e^{2\omega}\operatorname{Id}$, where $\omega$ is a smooth real-valued function on $M$, the equation $\Phi_g(h)=0$ is equivalent to anti-self-duality and constant scalar curvature $6\sqrt{2}$ of the conformal metric $\hat g=e^{2\omega}g$. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of $\Phi_g(h)=0$.

The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber $\kappa^2$ complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to in vivo brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of $0.77$ with measurement, above a $0.75$ target. The gain arises from the multiscale kernel rather than from real–imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.

On a spacetime endowed with a time function (such as a globally hyperbolic manifold) we solve the global Cauchy problem for hyperbolic D-modules in the framework of Sato's hyperfunctions, illustrating the effectiveness of microlocal-sheaf theoretic methods in linear problems.

Adriano Mario Garsia was born in Tunis on August 20, 1928, to a Tunisian-Italian family. He lived on a farm there until the end of World War II, then moved to Rome. After finishing high school, he was sent to the United States to live with relatives in Woyming and eventually made his way to California, becoming a student of Charles Loewner at Stanford in the early 1950s. Following his Ph.D., Adriano held positions at MIT, the University of Minnesota, and Caltech before joining the nascent mathematics department at the University of California, San Diego, in 1966 where he spent the remainder of his career. He passed away in San Diego on October 6, 2024, at the age of 96.

Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint. The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>\Phi(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at https://github.com/abhmul/weak-simplex-conjecture-lean.

In this article, we study scattering and inverse scattering problems for 3D nonlinear Schrödinger equations of the form, $$ \mathrm{i}\partial_t u + \Delta u = a(x)|u|^2u.$$ For $a(x)$ in a suitable class, such equations are globally well-posed and admit a large-data scattering theory. In this work, we prove that the large-data scattering behavior uniquely determines the inhomogeneity $a(x)$.

Single-file diffusion is a ubiquitous phenomenon in low-dimensional systems, arising in transport inside narrow channels. Its natural continuum model is a one-dimensional gas of extended Brownian hard rods (BHR). Perhaps owing to the perceived intractability of this problem, much of the literature has traditionally focused on lattice exclusion models, where integrability methods have yielded remarkable, albeit limited, exact results. A major recent advance comes from a formal solution of macroscopic fluctuation theory (MFT) for the exclusion process. Yet, despite the formal solution, only a handful of properties have been made explicit. We show that the corresponding MFT of the extended BHR gas is in fact exactly solvable through a canonical transformation. We demonstrate this by explicit computation of the large-deviation statistics of the tracer-position and integrated-current in both annealed and quenched ensembles. We further show that an analogous canonical transformation applies to the MFT of lattice gases with finite-volume exclusion, yielding corresponding tracer and current statistics. We validate our results using rare-event simulations for both the continuum and the lattice models.

A set $A\subseteq\mathbb{N}$ is called $\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. Building on recent work of Bergelson and Simmons and that of Griesmer, we establish a new strong-completeness criterion exploiting a three-component partition of a given set. As an application, we show that $A$ is strongly complete whenever \[ \big|A\cap(2^k,2^{k+1}]\big|\ge5 \] for every sufficiently large $k\in\mathbb{N}$, and \[ \sum_{a\in A}\|a\theta\|=\infty, \quad\forall\theta\in\mathbb{R}\setminus\mathbb{Z}. \] In particular, this resolves a 1961 conjecture of Erdős. The new strong-completeness criterion also enables us to make progress on a 1996 problem of Burr, Erdős, Graham, and Li concerning strong completeness of mixed power sets by refining a previous result of Bergelson and Simmons. Besides, we study the polynomially perturbed ray set \[ \{\lfloor t\alpha^n\rfloor,\lfloor t\alpha^n\rfloor+P(n):n\in\mathbb{N}\}, \] which combines the polynomial set $\{P(n):n\in\mathbb{N}\}$ and the single-ray set $\{\lfloor t\alpha^n\rfloor:n\in\mathbb{N}\}$ both previously considered by Graham, and show that it is strongly complete for any $t>0$ and $\alpha\in(0,2)$ and any primitive integer-valued polynomial $P$. The machinery developed for the proof of this result also yields other interesting applications.

The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a $k$-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work, starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all $k \geq 3$, with no superfluous polylogarithmic factors. For the case of $k$ even, our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using the polynomial method. The argument is then extended to the case of $k$ odd by adapting a procedure in [GKM22].

Speckle noise is a multiplicative noise commonly encountered in coherent imaging modalities such as synthetic aperture radar, optical coherence tomography, and digital holography. Although deep learning methods, in practice, have achieved state-of-the-art performance for speckle denoising, their fundamental statistical limits remain largely unexplored. Unlike additive noise models, multiplicative speckle noise makes the regression function unidentifiable from the conditional mean, rendering conventional least-squares-based deep learning approaches inapplicable. We study the minimax estimation of smooth nonparametric regression functions using likelihood-based deep neural network (DNN) estimators under a model with both multiplicative speckle noise and additive Gaussian noise. Our framework accommodates both low-dimensional and sparse high-dimensional features. We establish finite-sample upper bounds on the estimation error of the proposed DNN estimators and derive minimax lower bounds for nonparametric function recovery under our model, showing that they match up to logarithmic factors in the sample size. Moreover, these minimax rates coincide, up to logarithmic factors, with those for nonparametric regression under additive Gaussian noise alone, demonstrating that the intrinsic difficulty of estimation remains essentially unchanged despite the challenges posed by multiplicative speckle noise. Numerical experiments further supports consistency of our DNN-based despeckling methods and demonstrate their effectiveness.

We present a new percolation model on the two-dimensional lattice, which can be seen as a conditioned version of continuous percolation on the plane. Let us place a point uniformly at random in each cell of the grid $\mathbb{Z}^2$. These points correspond to the vertices of our graph, and we connect two points by an edge if their distance is less than a fixed radius $R$. We are interested in the radius from which there exists almost surely an infinite connected component. We also study two other critical radii specific to the geometry of our model: the smallest radius such that there exists a positioning of the points for which there is an infinite connected component, and the radius from which all points are connected to each other.

We propose a general definition of morphism of representations up to homotopy of Lie algebroids, which is be mandated by the requirement that taking cohomology with coefficients in a representation up to homotopy behave functorially. With this definition in place, we show that such representations behave in a similar way to honest representations as far as homotopy invariance of cohomology and local triviality of smooth families are concerned, and for essentially the same reasons as in the classical case.

We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are \[ \PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)), \] respectively. The construction combines a skew brace with additive group \(A_5\) and multiplicative group \((C_5\rtimes C_4)\times C_3\), the splitting of the automorphism extension of \(\PSU_5(64)\), and a semidirect product of skew braces.

Expanding on the recent development of adaptive local kernel methods for approximating the action of linear operators, a local estimate of the error and an adaptive procedure for approximating solutions to the Poisson equation is developed. The error estimate is used in the midst of the adaptive procedure to determine locations in the solution domain where decreasing the spacing between nodes can decrease the error in the solution. The approach described here is essentially "meshless", with only local Delaunay triangulations leveraged as a convenience to provide information on the local geometry of the node set. Once this information is utilized, it is discarded. The experiments performed show close agreement between the error estimate and actual absolute forward error in the approximate solution along with a comparison to existing indicators for refinement. The combination of the adaptive procedure and error estimate provides an automated technique to resolve localized features of a solution without the, often intractable, expense of uniform refinement across the entire solution domain.

A triangle surface is an open algebraic surface completed by a triangle of contractible $(-1)$-curves. We establish a combinatorial description of their completions. We also show that affine triangle surfaces are exactly cubic surfaces of Markov type. Explicit description of their automorphism groups is provided.

In this paper, we investigate a discrete-time SIS epidemic model and the epidemic thresholds on complex networks. We focus on proposing a community-level epidemic threshold set and establishing a comparative result between the local epidemic thresholds and the global epidemic threshold. To verify our theoretical findings and structural properties, we conduct numerical experiments on one synthetic network (Network1) and one real-world network (the Haslemere contact network). Our numerical simulations, along with the computation and statistical ranking of the epidemic threshold sets, align accurately with our theoretical results.

Let $F_{\mathrm{bip}}(n)$ denote the maximum, over all $n$-vertex bipartite graphs without isolated vertices, of the ratio of the minimum local average degree to the global average degree. We prove that $F_{\mathrm{bip}}(n)=\frac14\sqrt n+\frac38+o(1)$. This answers a problem posed by Tuza. The lower bound is obtained from a new family of bipartite constructions, while the upper bound is proved by a new algebraic approach, different from the combinatorial methods used previously. The key ingredient is a spectral argument based on a normalized biadjacency matrix.

We show that a finitely generated group $G=\langle\Sigma\rangle$ satisfying the falsification by fellow traveller property also satisfies the shortlex falsification by fellow traveller property. This implies that there are finitely many shortlex cone types and, therefore, that the corresponding language of shortlex representatives $\text{SL}_G\subset (\Sigma\cup\Sigma^{-1})^*$ is a regular language. Following the example of M. Elder, we prove that the converse of the previous statements do not hold.

For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $\phi$ and its dual $\widehat{\phi}$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.

We show that a ribbon concordance minimal fibered knot $K$ in $S^3$ can generate ribbon concordance minimal links by the addition of any braid closure in $S^3 \setminus K$. As a corollary, we show that any link $L$ in $S^3$ may be made ribbon concordance minimal by adding a single unknot linked with $L$. Our proofs use link Floer homology together with classical techniques.

We prove a cyclic sieving result for the action of promotion on the staircase plane partitions of height two. Our proof has two major algebraic inputs: an interpretation of this promotion action in terms of tensor powers of the spin crystal that was recently studied by Pappe–Pfannerer–Schilling–Simone, and the bush basis of the degree two part of the coordinate ring of the space of electrical networks that was recently introduced by Gao–Lam–Xu. Moreover, we explain how the existence of an electrical canonical basis in all degrees would yield cyclic sieving for promotion of staircase plane partitions of all heights.

Many extremal problems can be viewed as finding large independent sets in an auxiliary hypergraph. We propose a generalization of this by looking for “large” independent sets $I$ in a hypergraph $\mathcal{F}$ where “large” is measured by how many edges $I$ induces in another hypergraph $\mathcal{H}$ on the same vertex set as $\mathcal{F}$. We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting $F$-free graphs. As applications, we prove new supersaturation bounds for generalized Turán problems, as well as supersaturation bounds for a new set of extremal problems inspired by work of Fox and Pohoata on finding subsets $A\sub\mathbb{N}$ which maximize the number of solutions to a given system of equations while avoiding solutions to another system.

Lambert series of the form $\sum_{n>0}a(n)q^n/(1-q^n)$ are ubiquitous in mathematical physics. In particular, 2-loop sunrise and 3-loop banana Feynman diagrams yield Lambert series with $a(n)$ of the form $\chi(n)/n^s$ where $\chi(n)$ is a Dirichlet character. Resurgence concerns the singular limit as $|q|$ approaches 1. In the Feynman cases we can control this limit, obtaining rapidly convergent expressions, since the Lambert series are iterated integrals of holomorphic Eisenstein series twisted by a character. We generalize this result, to include modular resurgent structures found in topological-string observables.

We answer negatively a finite-dimensional unitary-model question for complex symmetric operators. More precisely, we show that, for every \(n\geq 10\), not every \(n\times n\) symmetric matrix is unitarily equivalent to a direct sum of truncated Toeplitz operators. In order to do this, we first use semialgebraic dimension, a tool from real algebraic geometry, to prove a general theorem showing that, if \(\mathcal X\) is a semialgebraic family of complex symmetric matrices, then the set of complex symmetric matrices which are unitarily equivalent to an element of \(\mathcal X\) is semialgebraic and has dimension at most $\dim_{\mathbb R}\mathcal X+\frac{n(n-1)}2.$ We then apply this theorem to show that when $n\geq 10$ there exist irreducible symmetric $n \times n$ matrices which are not unitarily equivalent to a truncated Toeplitz operator. Although unitary equivalence is too restrictive, we prove that every finite-dimensional complex symmetric operator is complex-orthogonally equivalent to a coanalytic truncated Toeplitz operator. We also answer positively a refined representation question by showing that whenever a symmetric matrix is unitarily equivalent to a truncated Toeplitz operator, it is a matrix representation of that operator with respect to a conjugation-invariant orthonormal basis.

We formulate and prove internal versions of the Yoneda lemma and of the Yoneda embedding theorem in a finitely complete, locally Cartesian closed $\infty$-category $\mathscr{C}$: for every object $X\in \mathscr{C}$ and every universe $\mathscr{U}$ classifying the diagonal of $X$, the Yoneda map $\mathscr{Y}_X\colon X \to \mathscr{U}^X$ is a monomorphism. The proof uses only finite limits, dependent products and universes, and does not rely on the external Yoneda lemma. The result applies notably to every elementary $\infty$-topos, where it recovers a theorem of Rasekh [Ras18].

Inspired by the notion of tricolorability of knots, we introduce the concept of lean coloring for hypergraphs and the associated lean number of a hypergraph. Lean coloring often involves very few colors, yet still requires the methods of usual graph coloring, forcing the overall complexity to be NP-Hard. We provide two alternative formulations of the lean coloring problem that involve a type of coloring on abstract simplicial complexes and a partial coloring on bipartite graphs. We then provide bounds for the lean numbers of hypergraphs that are $k$-uniform, $k$-partite, wide-path connected, or $r$-complete. Python-like script is included to allow the implementation and study of a lean coloring algorithm. We conclude with some directions for future work and present the lean numbers of $130$ knots and links.

Covert communication enables Alice to transmit to Bob while making the transmission difficult for Willie to detect. We study a scalar Gaussian covert-overlay model in which Alice's low-power covert signal is superimposed on an aggregate public component generated by Alice or other trackable sources. Willie is given all physically obtainable side information, including protocol details, timing, pilots, channel estimates, and calibration information, and subtracts his best estimate of the public component before testing. Covertness is imposed on the resulting residual through a relative-entropy constraint with budget $\delta$ conditioned on Willie's side information. In the stationary case, the residual under no covert transmission has variance $\sigma_0^2=\sigma_W^2+\sigma_e^2$, where $\sigma_W^2$ is Willie's receiver-noise variance and $\sigma_e^2$ is the irreducible cancellation error. Over $n$ channel uses, the maximal reliably transmissible covert payload is $R_C^\star\sqrt{n}(1+o(1))$ bits, where $R_C^\star=\frac{\sigma_0^2}{\sigma_B^2\ln 2}\sqrt{\delta}$, and $\sigma_B^2$ is Bob's receiver-noise variance. Thus, the square-root-law (SRL) constant is governed by the variance at Willie's actual detector input, not by receiver noise alone. Low-power Gaussian signaling achieves this constant, and a matching converse establishes first-order optimality within the conditioned additive Gaussian innovation model. For known time-varying conditioned residual variances, we also derive the first-order allocation, which assigns more covert power to larger residual variances. The results require a Gaussian post-cancellation null residual with known conditioned variance; non-Gaussian residuals and fixed non-vanishing variance uncertainty are outside the scope of this paper.

Every even symplectic Hirzebruch surface contains a Lagrangian torus and every odd symplectic Hirzebruch surface contains a Lagrangian Klein bottle as their respective real locus. From a toric point of view, this is a visible Lagrangian which surjects to the full moment polytope under the moment map. In this paper, we prove that every Hirzebruch surface contains both a Lagrangian torus and a Lagrangian Klein bottle with this property. An interesting consequence is that the topology of visible Lagrangian submanifolds is not determined by their image under the moment map. The proof is based on a new construction, which we call the spread of a family of Hamiltonian diffeomorphisms.

Let $r$ be a ring of positive characteristic with a $p$-basis over a subring $k$. We explore necessary and sufficient conditions under which the module of $k$-derivations of $R$ coincides with the module of integrable $k$-derivations of $R$.