Papers

We present a tailored framework for solving regularized linear inverse problems using quantum optimization methods. By discretizing the solution space and encoding data fidelity and regularization terms into quadratic unconstrained binary optimization (QUBO) models, we formulate both Tikhonov- and sparsity-promoting regularized inverse problems within a unified quantum optimization framework. We further introduce a notion of quantum sensitivity that characterizes the effect of perturbations arising from approximate quantum evolution and discretization. We derive bounds relating these perturbations to stability properties of the underlying inverse problem, thereby establishing a theoretical connection between quantum solution variability and classical notions of ill-posedness. The framework is extended to variational inverse problems through wavelet-based representations and complemented by reduced-order modeling strategies in both parameter and Hamiltonian spaces to mitigate current hardware limitations. Numerical experiments on simulated and physical quantum hardware indicate that the resulting low-energy solution distributions retain information about the underlying inverse problem, while also revealing the limitations imposed by finite-time evolution, discretization, and hardware noise.

The polyconvex envelope, used in the calculus of variations and elasticity theory, was expressed by Dacorogna pointwise as a linear program on finitely atomic measures on the space of $m\times n$ matrices. Weizsäcker and Winkler proved that the corresponding linear program on Borel measures restricts to the extreme points without increasing the infimum. Combining the two, one obtains a speed-up of grid-based algorithms and a new proof that the polyconvex envelope can be computed by the moment sum-of-squares hierarchy. Motivated by these applications, we seize the essence of extreme points in moment problems. First, we characterize extreme points of an affinely constrained convex set by the injectivity of the constraint map on the smallest faces containing them. We then study finitely many moment constraints. The extreme points are finitely atomic measures that have an affine independence property, under natural assumptions. We retrieve this known result with a simplified proof and apply it to faces of the probability simplex, among them the face of Radon measures. In the converse, we find that the assumption of a simplex is redundant. The Richter-Tchakaloff theorem allows us to show that the infimum of an integral functional restricts to the extreme points without increasing the infimum, not just for the known case of Radon measures but for any convex set of probability measures that contains the point measures.

Hasunuma [Graphs Combin. 41:10 (2025)] proved that for $k\ge 2$, there exists a function $f(k)=O(k)$ such that every $k$-connected graph $G$ of order $n\ge f(k)$ with $\delta(G)\ge \frac{n}{2}$ contains a Hamiltonian cycle $H$ such that $G-E(H)$ is $k$-connected. In this paper, we show that for $k\ge 2$, if $G$ is a $k$-connected graph of order $n\ge 6k+6$ with minimum degree at least $\frac{n+1}{2}$, then for any two distinct vertices $u,v\in V(G)$, there exists a Hamiltonian $(u,v)$-path $P$ such that $G-E(P)$ is $k$-connected. Moreover, we further extend this result to $s$ internally disjoint spanning $(u,v)$-paths.

We study a finite-horizon optimal consumption and portfolio problem with labor supply flexibility and an irreversible early retirement option under a borrowing constraint. The agent chooses consumption, risky investment, and leisure before retirement, while after retirement labor income disappears and leisure is fixed at its maximal level. Preferences are described by a Cobb–Douglas utility, and wealth must remain nonnegative. {Using a dual martingale method, we transform the primal problem into a zero-sum stopper–singular-controller game. The associated dual value is characterized by a min–max parabolic variational inequality with obstacle and gradient constraints. We show that the maximal strong solution of the resulting variational inequality is the unique admissible strong solution whose gradient-constrained free boundary, namely the binding boundary, is monotone increasing in calendar time. A verification argument then identifies this strong solution with the value of the stopper–singular-controller game, and duality recovers the optimal retirement, consumption, leisure, and portfolio policies.} The numerical analysis recovers the value function and optimal policies, and illustrates how labor supply flexibility affects consumption, portfolio choice, and retirement timing under borrowing constraints.

Let $M^n\subset \overline{\B_R^{n+k}}\subset \R^{n+k}$ be a compact orientable free boundary $f$-minimal submanifold of the Gaussian-weighted Euclidean ball $\left(\overline{\B_R^{n+k}},g_{\rm can},e^{-f}\dd V\right), f(x)=\frac c2 |x|^2,c\ge 0.$ We prove a cohomology vanishing theorem under the pointwise pinching condition $ |A|^2\le \frac{n-p}{R^2},1\le p<n.$ More precisely, the space of tangential $f$-harmonic $p$-forms vanishes, and hence$H^p(M;\R)=0.$ The proof is based on three elementary ingredients in the Gaussian-weighted ball: a weighted Hardy inequality obtained from the identity $\divf(x^T)=n-c|x|^2$, a cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction showing that tangential $f$-harmonic forms satisfy the same local absolute-boundary algebra as in the unweighted case. The constant pinching threshold is independent of the Gaussian parameter $c$, and the argument also includes the unweighted case $c=0$; the strict interior positivity comes from the full Hardy–Weitzenböck coefficient rather than from the sign of $c$ alone.

An inverse source problem for a class of fractional elliptic equations with singular coefficients is investigated in this paper. For the corresponding direct problem, a formal solution is derived and the well-posedness of the solution is established. For the inverse problem, a Hölder-type conditional stability estimate is obtained in a Hilbert scale associated with exponential operators. Based on this stability framework, two regularization methods are proposed for reconstructing the unknown source term: the exponential-type Tikhonov regularization method and the exponential quasi-boundary value regularization method. Convergence estimates for the regularized solutions are derived under both a priori and a posteriori choices of the regularization parameter. In addition, finite-dimensional spectral approximation results show that the proposed methods are also applicable to general square-integrable source terms, without requiring the exact source to satisfy an exponential-type source condition. Numerical experiments demonstrate that the proposed methods provide stable and accurate reconstructions for both smooth and piecewise smooth sources even under low signal-to-noise ratio conditions.

In this paper we present construction systems – tuples $(X, BB, \oplus, \nu)$ comprising objects, building blocks, an assembly operation, and a joining multiplicity – as a general algebraic framework for studying how complex objects are built from simpler parts. To each construction system we associate a toric ideal, a toric variety, and a matroid, obtaining analytical bounds on the growth function $N(a)$ (the number of objects of construction complexity $\leq a$) purely from the design signature $(m, \nu, n_0)$. For systems equipped with a type system and valence bounds, we define the composition polytope $P_{\mathrm{val}} \subset \mathbb{R}^m$, whose integer points count the feasible compositions. Compositions outside $P_{\mathrm{val}}$ – termed zombies – are combinatorially valid but physically unrealisable. We prove that the zombie classification is sound (zero false positives) and conservative: the true infeasibility rate is at least as high as the polytope predicts. Specialising to the molecular graph assembly system of Morales Parra et al. ($m = 19$ bond types, $5$ atom types with valences $1$–$4$), we identify a composition polytope whose lattice points capture the physically realisable compositions, and show that the resulting growth exponent tightens from $0.73$ to the exact value $\log 2 \approx 0.693$. While the difference of $0.037$ appears small, in the doubly-exponential regime it corresponds to a tightening of the bound by a factor exceeding $10^{198}$ at assembly index $10$. The framework also recovers the bioorthogonal click-chemistry system of the author's prior work as a second instance, with $m = 8$ and a partition matroid.

We study the fluctuations of the free energy of the Sherrington-Kirkpatrick model at the critical inverse temperature $\beta=1$. We prove that \[ \operatorname{Var}(F_N(1))\leq \frac14\log N+C, \] with $C$ independent of $N$. This gives a logarithmic upper bound, in agreement with the order predicted in the physics literature. We also prove that the critical variance diverges: more precisely, \[ \operatorname{Var}(F_N(1)) \geq \frac12\log\log\log N - C . \]

We study restart schemes in stochastic optimization problems for non-smooth and weakly convex functions that satisfy a Kurdyka-Łojasiewicz (KŁ) inequality. Using restarts allows us to leverage KŁ inequalities to achieve optimal convergence rates, with acceleration depending explicitly on the KŁ exponent. Furthermore, for stochastic gradient descent (SGD), optimal restart schedules correspond to piecewise constant (step decay) step sizes. While regularity constants such as the KŁ exponent are typically unknown in practice, we prove that restart schemes are robust to significant misspecification of these constants, hence nearly adaptive. We detail numerical experiments on a number of problems where the KŁ exponent is controlled.

In this work, intended to be a companion note to a future preprint, we give a proof of the fact that the classical (biased) notion of symmetric monoidal category, the notion of unbiased symmetric monoidal category, and the notion of homotopy symmetric monoidal category are equivalent in a precise sense (in that suitably defined groupoid-enriched categories having, respectively, biased, unbiased, and homotopy symmetric monoidal categories as objects are equivalent as enriched categories).

In this paper, we define the notion of a spectral sequence in the context of semi-abelian categories in the sense of Janelidze, Márki and Tholen. We show that from a double simplicial object, one can construct an exact couple, which gives rise to a spectral sequence. This extends Quillen's result on double simplicial groups to every semi-abelian category.

We prove Schoen's conjecture on $L^\infty$-metrics for tori. More precisely, we show that any $L^\infty$-metric on a torus that is smooth and has non-negative scalar curvature away from an embedded submanifold with codimension at least three extends to a smooth flat metric. We also prove a LLarull-type theorem for $L^\infty$-metric. Our proof uses weighted scalar curvature and the relative index theorem.

In the current paper, we introduce the class of significant pointwise contractions. This family of mappings extends the classical Banach contractions and also contains nonexpansive mappings. We prove that in a complete metric space every significant pointwise contraction admits a fixed point that can be approximated through Picard iterations. We provide examples showing that the newly introduced class differs from the class of piecewise contractions and is not contained in it. Moreover, we point out that significant pointwise contractions are continuous and may possess multiple fixed points. We further introduce the class of significant pointwise shrinkings, which generalizes the class of shrinkings, and prove that any such mapping has a fixed point in a compact metric space. Moreover, we define the class of annular pointwise contractions, which includes Banach contractions but is distinct from the class of Suzuki mappings. Finally, we obtain a fixed point theorem for annular pointwise contractions and present an additional condition ensuring the existence of fixed points.

We study Braverman-Kazhdan's asymptotic Hecke algebra $\mathcal{J}(G)$ for inner forms $G$ of $p$-adic $\mathrm{GL}_n$. We show that $\mathcal{J}(G)$ and the property for a $G$-representation to extend to a $\mathcal{J}(G)$-module are defined over $\overline{\mathbb{Q}_\ell}$, and hence make sense in the context of the categorical local Langlands correspondence. We show a rudimentary form of compatibility with Hecke operators, allowing us discuss stalks of sheaves on $\mathrm{Bun}_n$ corresponding to the trivial vector bundles on the stack of $L$-parameters, in particular the Whittaker sheaf, in terms of $\mathcal{J}(\mathrm{GL}_n)$-modules. We provide explicit formulas in terms of reductive centralizer of $L$-parameters for many functions in $\mathcal{J}(G)$, and show that $\mathcal{J}(G)$ has the same Hochschild homology as $C_c^\infty(G)$, and that the Kazhdan-Lusztig bijection appears in the isomorphism. We proceed via Bushnell-Kutzko and Sécherre-Stevens types, generalizing a theorem of Suzuki for $\mathrm{GL}_n$ for which we provide a proof.

This article builds upon our recently published fully discrete Cartesian grid Active Flux method for the Euler equations by introducing a new evolution operator for the linearized Euler equations. The evolution of the point value degrees of freedom located at grid cell boundaries and used to compute numerical fluxes is a key component of fully discrete Active Flux methods. Our methods are based on local linearizations of the Euler equations and the use of truly multi-dimensional evolution operators for these linearized problems. Here, we propose to solve the linearized Euler equations in moving coordinates, reducing them to acoustics, which can be solved exactly. We compare this approach with our previous one, in which we used approximate evolution operators derived using the method of bicharacteristics. The moving-grid approach is closely related to recent methods proposed by Barsukow as well as Duraisamy. Our choice of local linearization, as well as correction of linearization errors, enables us to construct third-order accurate methods for smooth solutions of the nonlinear Euler equations. Numerical results illustrate the performance of these methods for different flow regimes, ranging from discontinuous solution structures with shock waves to vortex structures in the low Mach number regime.

A family of permutations of $[n]$ is called setwise distinguishable if for every permutation in the family there exists a subset of $[n]$ whose image under this permutation differs from its image under any other permutation in the family. We prove that there exists a setwise distinguishable family of $2^{(2-o(1)) \cdot n}$ permutations of $[n]$. The result is optimal up to the $o(1)$ term in the exponent and is achieved through an explicit construction. As an application, we obtain nearly tight conditional lower bounds on the kernelization complexity of graph coloring problems parameterized by the vertex-deletion distance to split graphs. This improves a result of Jansen and Kratsch (Inf. Comput., 2013).

Schmutz Schaller developed techniques for studying Teichmüller space using the systole function. These were presented in \cite{SchmutzMorse}, \cite{SchmutzVoronoi} as a hyperbolic analogue of Voronoi's theory of quadratic forms in the theory of Euclidean lattice packings and coverings \cite{VoronoiPDQF}. It is known that the packing density function on Euclidean space is a topological Morse function, \cite{TMFAsh}, and the same is true of the systole function on Teichmüller space, \cite{Akrout}, \cite{SchmutzMorse}. The study of hyperbolic packing and covering problems is technical, for example, the density depends on the scale, and very little is known about optimisers of the density \cite{tóth2022ballpackingshyperbolicspace}. At least in the Euclidean setting, there are also fewer techniques available for studying sphere covering as opposed to sphere packing problems, as the covering problems seem to have less discernible structure. One approach to studying efficient circle coverings in the hyperbolic plane is to study the critical points of the diameter function on Teichmüller space. This paper shows that the diameter function on Teichmüller space is a topological Morse function. As a mapping class group-equivariant topological Morse function, critical points of the diameter function are related to the homology of moduli space. It would seem that for small genus, the systole function and diameter function have a larger proportion of common critical points than at higher genus.

The directed support graph of a discrete-time, time-homogeneous Markov chain naturally defines a regular language whose words are the admissible trajectories from a fixed initial state. Applying the Parikh map associates with each trajectory its occupation vector, recording how many times each state is visited while disregarding the chronological order of the visits. Equivalently, each trajectory determines a monomial whose exponents are its occupation numbers, yielding a natural commutative representation of occupation patterns. For each trajectory length n, we define the occupation ideal generated by the corresponding Parikh monomials. The minimal generators of this ideal are in one-to-one correspondence with the distinct occupation patterns realized at that length, providing an algebraic encoding of the combinatorial structure of admissible trajectories. This construction naturally gives rise to three complementary measures of support complexity describing reachability growth, trajectory growth, and occupation-pattern growth. The resulting framework connects Markov support graphs, regular languages, Parikh images, and monomial ideals, establishing occupation ideals as a new algebraic object associated with Markov support dynamics. It provides a natural interface between stochastic processes, formal language theory, and combinatorial commutative algebra, and suggests new algebraic and geometric approaches to the study of support dynamics. Examples illustrate how occupation patterns reflect branching, recurrence, transience, and local oscillation in the underlying support graph.

We study the modules of minimal dimension for reduced enveloping algebras of Lie algebras of reductive algebraic groups using the theory of modular finite $W$-algebras. First of all we consider the case where the $p$-character lies in a unique sheet, and demonstrate that in classical cases and in most exceptional cases all minimal modules are parabolically induced from a Levi subalgebra and a rigid $p$-character. Secondly we consider the minimal modules which are invariant under twisting by the component group, showing that in classical cases and in most exceptional cases these are also parabolically induced from a Levi subalgebra and a rigid $p$-character.

By a well-known characterisation, in a presheaf topos every internal suplattice is a presheaf of suplattices, but not every presheaf of suplattices is an internal suplattice (and similarly for frames). In this paper, we construct the free internal suplattice/frame on an presheaf of suplattices/frames, yielding a left adjoint to the forgetful functor from the respective internal structures to presheaves of structures. The description of this left adjoint has also appeared in recent work of Henry and Townsend, in connection to a different universal property, namely that of turning a lax natural transformation between poset-enriched functors to a strict one. As an application of our construction, we investigate conditions on frames internal to a presheaf topos, such as being locally compact, compact, stably locally compact or Hausdorff, in terms of properties of their sections in the base topos. In the first three cases, it is necessary that all the sections have the respective properties, while the Hausdorff property is not transferred to the sections. Moreover for local compactness it is necessary that the transition maps preserve the way-below relation. Finally, for an internal locally compact frame in presheaves we analyse the connection of its way-below relation to the respective relations of its sections.

In this article we set up and showcase cellular computations for (co)homology with values in strictly $\mathbb{A}^1$-invariant sheaves. These computations encapsulate many classical invariants like Chow groups and singular cohomology of the real points. They also extend enumerative arguments from algebraically closed fields to more general fields. The spaces considered here have to admit a cellular structure. Instead of using the classical notion of cellularity, i.e. having a stratification by affine spaces, more general stratifications by cohomologically trivial spaces are used, following Morel–Sawant. Examples of cellular spaces include projective spaces and their products, but also spaces such as $\overline{M_{0,n}}$, the moduli space of stable genus $0$ curves with $n$ marked points. For these examples, we showcase the computations and show how to derive the classical results. Hopefully, the following text provides enough evidence to be convincing that such computations are doable and is encouraging to start computing the cohomology for more cellular spaces. This is part of the author's PhD thesis.

We consider two-term even-order operators on the unit interval with homogeneous delay and with Dirichlet and Neumann boundary conditions. The main result provides to the eigenvalue asymptotics of this operator with respect to all indices enumerating the eigenvalues. This asymptotic formula is uniform in the parameter of the homogeneous delay. We also discuss the nontrivial high-frequency phenomenon demonstrated by the uniform spectral asymptotics.

The anti-ferromagnetic multi-state Potts model, a generalization of the Ising model, is one of the most fundamental models in statistical physics. It was conjectured by Kotecký (Phys. Rev. B, 1985) that the model undergoes a phase transition from a disordered phase at infinite temperature to an ordered phase at sufficiently low temperature on lattices. Such phase transitions are believed to play an important role in computational complexity theory and remain closely connected to the problem of approximating the partition function of the system. For proper three-coloring models (corresponding to the zero-temperature), torpid mixing of a family of local-update Markov chains on lattices was established by Galvin, Kahn, Randall and Sorkin (SIDMA, 2015), coinciding with the presence of phase coexistence following shown by Feldheim and Spinka (J. Eur. Math. Soc., 2019). In this work, we study approximating the partition function of the anti-ferromagnetic multi-state Potts model at low temperature on random regular bipartite graphs, which are with high probability good bipartite expanders. On the negative side, we generalize the result by Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026) for anti-ferromagnetic Ising models to show that when the temperature is sufficiently low relative to the degree of the underlying graph, the celebrated single-site Glauber dynamics has exponentially slow mixing time. On the positive side, we design a deterministic algorithm that yields an approximation to the partition function of the model via the framework of abstract polymer models as Jenssen, Keevash and Perkins (SICOMP, 2020), Liao, Lin, Lu and Mao (Theor. Comput. Sci., 2022), Galanis, Goldberg and Stewart (TOCT, 2021) and Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026).

We consider integral categories, a class of categories that gained importance in the last three decades in the categorical foundations of algebra and functional analysis. We discuss some familiar criteria for integrality and prove new criteria for one- and two-sided integrality. For the first time in the literature, an example is given of a right but not left integral category.

We investigate the structure of the poset of torsion classes for a finite dimensional algebra admitting a simple projective module. We generalise a result of Ladkani by showing that if two algebras are related by a 1-APR tilt, then their posets of torsion classes are related by a flip-flop. This implies that the incidence algebras of the posets are derived equivalent. We give two different proofs of this result. The first one applies to functorially finite torsion classes, and a key ingredient is to see the two posets we want to relate as subposets of a common poset of silting objects. The second proof applies to arbitrary torsion classes, and we use a similar strategy, this time embedding the two posets into a common poset of s-torsion pairs.

Physics-Informed Neural Networks (PINN) struggle with stiff, regime-changing transport equations due to instability, loss imbalance, and violations of physical consistency. This paper investigates these failures through the Marshak wave equations, a canonical benchmark from radiative transport, where initial and boundary conditions differ by up to 12 orders of magnitude, and proposes modifications to the standard PINN framework to overcome them. Three modifications are introduced: a ScaledSigmoid final activation enforcing physical bounds and positivity of the unknowns; a logarithmic MSE loss replacing the standard quadratic loss for initial and boundary conditions, enabling training across extreme scale disparities; and explicit enforcement of global conservation laws as an additional physics loss term. Monte Carlo sampling with exponential time weighting is used throughout. The proposed framework recovers the Marshak wave dynamics - including the hot, cold, and wave-front regions - in good agreement with a reference Implicit Monte Carlo solution (sub-millimeter absolute error on the front position, although the relative error remains substantial at early times), with run times under 5 minutes. Ablation studies confirm that each ingredient is essential: linear activation, absence of the logarithmic loss, or removal of the PDE term each independently cause the method to fail qualitatively. This work identifies and resolves three concrete failure modes of standard PINN on stiff hyperbolic systems with nonlinear coupling. The combination of bounded activations, scale-aware loss functions, and conservation law enforcement constitutes a framework validated on the present one-dimensional Marshak wave problem, with potential extension to other coupled stiff transport systems in engineering.

Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\operatorname{up}}_{r-1}(K)$. It is known that \[ \lambda_{\max}\bigl(L^{\operatorname{up}}_{r-1}(K)\bigr)\le n. \] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\widetilde H_{r-1}(K^c,\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected. Secondly, we prove a sharper upper bound for $\lambda_{\max}(L^{\operatorname{up}}_{r-1}(K))$: \[ \lambda_{\max}(L^{\operatorname{up}}_{r-1}(K)) \le \max_{F\in S_r(K)} \bigl|\bigcup_{E \in \partial F} N_K(E) \bigr| \le n,\] where, for an $(r-1)$-face $E$, $N_K(E)$ denotes the set of vertices $u$ outside $E$ such that the union $E \cup \{u\}$ is an $r$-face of $K$. This is the high-dimensional analog of the graph Laplacian bound. We give an explicit characterization of the equality case, and construct a broad family attaining the bound, namely, the partite semiregular complexes with admissible additions.

Motivated by the operation of decoders for DNA storage, we consider the problem of unsupervised clustering of noisy short sequences, each generated from one of multiple possible unknown source sequences after passing through a memoryless channel. Focusing on the statistically optimal clustering rule, we derive upper and lower bounds on the probability of incorrect clustering as a function of the sequence length, the number of reads, and the channel statistics.

We study linear subspaces of matrices whose inverse spaces are also linear. Based on the fact that any linear space containing the identity matrix and whose inverse space is linear must be a self-inverse space, we introduce such spaces as self-inverse spaces. In fact, as we will show, self-inverse spaces are finite-dimensional complex unitary Jordan algebras. We provide an algebraic classification of all self-inverse spaces of small type and a classification of small-dimensional self-inverse spaces up to isomorphism.

The measurement problem is one of the longest-standing open problems in quantum mechanics: noncommutative observables are incompatible with the Born and collapse rules, prescribing that the outcome of their observation is distributed according to the laws of classical probability. In this paper, we solve the measurement problem through the correspondence principle with a semiclassical approach. We prove that Born rule, the wavefunction collapse, and macroscopic observability all hold true, approximately, but with arbitrary precision, whenever a measuring device (in a semiclassical state) is coupled adiabatically to the observed system. Our measurement device is based on the propagation properties of semiclassical bosonic coherent states through an adiabatic two level system on scale times of the order of the Ehrenfest time.