Papers

We consider the problem of constructing a $K$-class classifier from the combination of $O(\log K)$ simple binary classifiers – this is a natural paradigm to construct a sophisticated classifier in a distributed manner with each agent performing a relatively straightforward task. We study the fundamental performance limits of such a classifier when the corresponding binary classifiers are hyperplanes. For a stylized Gaussian setting where the $K$ class centers are independent Gaussian points in $\mathbb R^d$ and the observations are corrupted by Gaussian noise, we derive explicit performance bounds across several decoding and dimensional regimes. Extensive simulation experiments provide strong empirical validation of the presented theoretical results.

We prove a strict dominant energy deformation theorem for asymptotically flat initial data with corners along a hypersurface $\Sigma$. The deformation preserves a corner condition on the Bartnik data across $\Sigma$. We show that if the dominant energy condition holds on each side of $\Sigma$ and the Bartnik data satisfy this corner condition, then the exterior end satisfies $E \ge |P|$ in every dimension $n \ge 3$.

A Cayley graph $\operatorname{Cay}(\Gamma,S)$ over a finite group $\Gamma$ is said to be normal if its connection set $S$ is a union of some conjugacy classes of $\Gamma$. This paper investigates perfect state transfer in Grover walks on normal Cayley graphs. The Grover walk is a widely studied discrete-time quantum walk. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on normal Cayley graphs. As applications, we derive explicit spectral criteria for perfect state transfer on Cayley graphs over abelian groups, dicyclic groups, and dihedral groups. These results yield several infinite families of Cayley graphs exhibiting perfect state transfer. We further obtain simple combinatorial characterizations of the existence of perfect state transfer on Cayley graphs over dihedral and dicyclic groups. Our general characterization also recovers a number of previously known results as special cases. As a further consequence, we obtain a complete characterization of perfect state transfer on unitary Cayley graphs. In particular, we prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.

Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.

Controlling dynamical systems in real-time across multiple scenarios is critical to enabling adaptive control strategies, ensuring stability and efficiency. However, to tailor control actions in response to varying scenarios, traditional optimal control problems typically require several system simulations, which are often computationally demanding due to the high-dimensionality of the underlying spatio-temporal dynamics. In this work, we exploit SHallow REcurrent Decoder networks-based Reduced Order Modeling (SHRED-ROM) to synthesize a real-time closed-loop controller for high-dimensional and parametric dynamics, relying solely on limited state sensor readings. After training the model on a few optimal examples given by an expert demonstrator, SHRED-ROM mimics the expert behavior with effective distributed control actions in new scenarios, alleviating the curse of dimensionality. Moreover, a sensor forecaster is synthesized and used to close the loop at the latent level, thus efficiently mitigating possible sensor failures or delays. The performance of the proposed optimal control strategy is finally assessed on three challenging high-dimensional cases dealing with either parametric density control or fluid flow control.

We extend the previous compactifications of the fifth Painlevé foliation studying the behaviour of the leaves for the time parameter $t$ being equal to $0$ or $\infty$. In particular, we find, for each boundary component, a first integral of the Hamiltonian vector field, providing an explicit description of the foliation.

Essentially non-oscillatory (ENO) reconstruction provides a key mechanism for designing high-order entropy-stable schemes for hyperbolic conservation laws, with its sign property ensuring nonnegative local dissipation for a prescribed entropy. However, two fundamental questions concerning convergence remain open: whether this dissipation provides the coercivity required for weak-BV compactness, as posited by the ENO–TV conjecture, and whether entropy stability transfers from the prescribed entropy pair to additional pairs. This paper resolves the ENO–TV conjecture by establishing a sharp parity dichotomy: it holds if and only if the reconstruction order $k=2$ or $k$ is odd, and fails for all even orders $k\ge 4$.The key to our proof is a localization principle that eliminates dependence on nonlinear adaptive stencil selection, establishing a two-sided equivalence between ENO dissipation and a canonical finite-difference functional. For odd orders, the conjecture is proved via a hidden quadratic energy and novel discrete Gagliardo–Nirenberg inequalities. For even orders $k\ge 4$, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every $k\ge 2$, the conjecture can fail on non-quasi-uniform meshes. Addressing the above second open question, we discover on ENO null modes that local entropy transfer is governed by the first cohomology of a unipotent shift. Using apolar duality and binary covariants, we compute the dimensions of the associated cohomology subspaces and prove that smooth local entropy transfer encounters generic obstructions for every $k\ge 4$. By revealing how ENO null modes link global coercivity and local entropy compatibility, this work provides a structural foundation for the compactness and convergence analysis of high-order entropy-stable discretizations.

We study the nuclear-norm error of a column-selected Nyström approximation to $K=(L+\gamma I)^{-1}$, where $L$ is symmetric diagonally dominant and $\gamma>0$. Our central question is whether this error has diminishing returns. A Schur-complement identity reduces the question to traces of inverses of principal submatrices. Existing $M$-matrix results settle the case in which $L$ is a symmetric diagonally dominant $M$-matrix (SDDM). However, diagonal dominance alone is not enough: failure occurs already in dimension three. We construct an exact one-parameter SDD family and determine its sharp failure interval. A $2\times2$ identity proves that dimension three is minimal within the SDD class. We then show that failure persists under strict diagonal dominance; with a nonempty selected base set, dimension four is minimal. Finally, we prove invariance under signature switching, derive a three-dimensional formula showing how a signed triangle causes failure, and give an example in which greedy column selection misses the optimal pair. Together, these findings complete the answer to Problem 4.6 in a recent Simons workshop report.

We prove Norine's conjecture: every red–blue edge-coloring of the \(n\)-dimensional hypercube \(Q_n\), \(n\geq2\), in which antipodal edges have opposite colors contains a monochromatic path joining some vertex to its antipode. From a hypothetical counterexample we construct an antipodally equivariant, augmentation-preserving chain map from the cellular chains of the cubical boundary of a cube to subdivision-invariant polyhedral chains on a sphere of one lower dimension. A purely algebraic chain-level Borsuk–Ulam obstruction rules out this map.

The existence of non-trivial solutions of homogeneous partial differential equations with prescribed support properties plays a fundamental role in the theory of linear partial differential operators. In this article, we establish the existence of smooth zero solutions with partially bounded supports for a broad class of constant coefficient partial differential operators. As applications, we obtain new geometric results concerning solvability and approximation for partial differential equations. In particular, we derive geometric characterizations of $P$-convexity for supports for a large class of non-elliptic operators, thereby extending classical geometric ideas of Hörmander. By a theorem of Malgrange, $P$-convexity for supports is equivalent to the surjectivity of the differential operator on spaces of smooth functions and, more generally, on local subspaces of distributions of finite order. As a further consequence, we prove that the kernel of every surjective semi-elliptic operator satisfies condition ($\Omega$) implying parameter dependence results. We also investigate Runge-type approximation phenomena. We obtain geometric characterizations of Runge pairs for smooth functions, distributions, and spaces of smooth Whitney jets. Finally, we develop a general framework for Runge approximation for square systems of constant coefficient partial differential equations. As applications, we recover by alternative methods known Runge approximation results for Beltrami fields and for the three-dimensional unsteady Stokes system, and we complement them by corresponding approximation theorems in spaces of smooth Whitney jets. In particular, this yields approximation up to the boundary for solutions on suitable domains with Hölder continuous boundary.

Applying policy optimization to Model Predictive Control (MPC) yields high-performance and reliable controllers. However, the resulting controllers often overfit their training conditions and suffer significant performance degradation in unseen tasks. We propose a novel framework combining policy optimization with meta-learning to train highly adaptable MPC controllers. Our approach enables rapid adaptation to unseen tasks, maintaining high performance at a fraction of the computational cost required for full retraining. Furthermore, we integrate system identification into the pipeline to continuously refine both the MPC hyperparameters and the underlying predictive models. We validate our proposed methodology on a Ball-on-Plate system, demonstrating superior adaptability across various parameterized trajectory-tracking tasks.

Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.

In this paper, we construct a low-rank, structure preserving discontinuous Galerkin (DG) method to simulate the Vlasov-Poisson (VP) system coupled with the Dougherty Fokker-Planck (DFP) collision operator. When Coulomb collisions occur in dense or weakly-collisional plasmas, electrons get pushed to a low-rank steady state. In many cases, the plasma arrives to this steady state quickly, meaning that for most of the run-time, the plasma consists mainly of numerical low-rank structures. Our new low-rank scheme is constructed to exploit these numerical low-rank structures to greatly reduce the needed storage complexity of simulations for the VP-DFP system. It is constructed as an extension of the previously established Local Macroscopic Conservative (LoMaC) method by incorporating Coulomb collisions into the system. The LoMaC property ensures local conservation of macroscopic mass, momentum, and energy at the discrete level. Details of the new method are discussed in this paper. Numerical experiments are performed to show the efficacy of the method.

In this paper, we propose a weighted inverse Lax-Wendroff (WILW) boundary treatment for the discontinuous Galerkin (DG) method on unfitted meshes to efficiently solve hyperbolic conservation laws in complex geometries. The proposed method employs the standard DG scheme for interior cells and reconstructs high-order approximation polynomials via the ILW principle for cut cells near boundaries to impose numerical boundary conditions, effectively eliminating the time-step restriction typically caused by small cut cells. In particular, to address the sensitivity of numerical errors to the geometric size of cut cells in the basic ILW scheme, we raise the reconstruction order at the boundary, ensuring that accuracy becomes independent of the cut-cell size. Furthermore, it incorporates a weighted least-squares reconstruction to reduce the need for complex high-order boundary derivatives during construction. As a result, the method maintains high-order accuracy while significantly improving computational efficiency for multi-dimensional problems. Finally, the stability of the proposed method is theoretically validated through linear stability analysis, and the effectiveness and robustness of the proposed scheme are numerically verified through a series of one-dimensional and two-dimensional numerical experiments for scalar and system equations.

We prove that a smooth projective non-uniruled surface of Picard rank one, generated by an ample and base point free line bundle, cannot be covered by an isotrivial family of smooth curves in the primitive polarisation.

Let $k$ be an algebraically closed, complete non-Archimedean field of residue characteristic $0$. Let $f$ be a polynomial of degree at least $2$ over $k$ which does not have potential good reduction. We prove that if $g$ is any other polynomial with the same Julia set, then $f$ and $g$ must be dynamically related. As a consequence, we show that for any two complex polynomials $f,g$ of degree at least $2$, either their sets of preperiodic points coincide, or the number of their common preperiodic points is uniformly bounded above by a constant depending only on the degrees, thereby answering a conjecture of DeMarco–Krieger–Ye for polynomials. We also establish relative results, allowing us to prove special cases of the DeMarco–Mavraki conjecture.

We extend the definitions of kappa, psi and lambda classes from the case of moduli spaces of stable curves to the higher-dimensional case of KSBA spaces, and prove a number of their basic properties.

For a class of superlinear SPDEs driven by multiplicative noise, we prove an (essentially) sharp uniform-in-time (UIT) weak convergence rate for the nonlinearity-explicit Galerkin tamed Euler method (GTEM). Under standard monotonicity assumptions, the proof combines Malliavin calculus with regularity theory for the associated backward Kolmogorov equation (BKE), leading to UIT moment, Hölder, and Malliavin estimates, along with regularity estimates for the BKE solution. These estimates, together with a weak error decomposition and Malliavin integration by parts (IBP) formula, then yield a UIT weak convergence rate $\tau^\rho+\lambda_N^{-(\rho+\gamma/2)}$ for any $\rho \in (0,1)$, where $\gamma\in[0,1)$ quantifies the assumed spatial Sobolev regularity. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.

For any group $\Gamma$ and any free factor system $\mathscr A$ of $\Gamma$, the relative outer automorphism group $\text{Out}(\Gamma;\mathscr A)$ acts naturally on the relative free splitting complex $\mathcal{F\!S}(\Gamma;\mathscr A)$ and on the complex of relative free factor systems $\mathcal{F\!F}(\Gamma;\mathscr A)$, generalizing the well known actions of $\text{Out}(F_n)$ on the absolute free splitting complex $\mathcal{F\!S}(F_n)$ and the absolute free factor complex ${\mathcal{F}}(F_n)$ of the rank $n$ free group $F_n$. This overview summarizes a three part work regarding the large scale geometry of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and $\mathcal{F\!F}(\Gamma;\mathscr A)$ and the geometric dynamics of the actions on these complexes by elements of $\text{Out}(\Gamma;\mathscr A)$. In Part I arXiv:1407.3508 we prove hyperbolicity of $\mathcal{F\!S}(\Gamma;\mathscr A)$ and of $\mathcal{F\!F}(\Gamma;\mathscr A)$. In Parts II and III arXiv:2212.09907, arXiv:2503.07532 we study the relation between the geometric dynamics of an element of $\text{Out}(\Gamma;\mathscr A)$ and the dynamics of its relative train track representatives. The main tool in Part II is the Two Over All Theorem, expressing an exponential flaring property of Stallings fold paths in $\mathcal{F\!S}(\Gamma;\mathscr A)$. The main tools in Part III are "filling paths", used to formulate and prove a strong version of the Two Over All Theorem.

In this paper, we study the Hardy-Hénon parabolic equation \begin{equation*} \partial_t u=\Delta u+a|x|^{-\gamma}|u|^{\beta}u, \qquad t>0,\; x\in\mathbb{R}^{n}\setminus\{0\}, \quad \beta>0,\; a\in\mathbb{R}, \end{equation*} under suitable assumptions on the parameter $\gamma$. We establish the local and global well-posedness of mild solutions in homogeneous Herz spaces.

We consider the design of optimal fixed-step first-order methods for $M$-Lipschitz convex optimization given $\|x_0-x_\star\|\leq D$. Prior works have identified several distinct fixed-step methods, parameterized by a matrix of stepsizes $W$, with the (information-theoretic) minimax optimal rate $MD/\sqrt{N+1}$ of objective gap convergence. We provide a complete characterization of every optimal fixed-step method. Moreover, we show every optimal fixed-step method can be derived from the constructive approach of~\cite{constructive_approach} and provide a polyhedral representation of the set of optimal methods through proof multipliers. From this characterization, we show that no anytime optimal fixed-step subgradient methods exist.

We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to 0 as $\varepsilon \to 0$. We show that, as $\varepsilon \to 0$, these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of $ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.

In incidence geometry, a triality is a symmetry cyclically exchanging triples of types of elements. Requiring geometries with trialities to satisfy standard regularity conditions makes their construction highly non trivial, and known examples are rare. In this paper, we present two infinite families of flag transitive, thin and residually connected geometries admitting trialities and no dualities with type preserving automorphism groups isomorphic to Sym(n) or Alt(n). We also develop general methods that extend to other settings. The residues of the two families are fundamentally different. In the first family, the maximal parabolics have constant size while in the other their size grows essentially linearly with the degree of the group.

We show that, for almost-minimizers of the Griffith energy in the plane, the complement of the crack is locally covered by a finite number of John domains where the displacement satisfies Hölder-type estimates. As a consequence, we derive the integrability of the full gradient and the finiteness of the traces along the crack. In particular, we show that any Griffith almost-minimizer in dimension two locally belongs to the space SBV^2.

Heavy-tailed diffusion models replace Gaussian noise by a Gaussian variance mixture: denoising Levy probabilistic models (DLPM) take the mixing variables i.i.d. across coordinates, while Student-t EDM shares one mixing variable per sample. Neither has dynamics, yet temporal dependence of the noise amplitude - volatility clustering - is the defining stylized fact of financial returns. We introduce the Denoising Subordinated Probabilistic Model (DSPM), whose mixing vector is a stationary AR(1) chain driven by tempered-stable subordinator increments (the discrete Barndorff-Nielsen-Shephard volatility process) along the data axis. Conditionally on the chain the DDPM machinery survives verbatim; kurtosis and squared-noise autocorrelation are closed-form in the chain parameters, giving an exactly identified, analytically invertible calibration; DDPM, DLPM and Student-t noise are boundary cases of one memory parameter. We then prove a delimiting result: when the denoiser is conditioned on the mixing variables, their law is a nuisance - in the exact-denoiser limit the generated distribution is invariant to it and interventions on the chain do nothing. Experiments confirm both halves: conditioned models match the data's clustering whatever the mixing law, a designed x8 volatility shock moves the envelope by under 13%, while blind models transmit the mechanism exactly as calibrated. Finally, coupling the chain to the data by a variational volatility encoder - trained with the stochastic-volatility likelihood whose log-determinant the simplified denoising loss provably drops - restores control (shock response 3.07 vs. naive 2.83), recovers latent volatility (correlation 0.76), and learns the prior memory toward the true persistence.

In this work, we study a single-source single-server continuous-time status update system where the updates arrive according to a Poisson process and update service times are generally distributed. In our proposed setting, a preemption policy refers to one where a new update preempts the ongoing one with a probability depending on the age of the update in service. We first propose an analytical method to derive the average age of information (AoI) and average peak AoI (PAoI) for any such preemption policy. This analysis is then utilized to tune two particular preemption policies: (i) probabilistic preemption (PP), in which preemption takes place according to a fixed probability regardless of the update age, (ii) threshold-based preemption (TP), for which preemption is incurred when the update age exceeds a certain threshold, both using one-dimensional line search. The effectiveness of policy tuning for the PP and TP policies is validated using lognormal-distributed update service times.

Quasi-polynomials are ubiquitous in combinatorics and algebra, as they arise in a variety of enumeration problems. Because quasi-polynomials consist of constituent polynomials, their behavior is more subtle than for a single polynomial. In particular, unlike for a polynomial, it is possible for a quasi-polynomial defined on the positive integers to have infinitely many points at which it is decreasing. In this work, we characterize quasi-polynomials of degree $d$ and period dividing $p$ that are eventually nondecreasing, i.e., that have only finitely many values at which they decrease. We then give a detailed analysis of the space of eventually nondecreasing quasi-polynomials with fixed degree $d$ and period dividing a fixed $p$ such that the $h$-vector of the quasi-polynomial is nonnegative. Using this analysis, we determine the rate of growth of the number of such quasi-polynomials as a function of the sum of the $h$-vector entries for the $0$-th constituent polynomial.

Two misunderstandings, frequently arising in Bayesian predictive inference, are discussed. The first deals with the data generating mechanism, while the second consists in overestimating the role played by asymptotic exchangeability. Some consequences of such misunderstandings are highlighted through examples.

The Greenland shark (Somniosus microcephalus) is among the longest-lived vertebrates known, with female maturity estimated at about 156 years and maximum ages approaching four centuries. We use the entropic theory of life histories and an open-group Leslie model to determine which reproductive windows are compatible with this extreme demographic profile. The model yields generation times, reproductive quantiles and entropy-maximizing reproductive endpoints. A critical value of reproductive persistence, approximately 0.9763, separates two regimes: below it, evolutionary entropy increases towards an asymptote; above it, entropy is maximized at a finite reproductive endpoint. Longevity-calibrated scenarios lie on both sides of this threshold and imply generation times near or above two centuries, with reproductive contributions extending into the third, fourth and fifth centuries of life. As an independent benchmark, an empirically parametrized Leslie model for the bowhead whale yields comparable entropy-based demographic quantities while allowing the reproductive and survivorship parameters to be calculated directly. The method provides quantitative constraints on reproductive organisation in exceptionally long-lived species for which detailed demographic data are unavailable.

We prove that $(k+1)$-uniform hypergraphs definable in an NIP strongly $k$-distal structure satisfy a homogeneous regularity lemma – they can be partitioned into a bounded number of simplicial complexes, most of which are homogeneous (meaning that the restriction of the hypergraph to the simplicial complex is either complete or empty). Furthermore, the parts of the partition can be chosen uniformly definably, and the size of the partition is polynomial in the reciprocal of the error parameter. This extends the homogeneous regularity lemma proven by Chernikov and Starchenko for hypergraphs definable in a distal structure. We prove this by introducing $k$-strong honest definitions and showing that an NIP structure is strongly $k$-distal if and only if every formula $\varphi(x_1, ..., x_k; y)$ has a $k$-strong honest definition. This extends the theory of strong honest definitions in distal structures to the higher-arity setting.