Papers

We show that there is a universal compact branched 3-manifold $W$ such that a closed 3-manifold $M$ immerses into $W$ if and only if $M$ admits a Sol structure. Equivalently, a closed 3-manifold is Sol if and only if it has a certain type of triangulation. The construction of $W$ is based on a regular language that characterizes Sol manifolds.

We study sequential testing and Bayesian quickest detection for diffusion observations whose drift changes between two alternatives while the signal-to-noise ratio may depend on the current observation. In this setting the posterior probability is generally not a closed one-dimensional Markov statistic: the natural sufficient state is the augmented process consisting of the posterior (or likelihood ratio) and the observed diffusion. We formulate both testing and quickest detection within this common filtering framework and identify the corresponding degenerate free-boundary problems. The main contribution is a delay-penalty comparison principle. For a common terminal false-alarm or terminal decision cost, a pointwise larger running delay penalty increases the value of continuation, shrinks the continuation region, and yields earlier stopping. When the stopping set has a one-sided posterior representation, this gives an order relation for the optimal alarm boundaries. The result applies to linear delay costs and to nonlinear marginal delay penalties after the appropriate Markovian augmentation, and is illustrated by a constant signal-to-noise Shiryaev example in which the alarm threshold is computed numerically and shown to be monotone in the delay cost. The framework clarifies how state-dependent information and nonlinear delay costs jointly affect the geometry of sequential testing and quickest-detection rules.

Let $G$ be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight $\lambda$, there is the associated principal indecomposable $G_1$-module $Q_1(\lambda)$, where $G_1$ is the first infinitesimal subgroup of $G$. The assertion that, for every such $\lambda$, there exists a $G$-module whose restriction to $G_1$ is isomorphic to $Q_1(\lambda)$ is known as the Humphreys–Verma Conjecture. For groups of rank $2$, it was shown in \cite{BNPS1} that the Humphreys–Verma Conjecture holds in all cases except one, namely when $G$ is of type $G_2$, the characteristic is $2$, and $\lambda=0$. This case remained completely open. Moreover, in every previously resolved case, the module $Q_1(\lambda)$ could be realized as the restriction of a suitable tilting module. However, in \cite{BNPS2} it was shown that $Q_1(0)$ for $G_2$ in characteristic $2$ cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a $G$-module whose restriction to $G_1$ is $Q_1(0)$, thereby establishing the Humphreys–Verma Conjecture in the last remaining rank $2$ case. Our construction provides the first known example of a $G$-structure on a principal indecomposable $G_1$-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys–Verma Conjecture and suggests new directions for understanding $G$-structures on principal indecomposable $G_1$-modules.

In this work, we present an efficient method to compress a high-dimensional discrete probability function, i.e., a probability tensor, into a non-negative hierarchical Tucker format. The methodology is a two-stage procedure. In the first stage, we take an existing interpolation method to compress the target tensor into a hierarchical Tucker (HT) in a manner similar to the CUR decomposition for low-rank matrix reconstruction. In the second stage, we fit the first-stage output against a non-negative hierarchical Tucker ansatz using a second-order method tailored specifically for this setting. When the tensor is of order \(d\), both stages admit an \(\mathcal{O}(d)\) computational complexity, and therefore the proposed methodology readily extends into high-dimensional settings. Numerical experiments show success in compressing various high-dimensional probability tensors.

Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are subject to convex constraints and the eliminated variables arise from a lower-level least-squares problem. In particular, by interpreting variable projection as a collapsed bilevel optimization problem, we derive exact reduced-gradient formulas compatible with automatic differentiation and propose a conditional-gradient algorithm for the resulting constrained reduced problem. We establish convergence guarantees under standard smoothness and compactness assumptions, and discuss extensions to structured lower-level variables. Numerical experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve wall-clock efficiency and data efficiency relative to natural joint-optimization baselines.

At the core of most socio-technical systems lies a scarce resource that is allocated among agents: highway lanes, public transit, road space, water rights, energy access, grid capacity, user attention, pollution rights, etc. With further automation of the underlying allocation processes, control engineers are increasingly tasked with making decisive assumptions regarding what society wants. In practice to date, design choices are largely driven by industry norms and conventions rather than a result of conscientiously responsible and ethical design. In this paper, we look at tools available to control engineers to design systems in a more principled manner in order to match the societal mandate. We consider three control design paradigms: online feedback optimization, control of Markov decision processes, and model predictive control. Beginning with aggregating individual agents' preferences into control design objectives, subsequently ensuring and certifying the fulfillment of those specifications, we argue that the feedback nature of control systems enables appropriate allocation of the shared resources in ways hitherto unparalleled.

The search for free boundary minimal submanifolds in Euclidean space with boundaries on a collection of concentric spheres naturally extends the classical problem in the unit Euclidean ball. A key feature of this setting is that the coordinate functions of such submanifolds satisfy a Steklov problem with an indefinite weight. This framework allows us to introduce a spectral index, which in turn yields both upper and lower bounds for the Morse index. As a concrete application, we compute the exact Morse index of an $m$-dimensional flat annulus in an $n$-dimensional spherical shell, showing that it equals $n-m$. Moreover, we study in detail free boundary minimal immersions from 2-dimensional annuli into Euclidean space whose boundaries lie on concentric spheres. We show that the images of these free boundary minimal immersions (FBMI) lie in an $m$-dimensional subspace with $2\leqslant m\leqslant 4$, and list the explicit forms of these FBMI. We also demonstrate how to find examples of FBMIs for which the ratio between the radii of the concentric spheres containing the boundaries is arbitrarily large.

We continue the local-global study of obstructed deformation problems for two-dimensional residual Galois representations arising from weight $2$ newforms of level $N$, initiated in \cite{Bin26}. Using the Greenberg-Wiles formula and the explicit classification of inertial Weil-Deligne types due to Dembélé-Freitas-Voight \cite{DFV22}, we systematically compute the local obstruction groups $H^0(G_p, \bar{\varepsilon}_\ell \otimes \mathrm{ad}^0\bar{\rho})$ for every inertial type arising at primes $p$ with $p^2 \mid N$ and $p \neq \ell$. For each type and in each of three arithmetic cases ($p \not\equiv \pm 1$, $p \equiv -1$, and $p \equiv 1 \pmod{\ell}$), we give the dimension of the local obstruction group and an explicit presentation of the universal deformation ring as a power series ring over the Witt vectors modulo explicit relations. We treat in detail the twisted Steinberg case ($\tau \simeq \tau_{\mathrm{St},p} \otimes \varepsilon_p$), the principal series cases, and the non-exceptional supercuspidal cases, including the full family of types at $p = 3$.

Let $\gamma_s(G)$ and $Z_s(G)$ denote the $s$-th terms of the lower and upper central series of a group $G$, respectively. A classical theorem by R. Baer states that if $Z_s(G)$ has finite index $n$ in $G$, then $\gamma_{s+1}(G)$ is also finite. In this paper, we prove that if $G$ is a generalized soluble group such that the quotient $\gamma_s(G)/(\gamma_s(G) \cap Z_t(G))$ has finite rank $r$ for some $s,t$, then the rank of $\gamma_{s+t}(G)$ is finite and $(r,s,t)$-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition to be a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer's theorem.

This thesis explores the notion of fundamental groups across three mathematical settings. We begin with the classical topological theory of covering spaces, highlighting its structural analogy with Galois theory. We then follow Grothendieck in transporting these ideas to algebraic geometry. The inadequacy of the Zariski topology motivates the étale topology, from which the étale fundamental group is constructed and compared to its topological counterpart via transcendental methods. Finally, we linearise the theory through Tannakian duality, where fundamental groups are recovered as automorphism groups of fibre functors on certain monoidal categories, a framework broad enough to encompass étale, topological, and motivic Galois groups alike.

This note is a detailed review of the geometry behind the Batalin-Vilkovisky formalism and how it fits into the framework of the quantum odd symplectic category and the odd quantization functor.

We identify the Poisson boundary of the one-point Schreier-chain random walk obtained by projecting the simple symmetric random walk on Thompson's group $F$ to the dyadic orbit point $1/2$. For the associated simple labelled-generator walk on the dyadic Schreier graph, the full Poisson boundary is the skeleton end boundary. The proof combines the known description of this Schreier graph as a binary-tree skeleton with recurrent one-dimensional ray attachments with an explicit trace computation. After tracing to the grey skeleton and deleting holding probabilities, the walk becomes a reversible nearest-neighbor walk on the rooted binary tree with two unequal classes of edge conductance. This reduces the boundary identification to standard Poisson–Martin theory for transient walks on trees and leaves a finite electrical-network calculation for the harmonic measure. Following Kaimanovich's coding of skeleton ends by odd 2-adic integers [{Groups, Graphs and Random Walks}, London Math. Soc. Lecture Note Ser.~436, pp.~300–342, 2017], the hitting measure is a biased Bernoulli product measure with explicitly computed bias. It is singular with respect to Haar measure, has full topological support, and is exact-dimensional; these properties and the exact constants are proved here.

The mass spectrum of affine Toda theory is known to be expressible in terms of a suitable eigenvector of the relevant Cartan matrix. The particles correspond in a precise manner to the Coxeter element orbits in the set of roots. Recently, variants of affine Toda theory have been constructed for many different Weyl group elements. Again, the particles correspond to orbits in the set of roots and this allows the calculation of the classical mass spectrum. We show how these spectral calculations generalize the affine Toda relation with the Cartan matrix. As an example, we calculate the spectrum for the unique non-Coxeter infinite family $\textrm{D}_{2n}(a_{n-1})$ of primitive regular conjugacy classes in the Weyl groups of complex simple Lie algebras.

In this paper we propose estimators of the parameter functions in an Aalen additive hasard regression model. The estimators are the individual and component wise $l^2$ projections of the naive estimators resulting from the ordinary least squares estimator in the Aalen additive hazard model on the space of monotone functions. The explicit restrictions imposed on the parameter function by our method have a clear clinical interpretation as they can be used to model early onset or late onset covariate effects on complex diseases. We provide point wise limit distribution results for the resulting estimators, that exhibit $n^{1/3}$ rate of convergence and the Chernoff distribution as the limit distribution. We also provide an asymptotic minimax lower bound showing that our suggested estimators are rate optimal, as well as provide an asymptotic minimax lower bound for the more general problem of estimating a monotone hasard function.

We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra. As a connection, it has a canonical Katz extension from the disc to the sphere. We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class. As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.

The forest polynomials $\mathfrak{P}_a$ of Nadeau-Tewari form a $\mathbb{Z}$-basis of $\mathbb{Z}[x_1,x_2,\ldots]$ whose role for the cohomology of the quasisymmetric flag variety parallels that of Schubert polynomials for the classical flag variety. Nonnegativity of the structure constants $\beta_{a,b}^c$ in $\mathfrak{P}_a \mathfrak{P}_b = \sum_c \beta_{a,b}^c \mathfrak{P}_c$ is known, but no Littlewood-Richardson-style enumerative rule has been available. We give such a rule: $\beta_{a,b}^c$ counts pairs of forest RC graphs of forest-codes $a$ and $b$ whose lift product lands on a forest RC graph of forest-code and weight both equal to $c$. The same rule descends to the cup product on $H^\bullet(\mathrm{QFl}_n)$. The proof introduces a squash product $\boxtimes$ on RC graphs, which on full Grassmannian RC graphs of a fixed height recovers the plactic monoid, a multiplactic algebra $\mathcal{M}_n$ that models Schubert polynomial multiplication, and a lift product $*$ on RC graphs whose quasi-crystal structure, in the framework of Cain-Guilherme-Malheiro, transports the Nadeau-Tewari forest equivalence through the computation. The row-decomposition machinery on RC graphs that underlies these constructions is developed in a companion paper.

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis–Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.

We establish a uniform-in-time quantitative central limit theorem (QCLT) for a nonlinear slow-fast stochastic system. We identify significant weaker sufficient conditions that enable us to obtain time-independent bounds for the Wasserstein distance between the fluctuation process and a centered Gaussian random variable. To prove our main result, we utilize tools from Malliavin calculus, specifically the second-order Poincaré inequality. In this context, applying the Poincaré inequality requires demonstrating uniform bounds over time for both the first- and second-order Malliavin derivatives.

This paper relates algebraic $K$-theory of fields to polylogarithms via general linear groups. We introduce the Goncharov Lie coalgebra, defined in terms of the $E_\infty$-homology of general linear groups. Using Steinberg modules, we find a presentation, compute its Lie cobracket, and construct motivic and Hodge realisations. Combining these results with the Rognes rank spectral sequence, we give symbolic descriptions of the rationalisation of the algebraic $K$-theory of fields beyond the cases studied by Matsumoto-Milnor and Bloch-Suslin: we express $K^{(3)}_4(F)$ and the indecomposable part of $K^{(3)}_5(F)$ in terms of Goncharov's polylogarithmic complex of weight 3.

This paper studies strict majority reasoning in finite electorates using so-called $\textit{social decision frames}$: finite sets of voters equipped with distinguished families of coalitions interpreted as those voting blocs evaluated to form a strict majority. A coherence criterion for qualitative majority judgments is identified and shown to give an exact characterization for representability of strict majorities by finitely additive measures. In addition, a minimal natural logic for reasoning about strict majorities is shown to be sound and complete. These developments motivate examination of associated combinatorial questions concerning incoherence in finite families of sets; partial results and a conjecture are given. Finally, the results of this paper are applied to correct a classical representation theorem for weak qualitative probability structures due to Patrick Suppes and to establish a May-type characterization for ordinary strict majority rule for social decision frames.

We develop general strategies for transferring soundness and completeness from more expressive modal languages to less expressive ones, unifying several existing notions of operator definability along the way. For soundness, we exploit semantic insensitivity: if a less expressive language is insensitive to a frame operation, soundness extends to the operation's closure of the original frame class. For completeness, restricting to relational semantics and languages with a single operator, we present strategies for relating the target logic's canonical model to that of a normal modal logic via a truth-preserving translation. Three of those dispense entirely with specifying an accessibility condition for the target logic, inheriting it from a normal modal logic instead.

We study the extension profile of a pair of jointly distributed finite-valued random variables $(X,Y)$, defined as the set of all triples of numbers \[ ( H(X|W), H(Y|W), I(X:Y|W) ) \] obtained by extending the pair with an auxiliary random variable $W$. This object captures structural properties of a joint distribution that are not determined by the entropies of $X$ and $Y$ and their mutual information. We associate with the extension profile its shape function defined in terms of the Legendre–Fenchel transform of the profile's lower boundary. We establish upper and lower bounds on the shape function in terms of classical information-theoretic quantities. For random pairs $(X,Y)$ whose joint distribution is uniform on its support, we relate the shape function to combinatorial and spectral properties of the associated bipartite support graph. We show that for random pairs uniformly distributed on the edges of balanced expander graphs, the shape function is within $O(\log H(XY))$ of the pointwise infimum over all pairs, which proves the conjecture stated in [MR26]. These results connect the extension profile with non-Shannon-type information inequalities, Gács–Körner common information, and spectral graph theory. We apply these techniques to study an approximate version of a question posed by Zhen Zhang ([Zha03]) concerning the characterization of pairs with maximally non-extractable mutual information.

We construct a smooth nonsingular periodic flow on a compact manifold with high emergence, in sharp contrast with the low statistical complexity of periodic self-maps. The construction is based on a modification of the Epstein–Vogt counterexample to the Periodic Orbit Conjecture and on the high-emergence mechanism of Berger–Bochi.

We study the geometry of unbounded wandering domains of entire functions using Arakelian approximation. First, we show that, given a uniformly accessible closed set contained in a strip, the connected components of its interior can be realized as escaping or oscillating wandering domains of some entire function. The iterates of the function are univalent on these wandering domains, and any unbounded wandering domain remains unbounded under iteration. Next, we construct escaping or oscillating wandering domains by lifting bounded wandering domains with prescribed geometry via the exponential map. In particular, there exists an entire function that has a half-plane as a wandering domain. Finally, we show that, in some precise sense, any simply connected open set can be approximated by escaping or oscillating wandering domains. As a direct consequence, we obtain wandering domains whose complements have arbitrarily small areas.

Flexible continuous univariate distributions with bounded support are essential for accurate input modeling in stochastic simulation and decision analysis. Although Bézier distributions provide a powerful family capable of representing complex shapes, their adoption has been hindered by the lack of efficient fitting procedures and modern software implementations. This paper develops a computational framework for fitting Bézier distributions to empirical data via both minimum error and maximum likelihood estimation, leveraging first-order optimization methods and exploiting the geometry of the parameter space. We identify provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression and develop first-order algorithms that reduce computational runtime by three to four orders of magnitude compared to traditional derivative-free methods, while delivering consistent fits across real-world data. When benchmarked against the nonlinear solver IPOPT, our methods prove three orders of magnitude faster on average and more robust, while achieving comparable accuracy. To bridge the gap between theory and practice, we introduce bezierv, an open-source Python package providing a unified interface for fitting, analyzing, and convolving Bézier distributions.

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

We prove that a biased graph is gainable over the group $\mathsf{Z}_{3}$ if and only if it contains no minor isomorphic to $(4K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over $\mathsf{Z}_{2}$ and $\mathsf{Z}_{3}$. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to $(3K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$.

An edge-coloring of a graph $G$ is called a strong edge-coloring if all its color classes are induced matchings in $G$; the minimum number of colors required for such a coloring, denoted by $\chi_{s}'(G)$, is known as the strong chromatic index of $G$. For each vertex $v$ of a graph $G$, let $d_G(v)$ denote the degree of $v$ in $G$. Let $G$ be a bipartite graph with partite sets $A$ and $B$, and let $\Delta_A=\max\{d_G(a): a\in A\}$ and $\Delta_B=\max\{d_G(b): b\in B\}$. A conjecture of Brualdi and Quinn Massey asserts that \( \chi_s'(G) \le \Delta_A \Delta_B\). In this paper, we show that \(\chi_s'(G) \le 1.676\, \Delta_A \Delta_B\) provided that the product $\Delta_A\Delta_B$ is sufficiently large.

Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human–AI collaboration on two such computations. For a singular self-adjoint Schrödinger operator, a verified zero count and Dirichlet–Neumann bracketing certify the complete negative spectrum to ten decimal places. For a delicate non-normal atom–molecule benchmark, a previously unresolved resonance pair is separated, with each member enclosed to ten digits. The second result is achieved not by increasing the precision of one-way shooting, but by reformulating the problem as a global matching system for projective solution lines. The infinite tail is encoded as uncertainty in the terminal projective data, and a componentwise, tail-robust Krawczyk–Brouwer inclusion supplies the certificate. This gives a reusable architecture for analytic boundary-value systems with ill-conditioned propagation and uncertain asymptotic data. The collaboration also exposes the strengths and limits of AI assistance. AI rapidly produced accurate candidates and plausible proof strategies, but several failed, including one apparently complete tail argument that omitted the componentwise check required by a nonuniform polydisc. Validated computation is a stringent test of AI-assisted mathematics: the output is not merely a number, but a number with a proof. These examples show why the proof object matters, and why human mathematical judgment remained decisive. More broadly, as AI makes code, exposition, and plausible numerical claims inexpensive, standards for verification, attribution, peer review, and training must adapt. The implications are unsettling; the opportunity is extraordinary.

We use the correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank to classify polynomials of minimal border rank of sufficiently high degree in up to 7 variables.