We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are $\mathbb{Z}/2$-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the 4-manifold is simply-connected and the surfaces are closed, non-orientable, and cobordant, we show that they are in fact concordant. This completes the classification of closed surfaces in simply-connected 4-manifolds up to concordance. Our methods give new constructions of cobordisms with prescribed boundaries, and completely determine when a given cobordism between the boundaries extends to a cobordism or concordance between the surfaces. We obtain our concordance results by extending Sunukjian's method of ambient surgery to the unoriented case using Pin$^-$-structures. We also discuss conditions for an arbitrary codimension 2 properly embedded submanifold to admit an unoriented spanning manifold with prescribed boundary. All results hold in both the smooth and topological categories.

Given a finite vector space $V=\mathbb{F}_q^n$, the $q$-analogue of a graph, called a $q$-graph, is a pair $Γ=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of $1$-dimensional subspaces of $V$ and $\mathcal{E}$ is a subset of the $2$-dimensional subspaces of $V$. Elements of $\mathcal{V}$ and $\mathcal{E}$ are called vertices and edges, respectively. If the edges through a vertex $X$ consist of all $2$-spaces of a $(k+1)$-dimensional space which contain $X$, regardless of the choice of vertex, then $Γ$ is $k$-regular. Moreover, $Γ$ is flag-transitive if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all incident vertex-edge pairs; and symmetric if there is a subgroup of $Γ{\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all ordered pairs of adjacent vertices. This paper classifies all $k$-regular $q$-graphs that are either flag-transitive or symmetric. The $q$-graphs in the classification are constructed from familiar objects in finite geometry, including spreads, symplectic polar spaces, and generalised hexagons. The classification depends essentially on the classification of transitive linear groups, and thus ultimately on the classification of finite simple groups.

We study the long-time behavior of a particle in $\mathbb{R}^d$, $d \geq 2$, subject to molecular diffusion and advection by a random incompressible flow. The velocity field is the divergence of a stationary random stream matrix $\mathbf{k} $ with positive Hurst exponent $γ> 0$, so the resulting random environment is multiscale and self-similar. In the perturbative regime $γ\ll 1$, we prove quenched power-law superdiffusion: for a typical realization of the environment, the displacement variance at time $t$ grows like $t^{2/(2-γ)}$, the scaling predicted by renormalization group heuristics. We also identify the leading prefactor up to a random (quenched) relative error of order $γ^{\frac12}\left| \log γ\right|^3$. The proof implements a Wilsonian renormalization group scheme at the level of the infinitesimal generator $\nabla \cdot (νI_d + \mathbf{k} ) \nabla$, based on a self-similar induction across scales. We demonstrate that the coarse-grained generator is well-approximated, at each scale $r$, by a constant-coefficient Laplacian with effective diffusivity growing like $r^γ$. This approximation is inherently scale-local: reflecting the multifractal nature of the environment, the relative error does not decay with the scale, but remains of order $γ^{\frac12}\left| \log γ\right|^2$. We also prove anomalous regularization under the quenched law: for almost every realization of the drift, solutions of the associated elliptic equation are Hölder continuous with exponent $1 - Cγ^{\frac12}$ and satisfy estimates which are uniform in the molecular diffusivity $ν$ and the scale.

We construct a quantum Schwinger-Keldysh (SK) effective field theory for the diffusive hydrodynamics of a conserved scalar field. Quantum corrections within the SK framework are guided by fluctuation-dissipation relations, enforced via a dynamical Kubo-Martin-Schwinger (KMS) symmetry. We find that the KMS symmetry necessarily generates fluctuation contributions in the SK effective action at all orders in the noise field, thereby giving rise to intrinsically non-Gaussian noise. We use our results to compute one-loop quantum corrections to the two-point density-density retarded correlation function, leading to a quantum generalization of hydrodynamic long-time tails. Our results apply at arbitrarily high orders in $\hbar$. The one-loop results for retarded correlation functions have been expressed in terms of a family of polynomials. We also provide a closed-form expression for the one-loop results at leading order in the wavevector expansion.

Matrix functions such as square root, inverse roots, and orthogonalization play a central role in preconditioned gradient methods for neural network training. This has motivated the development of iterative algorithms that avoid explicit eigendecompositions and rely primarily on matrix multiplications, making them well suited for modern GPU accelerators. We present PRISM (Polynomial-fitting and Randomized Iterative Sketching for Matrix functions computation), a general framework for accelerating iterative algorithms for computing matrix functions. PRISM combines adaptive polynomial approximation with randomized sketching: at each iteration, it fits a polynomial surrogate to the current spectrum via a sketched least-squares problem, adapting to the instance at hand with minimal overhead. We apply PRISM to accelerate Newton-Schulz-like iterations for matrix square roots and orthogonalization, which are core primitives in machine learning. Unlike prior methods, PRISM requires no explicit spectral bounds or singular value estimates; and it adapts automatically to the evolving spectrum. Empirically, PRISM accelerates training when integrated into Shampoo and Muon optimizers.

Given some vector fields on a smooth manifold satisfying Hörmander's condition, we define a bi-graded pseudo-differential calculus which contains the classical pseudo-differential calculus and a pseudo-differential calculus adapted to the sub-Riemannian structure induced by the vector fields. Our approach is based on geometric constructions (resolution of singularities) together with methods from operators algebras. We develop this calculus in full generality, including Sobolev spaces, the wavefront set, and the principal symbol, etc. In particular, using this calculus, we prove that invertibility of the principal symbol implies microlocal maximal hypoellipticity. This allows us to resolve affirmatively the microlocal version of a conjecture of Helffer and Nourrigat.

A question posed independently by Letzter and Pokrovskiy asks: how many vertex-disjoint monochromatic cycles are needed to cover the vertex set of an $r$-edge-coloured graph, as a function of its minimum (uncoloured) degree? We resolve this problem up to a $(\log r)$-factor. Specifically, we prove that, for any $r \geq 2$ and $δ\in (0,1/2)$, any $n$-vertex $r$-edge-coloured graph $G$ with $δ(G) \geq (1- δ)n$ can be covered with $\mathcal{O}(r \log r \cdot \lceil r/\log(1/δ)\rceil)$ vertex-disjoint monochromatic cycles. We construct graphs that show this is tight up to the $(\log r)$-factor for all values of $r$ and $δ$, and along the way disprove a conjecture of Bal and DeBiasio about monochromatic tree covering.

We study the convergence of the empirical measure of moderately interacting particle systems subject to singular forces derived by Lennard-Jones potential. Although the classical Lennard-Jones force is widely used in molecular dynamics, analytical results are not available. We consider a Lennard-Jones potential with free parameters in the McKean-Vlasov framework and proceed with a regularization at the mesoscale letting the particles interact moderately. We prove the well-posedness of the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards the solution of the McKean-Vlasov Fokker-Planck PDE, by means of a semigroup approach. We derive both the range of parameters characterizing the aggregation and repulsive force and the mesoscale order for which the convergence is achieved, by obtaining the right integrability regularity of the drift.

This paper investigates information freshness in a remote estimation system in which the remote information source is a continuous-time Markov chain (CTMC). For such systems, estimators have been mainly restricted to the class of martingale estimators in which the remote estimate at any time is equal to the value of the most recently received update. This is mainly due to the simplicity and ease of analysis of martingale estimators, which however are far from optimal, especially in query-based (i.e., pull-based) update systems. In such systems, maximum a-posteriori probability (MAP) estimators are optimal. However, MAP estimators can be challenging to analyze in continuous-time settings. In this paper, we introduce a new class of estimators, called structured estimators, which can seamlessly shift from a martingale estimator to a MAP estimator, enabling them to retain useful characteristics of the MAP estimate, while still being analytically tractable. Particularly, we introduce a new estimator termed as the $p$-MAP estimator which is a piecewise-constant approximation of the MAP estimator with finitely many discontinuities, bringing us closer to a full characterization of MAP estimators when modeling information freshness. In fact, we show that for time-reversible CTMCs, the MAP estimator reduces to a $p$-MAP estimator. Using the binary freshness (BF) process for the characterization of information freshness, we derive the freshness expressions and provide optimal state-dependent sampling policies (i.e., querying policies) for maximizing the mean BF (MBF) for pull-based remote estimation of a single CTMC information source, when structured estimators are used. Moreover, we provide optimal query rate allocation policies when a monitor pulls information from multiple heterogeneous CTMCs with a constraint on the overall query rate.

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic growth, we prove that there are no proper surfaces contained either in two transverse vertical half-spaces of $\mathbb{R}^3$ or in a half-space determined by a non-vertical plane. We show that this second result holds in a more general context, namely, for a class of stochastically complete weighted minimal surfaces. In this setup, we also prove a result for surfaces contained in regions bounded by cones. Furthermore, for stochastically complete weighted minimal surfaces satisfying restrictions on their principal curvatures, we establish a version of the classic strong half-space result due to Hoffman-Meeks.

We investigate the connection between bijective, not necessarily finite, set-theoretic solutions of the pentagon equation and Hopf algebras. Firstly, we prove that finite solutions correspond to Hopf algebras with the positive basis property. As a corollary we generalise Lu-Yan-Zhu classification to arbitrary characteristic $0$ fields $k$. Secondly, we study the general problem of when a Hopf algebra has a basis yielding a set-theoretic solution. Finally, we classify all (co)commutative bijective solutions. This result requires to obtain a description of all bases of a group algebra $k[G]$ yielding a set-theoretic solution. We namely show that such bases correspond, through a Fourier transform, to splittings $A \rtimes N$ of $G$ with $A$ a finite abelian group.

We study the magnetic two-component Hunter-Saxton system (M2HS), which was recently derived in \cite{M24} as a magnetic geodesic equation on an infinite-dimensional configuration space. While the geometric framework and the global weak flow were outlined there, the present paper provides the analytical foundations of this construction from the PDE perspective. First, we derive an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding an alternative proof of the blow-up criterion together with an explicit expression for the blow-up time. Second, we rigorously construct global conservative weak solutions by developing the analytic theory of the relaxed configuration space and the associated weak magnetic geodesic flow, thereby realizing the geometric program proposed in \cite{M24}.

Voltage (Volt) and reactive-power (VAR) control in transmission networks is critical for reliability and increasingly needs fast, implementable decisions. This paper presents a transmission Volt/VAR Optimization (VVO) framework that co-optimizes discrete control of on-load tap-changing transformers (OLTCs) and capacitor banks (CBs) with AC power flow (ACPF) physics to improve voltage stability and minimize VAR generation. The framework follows a relax-round-resolve pipeline: a continuous relaxation proposes targets, a rounding step selects feasible discrete settings, and a final solve enforces AC power flow physics. Extensive experiments on IEEE, PEGASE, and RTE systems show consistent improvements in voltage and VAR quality metrics with modest generator redispatch while preserving economic operation and achieving compatible runtimes with real-time transmission operations.

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general $L^2$ initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.

Given a closed subscheme $Z$ in a smooth variety $X$, defined by the maximal minors of an $s\times r$ matrix of regular functions, with $s\geq r$, we consider the corresponding incidence correspondence $W$ in $Y=X\times {\mathbf P}^{r-1}$, and relate the log canonical thresholds of $(X,Z)$ and $(Y,W)$. In particular, when $r=s$, we show that ${\rm lct}(X,Z)=1$ if and only if ${\rm lct}(Y,W)=r$. Moreover, in this case, we show that $Z$ has rational singularities if and only if $W$ has pure codimension $r$ in $Y$ and has rational singularities. As a consequence, we deduce that for a configuration hypersurface with a connected configuration matroid, the corresponding configuration incidence variety has rational singularities.

In 1960, Smale defined a filtration of a closed smooth manifold by the unstable manifolds of fixed points and closed orbits of a Morse-Smale vector field defined on it, and derived generalized Morse inequalities. This suggests that, similarly to the Morse chain complex of a gradient-like vector field, even in the presence of closed orbits, Morse-Smale vector fields admit canonical chain complexes, invariant under topological equivalence, from which one can algebraically derive Morse inequalities. In this paper we show that this is actually the case, improving the state of the art that only offers non-canonical chain complexes. Technically, we achieve this result considering the Čech homology spectral sequence of the unstable manifolds filtration. In particular, we turn bounded exact couples into chain complexes such that the limit page of the spectral sequence associated with an exact couple gives the homology of the chain complex. We showcase our construction with examples.

We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even if the deterministic part of equation has multiple steady states.

We study the homogeneous involutions on the full square matrices over an algebraically closed field endowed with a division grading with commutative support. We obtain the classification of the isomorphism and equivalence classes for the Pauli grading. We also investigate the homogeneous involutions on the full square matrices with entries in a finite-dimensional graded-division algebra over an algebraically closed field of characteristic not $2$ endowed with an arbitrary grading by an arbitrary group.

In this article we investigate the properties of isogonal conjugation in isosceles tetrahedron. Particularly we reveal three hyperbolic paraboloids each of which is formed by pairs of isogonal conjugate points symmetric in the respective bimedian, as well as we prove that the circumsphere of an isosceles tetrahedron is invariant under isogonal conjugation in that tetrahedron.

Various statistical tasks, including sampling or computing Wasserstein barycenters, can be reformulated as fixed-point problems for operators on probability distributions. Accelerating standard fixed-point iteration schemes provides a promising novel approach to the design of efficient numerical methods for these problems. The Wasserstein geometry on the space of probability measures, although not precisely Riemannian, allows us to define various useful Riemannian notions, such as tangent spaces, exponential maps and parallel transport, motivating the adaptation of Riemannian numerical methods. We demonstrate this by developing and implementing the Riemannian Anderson Mixing (RAM) method for Gaussian distributions. The method reuses the history of the residuals and improves the iteration complexity, and we argue that the additional costs, compared to Picard method, are negligible. We show that certain open balls in the Bures-Wasserstein manifold satisfy the requirements for convergence of RAM. The numerical experiments show a significant acceleration compared to a Picard iteration, and performance on par with Riemannian Gradient Descent and Conjugate Gradient methods.

The category $\mathbb{DRDL'}$, whose objects are c-differential residuated distributive lattices that satisfy the condition $\mathbf{CK}$, is the image of the category $\mathbb{RDL}$, whose objects are residuated distributive lattices, under the categorical equivalence (Kalman functor) $\mathbf{K}$. The main goal of this paper is to lift this equivalence $\mathbf{K}$ to the category $\mathbb{MRDL}$, whose objects are monadic residuated distributive lattices, and the category $\mathbb{MDRDL'}$, whose objects are pairs formed by an object of $\mathbb{DRDL'}$ and a center universal quantifier. Firstly, based on the variety of monadic FL$_\textrm{e}$-algebras, we introduce the concept of monadic residuated lattices and study some of their further algebraic properties, proving the classes of monadic residuated distributive lattices and monadic c-differential residuated distributive lattices are in one-to-one correspondence. Subsequently, based on this corresponding relation, we prove that there exists a categorical equivalence between the categories $\mathbb{MRDL}$ and $\mathbb{MDRDL'}$. The results of this paper not only generalizes the works of Sagastume and San Martín in [Mathematical Logic Quarterly, {\bf 60}(2014), 375--388], but also addresses and overcomes the limitations identified in the works of [Studia Logica, {\bf 111}(2023), 361--390]. Finally, this paper concludes with some applications regarding descriptions of a 2-contextual translation.

We define fully exact module categories, a subclass of exact module categories over a finite braided tensor category that is stable under the relative Deligne product. In contrast, we demonstrate with examples in both zero and non-zero characteristic of the base field that the class of exact module categories is not stable under this product. We also observe in examples that fully exact module categories form a dense subset in the class of exact ones. The monoidal 2-category of fully exact module categories strictly contains those of invertible and separable module categories. In fact, we show that each internal algebra of a fully exact module category is projectively separable, a generalization of separable algebras involving projective objects. In the semisimple case, a module category is fully exact if and only if it is separable. In general, fully exact module categories are not dualizable inside their class, but if they are, they are fully dualizable objects in the monoidal 2-category of finite module categories. We call such module categories perfect. We show that perfect module categories form a rigid monoidal 2-subcategory containing all fully dualizable objects. Therefore, we propose perfect module categories as a model for finite tensor 2-categories. If the braiding is symmetric, a module category is fully exact if and only if it is perfect. As a detailed example, we classify fully exact, and hence perfect, module categories over the symmetric tensor category of modules over Sweedler's four-dimensional Hopf algebra and compute their relative Deligne products, and the categories of 1-morphisms. For a general quasi-triangular Hopf algebra, we analyze when the category of finite-dimensional vector spaces is fully exact. We show that this is not the case for both Sweedler's Hopf algebra and Lusztig's factorizable small quantum group of type $A_1$ at an odd root of unity.

Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibit a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ using experimentally feasible SWAP-test-based estimators requires $Θ(N^{2-2/k})$ samples for constant $k$, and $Θ(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $Θ(N^2 \log N)$ samples. These results provide principled guidance for the design of loss functions in quantum machine learning, which we illustrate in the training quantum denoising diffusion probabilistic models.

In this paper we present criteria in terms of dual pairs of exceptional sequences for an abelian category to be highest weight. The criteria are applied in three situations of geometric origin. We give new proofs for the facts that the category of perverse sheaves of middle perversity on complex-analytic manifolds with suitable conditions on the stratification is highest weight and that the derived coherent category of any Grassmannian has a $t$-structure with highest weight heart. Also we show that the abelian null category of any proper birational morphism of regular surfaces is highest weight. For this null category, we give a geometric description of some special objects related to the highest weight structure, such as standard, costandard and characteristic tilting objects.

We present a batched first-order method for solving multiple linear programs in parallel on GPUs. Our approach extends the primal-dual hybrid gradient algorithm to efficiently solve batches of related linear programming problems that arise in mixed-integer programming techniques such as strong branching and bound tightening. By leveraging matrix-matrix operations instead of repeated matrix-vector operations, we obtain significant computational advantages on GPU architectures. We demonstrate the effectiveness of our approach on various case studies and identify the problem sizes where first-order methods outperform traditional simplex-based solvers depending on the computational environment one can use. This is a significant step for the design and development of integer programming algorithms tightly exploiting GPU capabilities where we argue that some specific operations should be allocated to GPUs and performed in full instead of using light-weight heuristic approaches on CPUs.

We study tropical Tevelev degrees arising from maps between certain tropical moduli spaces of curves. Building on work of Dawson and Cavalieri, who defined and computed tropical Tevelev degrees in the case of degree $d = g+1$ and $n = g+3$ marked points, we extend the theory by introducing an additional integer parameter $\ell$. In our framework the curve degree and number of marked points vary as $d = g + 1 + \ell$ and $n = g + 3 + 2\ell$, and we analyze the resulting tropical Tevelev degrees for both positive and negative values of $\ell$. This tropicalizes results of Cela, Pandharipande, and Schmitt on algebraic Tevelev degrees. We then further broaden the framework by introducing generalized tropical Tevelev degrees, providing the tropical counterpart to the generalized Tevelev degrees studied by Cela and Lian. These results establish a wider set of computational and structural patterns for intersection calculations on tropical moduli spaces and reveal new behavior beyond the classical setting.

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that any two such diagrams of the same knot have identical writhe. The proof relies on a generalization of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden, Karimi, and Sikora using a different generalized Jones polynomial; therefore, this text largely expands on one of the main proof tools.

Motivated by various equivalent versions of the SHGH conjecture for $\mathbb{P}^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces. We prove that this conjecture is true for the Hirzebruch surface $\mathbb{F}_e$ blown up at $r\leqslant e+4$ very general points.

The aim in these lectures is to study singularity formation, nonuniqueness, and topological change in motion by mean curvature.

Diffusion generative models synthesize samples by discretizing reverse-time dynamics driven by a learned score (or denoiser). Existing convergence analyses of diffusion models typically scale at least linearly with the ambient dimension, and sharper rates often depend on intrinsic-dimension assumptions or other geometric restrictions on the target distribution. We develop an alternative, information-theoretic approach to dimension-free convergence that avoids any geometric assumptions. Under mild assumptions on the target distribution, we bound KL divergence between the target and generated distributions by $O(H^2/K)$ (up to endpoint factors), where $H$ is the Shannon entropy and $K$ is the number of sampling steps. Moreover, using a reformulation of the KL divergence, we propose a Loss-Adaptive Schedule (LAS) for efficient discretization of reverse SDE which is lightweight and relies only on the training loss, requiring no post-training heavy computation. Empirically, LAS improves sampling quality over common heuristic schedules.