In recent years, large language models have greatly improved in their ability to perform complex multi-step reasoning. However, even state-of-the-art models still regularly produce logical mistakes. To train more reliable models, we can turn either to outcome supervision, which provides feedback for a final result, or process supervision, which provides feedback for each intermediate reasoning step. Given the importance of training reliable models, and given the high cost of human feedback, it is important to carefully compare the both methods. Recent work has already begun this comparison, but many questions still remain. We conduct our own investigation, finding that process supervision significantly outperforms outcome supervision for training models to solve problems from the challenging MATH dataset. Our process-supervised model solves 78% of problems from a representative subset of the MATH test set. Additionally, we show that active learning significantly improves the efficacy of process supervision. To support related research, we also release PRM800K, the complete dataset of 800,000 step-level human feedback labels used to train our best reward model.
Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non autonomous Hamiltonian equations.
We construct a quantization of the moduli space $\mathcal{GH}_Λ(S\times\mathbb{R})$ of maximal globally hyperbolic Lorentzian metrics on $S\times \mathbb{R}$ with constant sectional curvature $Λ$, for a punctured surface $S$. Although this moduli space is known to be symplectomorphic to the cotangent bundle of the Teichmüller space of $S$ independently of the value of $Λ$, we define geometrically natural classes of observables leading to $Λ$-dependent quantizations. Using special coordinate systems, we first view $\mathcal{GH}_Λ(S\times\mathbb{R})$ as the set of points of a cluster $\mathscr{X}$-variety valued in the ring of generalized complex numbers $\mathbb{R}_Λ= \mathbb{R}[\ell]/(\ell^2+Λ)$. We then develop an $\mathbb{R}_Λ$-version of the quantum theory for cluster $\mathscr{X}$-varieties by establishing $\mathbb{R}_Λ$-versions of the quantum dilogarithm function. As a consequence, we obtain three families of projective unitary representations of the mapping class group of $S$. For $Λ<0$ these representations recover those of Fock and Goncharov, while for $Λ\geq 0$ the representations are new.
The bulk scaling limit of eigenvalue distribution on the complex plane ${\mathbb{C}}$ of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the $D$-dimensional complex spaces ${\mathbb{C}}$, $D \in {\mathbb{N}}$, in which the Ginibre DPP is realized when $D=1$. This one-parameter family ($D \in {\mathbb{N}}$) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each $D$, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius $R$ in ${\mathbb{R}}^{2D} \simeq {\mathbb{C}}^D$. We prove that any DPP in the Heisenberg family is in the hyperuniform state of Class I, in the sense that the number variance behaves as $R^{2D-1}$ as $R \to \infty$. Our exact results provide asymptotic expansions of the number variances in large $R$.
The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally-unbounded infinitesimal generators. In conclusion the concept of module over a Banach algebra is proposed as the generalization of Banach algebra. As an application to mathematical physics, the rigorous formulation of rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.
For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^*$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.
We recall the fat-graph description of Riemann surfaces $Σ_{g,s,n}$ and the corresponding Teichmüller spaces $\mathfrak T_{g,s,n}$ with $s>0$ holes and $n>0$ bordered cusps in the hyperbolic geometry setting. If $n>0$, we have a bijection between the set of Thurston shear coordinates and Penner's $λ$-lengths and we can induce, on the one hand, the Poisson bracket on $λ$-lengths from the Poisson bracket on shear coordinates introduced by V.V.Fock in 1997 and, on the other hand, a symplectic structure $Ω_{\text{WP}}$ on the set of extended shear coordinates from Penner's symplectic structure on $λ$-lengths. We derive $Ω_{\text{WP}}$, which turns out to be similar to the Kontsevich symplectic structure for $ψ$-classes in complex-analytic geometry, and demonstrate that it is indeed inverse to the Fock Poisson structure.