Nathaniel Bottman

Detecting LLM reasoning failures at inference time without ground-truth labels has motivated a wide range of confidence baselines, including self-consistency, semantic entropy, and P(True), built on within-question sampling and self-evaluation. Operad theory, the formalism for systems built by iterated substitution, suggests a complementary diagnostic: a model's direct answer to a compositional query should agree with the answer it produces by composing a stated decomposition of the same query. We instantiate this idea as operadic consistency (OC), a per-question signal. Across twelve instruction-tuned LLMs (4B to 671B parameters, open-weights and closed-source) on four multi-hop QA datasets, OC is strongly correlated with accuracy on every dataset (Pearson \(r \in [0.86, 0.94]\), all \(p \leq 0.0004\)), and is the only signal we evaluate with \(r \geq 0.85\) uniformly across all four datasets. Chain-of-thought self-consistency (CoT-SC; Wang et al., 2023) matches OC on HotpotQA and DROP (\(r = 0.93, 0.87\)) but drops to \(r \approx 0.45\) on MuSiQue and StrategyQA. At the per-question level, OC contributes information beyond CoT-SC and semantic entropy on every dataset (cluster-robust \(p \leq 10^{-16}\) for the OC coefficient), and the conclusion is robust to additionally controlling for constructed decomposition-aware baselines (\(p \leq 10^{-13}\)). The same signal yields selective-prediction improvements (accuracy at fixed coverage) over a tuned CoT-SC baseline at the equal-cost \(K = 3\) budget (AUARC lifts of +0.086 to +0.096 and AUROC lifts of +0.092 to +0.164; 95% CIs exclude zero on every cell). On five frontier thinking models, where the decomposition is extracted from the model's own chain of thought, the same equal-cost comparison gives positive selective-prediction point-estimate lift on all 16 (dataset, budget, metric) cells tested, with 95% CIs excluding zero on 12 of the 16.

Question decomposition, i.e. breaking a complex query into simpler sub-queries whose answers are composed to produce a final answer, is a widely used strategy for improving LLM reasoning, yet it currently lacks a rigorous mathematical foundation. In this paper, we propose operads, mathematical structures that model many-in, one-out operations and compositions thereof, as a natural framework for describing question decomposition. We define the questions operad \(Q\), in which operations correspond to question templates and composition corresponds to substitution of sub-answers, and show how QA models can be interpreted as algebras over \(Q\). Beyond reframing existing practice, this operadic perspective points toward new methods, in particular a notion of operadic consistency, which measures whether a QA model's answers agree across the partial collapses of a question decomposition tree. Empirical evaluation of operadic consistency is reported in our companion paper (Bottman, Liu, and Richardson, 2026), which finds it strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets and outperforming standard temperature-based self-consistency baselines. We argue that operads are the natural mathematical home for question decomposition, and that invariants such as operadic consistency open new directions for analyzing and improving the reliability of multi-step reasoning.

What neural networks learn depends fundamentally on the geometry of the underlying loss function. We study how different regularizers affect the geometry of this function. One of the most basic geometric properties of a smooth function is whether it is Morse or not. For nonlinear deep neural networks, the unregularized loss function \(L\) is typically not Morse. We consider several different regularizers, including weight decay, and study for which regularizers the regularized function \(L_\epsilon\) becomes Morse.

This paper provides a blueprint for the construction of a symplectic \((A_\infty,2)\)-category, \(\mathsf{Symp}\). We develop two ways of encoding the information in \(\mathsf{Symp}\) — one topological, one algebraic. The topological encoding is as an \((A_\infty,2)\)-flow category, which we define here. The algebraic encoding is as a linear \((A_\infty,2)\)-category, which we extract from the topological encoding. In upcoming work, we plan to use the adiabatic Fredholm theory developed by us to construct \(\mathsf{Symp}\) as an \((A_\infty,2)\)-flow category, which thus induces a linear \((A_\infty,2)\)-category.

The notion of a linear \((A_\infty,2)\)-category developed here goes beyond the proposal of Bottman and Carmeli. The recursive structure of the 2-associahedra identifies faces with fiber products of 2-associahedra over associahedra, which led Bottman and Carmeli to associate operations to singular chains on 2-associahedra. The innovation in our new definition of linear \((A_\infty,2)\)-category is to extend the family of 2-associahedra to include all fiber products of 2-associahedra over associahedra. This allows us to associate operations to cellular chains, which in particular enables us to produce a definition that involves only one operation in each arity, governed by a collection of \((A_\infty,2)\)-equations.

We develop a robust functional analytic framework for adiabatic limits. This framework consists of a notion of adiabatic Fredholm family, several possible regularity properties, and an explicit construction that provides finite dimensional reductions that fit into all common regularization theories. We show that these finite dimensional reductions inherit global continuity and differentiability properties from the adiabatic Fredholm family. Moreover, we indicate how to construct adiabatic Fredholm families that describe the adiabatic limits for the nondegenerate Atiyah-Floer conjecture and strip-shrinking in quilted Floer theory.

For a symplectic 4-manifold \(M\) equipped with a singular Lagrangian fibration with a section, the natural fiberwise addition given by the local Hamiltonian flow is well-defined on the regular points. We prove, in the case that the singularities are of focus-focus type, that the closure of the corresponding addition graph is the image of a Lagrangian immersion in \((M\times M)^- \times M\), and we study its geometry. Our main motivation for this result is the construction of a symmetric monoidal structure on the Fukaya category of such a manifold.

The second author introduced 2-associahedra as a tool for investigating functoriality properties of Fukaya categories, and he conjectured that they could be realized as face posets of convex polytopes. We introduce a family of posets called categorical \(n\)-associahedra, which naturally extend the second author's 2-associahedra and the classical associahedra. Categorical \(n\)-associahedra give a combinatorial model for the poset of strata of a compactified real moduli space of a tree arrangement of affine coordinate subspaces. We construct a family of complete polyhedral fans, called velocity fans, whose coordinates encode the relative velocities of pairs of colliding coordinate subspaces, and whose face posets are the categorical n-associahedra. In particular, this gives the first fan realization of 2-associahedra. In the case of the classical associahedron, the velocity fan specializes to the normal fan of Loday's realization of the associahedron.

For proving that the velocity fan is a fan, we first construct a cone complex of metric \(n\)-bracketings and then exhibit a piecewise-linear isomorphism from this complex to the velocity fan. We demonstrate that the velocity fan, which is not simplicial, admits a canonical smooth flag triangulation on the same set of rays, and we describe a second, finer triangulation which provides a new extension of the braid arrangement. We describe piecewise-unimodular maps on the velocity fan such that the image of each cone is a union of cones in the braid arrangement, and we highlight a connection to the theory of building sets and nestohedra. We explore the local iterated fiber product structure of categorical \(n\)-associahedra and the extent to which this structure is realized by the velocity fan. For the class of concentrated \(n\)-associahedra we exhibit generalized permutahedra having velocity fans as their normal fans.

We define a family of convex polytopes called constrainahedra, which index collisions of horizontal and vertical lines. Our construction proceeds by first defining a poset \(C(m,n)\) of good rectangular preorders, then proving that \(C(m,n)\) is a lattice, and finally constructing a polytopal realization by taking the convex hull of a certain explicitly-defined collection of points. The constrainahedra will form the combinatorial backbone of the second author's construction of strong homotopy duoids. We indicate how constrainahedra could be realized as Gromov-compactified configuration spaces of horizontal and vertical lines; viewed from this perspective, the constrainahedra include naturally into the first author's notion of 2-associahedra.

For \(r \geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\setminus\{\mathbf{0}\}\), we construct a proper complex variety \(\overline{2M}_{\mathbf{n}}\). \(\overline{2M}_{\mathbf{n}}\) is locally toric, and it is equipped with a forgetful map \(\overline{2M}_{\mathbf{n}} \to \overline M_{0,r+1}\). This space is a compactification of \(2M_{\mathbf{n}}\), the configuration space of marked vertical lines in \(\mathbb{C}^2\) up to translations and dilations. In the appendices, we give several examples and show how the stratification of \(\overline{2M}_{\mathbf{n}}\) can be used to recursively compute its virtual Poincaré polynomial.

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya \(A_\infty\)-category, Floer cohomology, and symplectic cohomology. Beginning with seminal work of Wehrheim and Woodward in the late 2000s, several authors have developed techniques for functorial manipulation of these invariants. We survey these functorial structures, including Wehrheim–Woodward's quilted Floer cohomology and functors associated to Lagrangian correspondences, Fukaya's alternate approach to defining functors between Fukaya \(A_\infty\)-categories, and Bottman's ongoing construction of the symplectic \((A_\infty,2)\)-category. In the last section, we describe a number of direct and indirect applications of this circle of ideas, and propose a conjectural version of the Barr–Beck Monadicity Criterion in the context of the Fukaya \(A_\infty\)-category.

We define an operad in Top, called \(\text{FM}_2^W\). The spaces in \(\text{FM}_2^W\) come with CW decompositions, such that the operad compositions are cellular. In fact, each space in \(\text{FM}_2^W\) is the realization of a simplicial set. We expect, but do not prove here, that \(\text{FM}_2^W\) is isomorphic to the 2-dimensional Fulton-MacPherson operad \(\text{FM}_2\). Our construction is connected to the author's work on the symplectic \((A_\infty,2)\)-category, and suggests a strategy toward equipping the symplectic cochain complex with the structure of a homotopy Batalin-Vilkoviskiy algebra.

We show that the 2-associahedra are Eulerian lattices, by exploiting their recursive structure.

If \(G\) is a Lie group acting in a Hamiltonian fashion on a symplectic manifold \(M\), we may form the symplectic quotient \(M/\!/G\). Associated to this situation is a Lagrangian correspondence \(\Lambda_G\) from \(M/\!/G\) to \(M\). In this short paper, we construct in two related examples quilts with seam condition given by such a correspondence \(\Lambda_G\), in the case of \(S^1\) acting on \(\mathbb{CP}^2\) with symplectic quotient \(\mathbb{CP}^2/\!/S^1 = \mathbb{CP}^1\). First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain group \(CF(\gamma,S^1_{\text{Cl}})\) and \(CF(\mathbb{RP}^2,T^2_{\text{Cl}})\), where \(\gamma\) is the connected double-cover of \(\mathbb{RP}^1\). Second, we produce a figure eight bubble that was predicted by Akveld–Cannas da Silva–Wehrheim. The figure eight bubbles we construct in this paper are the first concrete examples of this bubbling phenomenon, which is of key importance to functoriality for the Fukaya category.

We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a relative 2-operad over the associahedra. Using this structure, we define the notions of an \((A_\infty,2)\)-category and \((A_\infty,2)\)-category in spaces and in chain complexes over a ring. Finally, we show that for any continuous map \(A\to X\), we can associate an \((A_\infty,2)\)-space \(\theta(A\to X)\), which specializes to \(\theta(\text{pt}\to X) = \Omega^2X\) and \(\theta(A \to \text{pt}) = \Omega A\times\Omega A\).

I show that the novel figure eight singularity in a pseudoholomorphic quilt can be continuously removed when composition of Lagrangian correspondences is cleanly immersed. The proof of this result requires a collection of width-independent elliptic estimates that allow for non-standard complex structures on the domain.

For \(r \geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\), I construct the compactified moduli space \(\overline{2\mathcal{M}}_{\mathbf{n}}\) of witch curves of type \(\mathbf{n}\). These are the domain moduli spaces for witch balls, analogous to the domain moduli spaces \(\overline{\mathcal{M}}_r\) for pseudoholomorphic polygons. I equip \(\overline{2\mathcal{M}}_{\mathbf{n}}\) with a stratification by the 2-associahedron \(W_{\mathbf{n}}\), and prove that \(\overline{2\mathcal{M}}_{\mathbf{n}}\) is compact, second-countable, and metrizable. In addition, I show that the forgetful map \(\overline{2\mathcal{M}}_{\mathbf{n}} \to \overline{\mathcal{M}}_r\) to the moduli space of stable disk trees is continuous and respects the stratifications.

For any \(r\geq 1\) and \(\mathbf{n} \in \mathbb{Z}_{\geq0}^r\setminus\{\mathbf0\}\) I construct a poset \(W_{\mathbf{n}}\) called a 2-associahedron. The 2-associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. I prove that the completion of \(W_{\mathbf{n}}\) is an abstract polytope of dimension \(|\mathbf{n}|+r-3\). There are forgetful maps \(W_{\mathbf{n}}\to K_r\), where \(K_r\) is the \((r−2)\)-dimensional associahedron, and the 2-associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, I work out the 2- and 3-dimensional 2-associahedra in detail.

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-0 effect, and predict its algebraic consequences — geometric composition extends to a curved \(A_\infty\)-bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schmäschke provides examples of nontrivial figure eight bubbles.

An important component of every country's COVID-19 response is fast and efficient testing — to identify and isolate cases, as well as for early detection of local hotspots. For many countries, producing a sufficient number of tests has been a serious limiting factor in their efforts to control COVID-19 infections. Group testing is a well-established mathematical tool, which can provide a serious and rapid improvement to this situation. In this note, we compare several well-established group testing schemes in the context of qPCR testing for COVID-19. We include example calculations, where we indicate which testing architectures yield the greatest efficiency gains in various settings. We find that for identification of individuals with COVID-19, array testing is usually the best choice, while for estimation of COVID-19 prevalence rates in the total population, Gibbs-Gower testing usually provides the most accurate estimates given a fixed and relatively small number of tests. This note is intended as a helpful handbook for labs implementing group testing methods.

The stability of the stationary periodic solutions of the integrable (one-dimensional, cubic) defocusing nonlinear Schrodinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additional argument using an appropriate Krein signature allows us to conclude the (nonlinear) orbital stability of all stationary solutions of the defocusing NLS equation with respect to so-called subharmonic perturbations: perturbations that have period equal to an integer multiple of the period of the amplitude of the solution. All results presented here are independent of the size of the amplitude of the solutions and apply equally to solutions with trivial and nontrivial phase profiles.

Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solutions of the Korteweg-deVries equation, the so-called cnoidal waves. In this paper, we exploit the squared-eigenfunction connection between the linear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrum of the linear stability problem for perturbations that are bounded on the real line. We find that this spectrum is confined to the imaginary axis, leading to the conclusion of spectral stability. An additional argument allows us to conclude the completeness of the associated eigenfunctions.