arXiv subjects

Joseph Cummings

Publications and source records attributed to Joseph Cummings.

15 recordsLinked to original sources

arXiv2026

Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves

We study the Green-Griffiths-Lang Conjecture for complements of smooth plane curves. We develop an effective method for computing a family of negatively twisted invariant logarithmic 2-jet differentials. By realizing the first logarithmic jet space as a hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$, we encode these jet differentials in a finitely generated bigraded module that can be computed explicitly. We use this description to give a computational criterion for the Green-Griffiths-Lang Conjecture and verify it for several families of smooth plane curves. In examples with sufficiently many independent jet differentials, we determine the exceptional locus explicitly.

math.AG↗

arXiv2026

Identifiability of phylogenetic networks and quintet concordance factors

Several statistical methods of phylogenetic network inference and testing for non-tree-like relationships are based on assessing genomic data through quartet Concordance Factors, the frequencies of 4-taxon topological relationships on gene trees. While such an approach obviates making several undesirable modeling assumptions, it also results in non-identifiability issues for network roots and for small cycles. In this work, an algorithm and accompanying Macaulay2 implementation are provided for computing $n$-tet Concordance Factors on any phylogenetic network. We employ this algorithm on quintet Concordance Factors, summarizing 5-taxon gene trees, to explore identifiability of level-1 networks under the Network Multispecies Coalescent model. We show some additional network features become identifiable that are not through quartets. As identifiability is a necessary prerequisite to inference by any method, this lays a foundation for future inference work.

q-bio.PE↗

arXiv2026

torchtune: PyTorch native post-training library

Modern LLMs typically require multistage training pipelines to achieve strong downstream performance, with post-training serving as the main interface for adapting open-weight models. We introduce torchtune, a PyTorch-native library designed to streamline the post-training lifecycle of LLMs, enabling efficient fine-tuning, experimentation, and deployment-oriented workflows. Unlike many existing fine-tuning frameworks, which often optimize for ease of use, specialized recipes, or hardware efficiency at the cost of transparency and extensibility, torchtune emphasizes modularity, hackability, and direct access to the underlying PyTorch components. In this paper, we present the design principles behind torchtune, describe how they are reflected in its model builders, training recipes, and distributed training stack, and evaluate the library across representative post-training settings. We compare against popular fine-tuning frameworks, including Axolotl and Unsloth, and show that torchtune provides strong performance and memory efficiency across many settings while remaining flexible enough for rapid research iteration. These results position torchtune as a practical foundation for reproducible LLMs post-training research.

cs.LG↗

arXiv2025

Routing functions for parameter space decomposition to describe stability landscapes of ecological models

Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical insight into these transitions. This paper introduces an algebraic framework for analyzing the stability landscapes of ecological models defined by systems of first-order autonomous ordinary differential equations with polynomial or rational rate functions. Using tools from real algebraic geometry, we characterize parameter regions associated with steady-state feasibility and stability via three key boundaries: singular, stability (Routh-Hurwitz), and coordinate boundaries. With these boundaries in mind, we employ routing functions to compute the connected components of parameter space in which the number and type of stable steady states remain constant, revealing the stability landscape of these ecological models. As case studies, we revisit the classical Levins-Culver competition-colonization model and a recent model of coral-bacteria symbioses. In the latter, our method uncovers complex stability regimes, including regions supporting limit cycles, that are inaccessible via traditional techniques. These results demonstrate the potential of our approach to inform ecological theory and intervention strategies in systems with nonlinear interactions and multiple stable states.

q-bio.PE↗

arXiv2025

Scalable and Performant Data Loading

We present SPDL (Scalable and Performant Data Loading), an open-source, framework-agnostic library designed for efficiently loading array data to GPU. Data loading is often a bottleneck in AI applications, and is challenging to optimize because it requires coordination of network calls, CPU-bound tasks, and GPU device transfer. On top of that, Python's GIL (Global Interpreter Lock) makes it difficult to gain performance improvement from multi-threading. We found that when data preprocessing functions release the GIL entirely, it is possible to execute them concurrently in a thread pool, thereby improving the workflow performance. Our benchmark shows that compared to the PyTorch DataLoader, SPDL can iterate through the ImageNet dataset 74% faster while using 38% less CPU and 50GB less memory. When training ViT-B/16 model, SPDL can send data to the GPU at a speed that does not starve the training. Additionally, when using SPDL on Python 3.13t, without changing any code, the throughput is further by improved by 33%, thanks to the disabled GIL. SPDL can improve the performance of current AI model training, and receives further performance improvements when Free-Threaded Python is adopted in production systems. SPDL is available at https://github.com/facebookresearch/spdl.

cs.DC↗

arXiv2024

Smooth connectivity in real algebraic varieties

A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. By adapting an approach for determining connectivity in complements of real hypersurfaces by Hong, Rohal, Safey El Din, and Schost, algorithms are presented for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When taking such real hypersurface to be the set of singular points, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety and several examples are included to demonstrate the approach.

math.AG↗

arXiv2023

The Pfaffian Structure of CFN Phylogenetic Networks

Algebraic techniques in phylogenetics have historically been successful at proving identifiability results and have also led to novel reconstruction algorithms. In this paper, we study the ideal of phylogenetic invariants of the Cavender-Farris-Neyman (CFN) model on a phylogenetic network with the goal of providing a description of the invariants which is useful for network inference. It was previously shown that to characterize the invariants of any level-1 network, it suffices to understand all sunlet networks, which are those consisting of a single cycle with a leaf adjacent to each cycle vertex. We show that the parameterization of an affine open patch of the CFN sunlet model, which intersects the probability simplex, factors through the space of skew-symmetric matrices via Pfaffians. We then show that this affine patch is isomorphic to a determinantal variety and give an explicit Gr{\"o}bner basis for the associated ideal, which involves only $\binom{n}{2}$ coordinates rather than $2^{n}$. Lastly, we show that sunlet networks with at least 6 leaves are identifiable using only these polynomials and run extensive simulations, which show that these polynomials can be used to accurately infer the correct network from DNA sequence data.

math.AG↗

arXiv2023

Computing Implicitizations of Multi-Graded Polynomial Maps

In this paper, we focus on computing the kernel of a map of polynomial rings $φ$. This core problem in symbolic computation is known as implicitization. While there are extremely effective Gröbner basis methods used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the map $φ$ is multigraded, we consider an alternative approach. We demonstrate how to quickly compute a matrix of maximal rank for which $φ$ has a positive multigrading. Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where Gröbner techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is \emph{embarassingly parallel} and a fully parallelized version of our algorithm will be forthcoming in OSCAR.

math.AG↗

arXiv2023

Multi-graded Macaulay Dual Spaces

We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.

math.AC↗

arXiv2022

A Fano compactification of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety

We show that a certain compactification $\mathfrak{X}_g$ of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety $\mathcal{X}(F_g, \mathrm{SL}_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by the second author, and separately by Biswas, Lawton, and Ramras. Part of the proof of this result involves the construction of a large family of integral reflexive polytopes.

math.AG↗

arXiv2021

The well-poised property and torus quotients

An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,β}$, are well-poised under Alexeev's embedding. Conversely, given an affine $T$-variety $X$ with polyhedral divisor $\mathfrak{D}$ on a well-poised base $Y$, we construct an embedding of $X \subseteq \mathbb{A}^N$ and provide conditions on $Y$ and $\mathfrak{D}$ which if met, imply $X$ is well-poised under this embedding. Then we show that any affine arrangement variety meets the specified criteria, generalizing results of Ilten and the second author for rational complexity 1 varieties. Using this result, we explicitly compute many Newton-Okounkov cones of $X$ and provide a criterion for the associated toric degenerations to be normal. Our final application combines these two results to show that hypertoric varieties have well-poised embeddings.

math.AG↗

arXiv2021

Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman Model

Phylogenetic networks can model more complicated evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models that are used to model evolution on trees can also be extended to networks and similar questions, such as the identifiability of the network parameter or the invariants of the model, can be asked. In this paper we focus on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks. We do this by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We then determine all quadratic invariants in the sunlet network ideal which we conjecture generate the full ideal.

q-bio.PE↗

arXiv2021

Machine-Assisted Script Curation

We describe Machine-Aided Script Curator (MASC), a system for human-machine collaborative script authoring. Scripts produced with MASC include (1) English descriptions of sub-events that comprise a larger, complex event; (2) event types for each of those events; (3) a record of entities expected to participate in multiple sub-events; and (4) temporal sequencing between the sub-events. MASC automates portions of the script creation process with suggestions for event types, links to Wikidata, and sub-events that may have been forgotten. We illustrate how these automations are useful to the script writer with a few case-study scripts.

cs.CL↗

arXiv2020

Generalized Cut Polytopes for Binary Hierarchical Models

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the existence of maximum likelihood estimates or the normality of the associated semigroup. Cut polytopes of graphs have been useful in analyzing binary marginal polytopes in the case where the simplicial complex underlying the hierarchical model is a graph. We introduce a generalized cut polytope that is isomorphic to the binary marginal polytope of an arbitrary simplicial complex via a generalized covariance map. This polytope is full dimensional in its ambient space and has a natural switching operation among its facets that can be used to deduce symmetries between the facets of the correlation and binary marginal polytopes. We find complete H-representations of the generalized cut polytope for some important families of simplicial complexes. We also compute the volume of these polytopes in some instances.

math.CO↗

arXiv2019

Developing Computational Models of Social Assistance to Guide Socially Assistive Robots

While there are many examples in which robots provide social assistance, a lack of theory on how the robots should decide how to assist impedes progress in realizing these technologies. To address this deficiency, we propose a pair of computational models to guide a robot as it provides social assistance. The model of social autonomy helps a robot select an appropriate assistance that will help with the task at hand while also maintaining the autonomy of the person being assisted. The model of social alliance describes how a to determine whether the robot and the person being assisted are cooperatively working towards the same goal. Each of these models are rooted in social reasoning between people, and we describe here our ongoing work to adapt this social reasoning to human-robot interactions.

cs.HC↗