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We extend the classical Poincar\'e-Birkhoff-Witt theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is the quotient of the associative operad by a right action of the spectral Lie operad. This statement, in turn, is a consequence of a fundamental relation between different $\mathbb{E}_n$-operads, which we articulate and prove. We deduce a variant of the Poincar\'{e}--Birkhoff--Witt theorem for relative enveloping algebras of $\mathbb{E}_n$-algebras. Our methods also give a simple construction and description of the higher enveloping $\mathbb{E}_n$-algebras of a spectral Lie algebra., Comment: 14 pages

We study deformations of Calabi-Yau varieties in characteristic $p$ using techniques from derived algebraic geometry. We prove a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau varieties in characteristic $0$ are unobstructed), and we show that ordinary Calabi-Yau varieties admit canonical lifts to characteristic $0$, generalising the Serre-Tate theorem on ordinary abelian varieties., Comment: 77 pages

Two of the many trends in neural network research of the past few years have been (i) the learning of dynamical systems, especially with recurrent neural networks such as long short-term memory networks (LSTMs) and (ii) the introduction of transformer neural networks for natural language processing (NLP) tasks. While some work has been performed on the intersection of these two trends, those efforts were largely limited to using the vanilla transformer directly without adjusting its architecture for the setting of a physical system. In this work we develop a transformer-inspired neural network and use it to learn a dynamical system. We (for the first time) change the activation function of the attention layer to imbue the transformer with structure-preserving properties to improve long-term stability. This is shown to be of great advantage when applying the neural network to learning the trajectory of a rigid body., Comment: Will be published as part of "Cemracs Proceedings 2023" (status: accepted)

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively expensive. In order to save computational cost, one can construct surrogate models by expressing the system in a low-dimensional basis, obtained from training data. This is referred to as model reduction. Past investigations have shown that, when performing model reduction of Hamiltonian systems, it is crucial to preserve the symplectic structure associated with the system in order to ensure long-term numerical stability. Up to this point structure-preserving reductions have largely been limited to linear transformations. We propose a new neural network architecture in the spirit of autoencoders, which are established tools for dimension reduction and feature extraction in data science, to obtain more general mappings. In order to train the network, a non-standard gradient descent approach is applied that leverages the differential-geometric structure emerging from the network design. The new architecture is shown to significantly outperform existing designs in accuracy.

One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is widely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold, the Grassmann manifold and the symplectic Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This global tangent space representation is used to perform all of the steps in the Adam optimizer and we are able to fully generalize the optimizer to manifolds without a projection step. The resulting algorithm is then applied to train a transformer for which orthogonality constraints are enforced up to machine precision and we observe significant speed-ups in the training process., Comment: 19 pages, 4 figures, was presented at Enumath2023

Individualized treatment decisions can improve health outcomes, but using data to make these decisions in a reliable, precise, and generalizable way is challenging with a single dataset. Leveraging multiple randomized controlled trials allows for the combination of datasets with unconfounded treatment assignment to better estimate heterogeneous treatment effects. This paper discusses several non-parametric approaches for estimating heterogeneous treatment effects using data from multiple trials. We extend single-study methods to a scenario with multiple trials and explore their performance through a simulation study, with data generation scenarios that have differing levels of cross-trial heterogeneity. The simulations demonstrate that methods that directly allow for heterogeneity of the treatment effect across trials perform better than methods that do not, and that the choice of single-study method matters based on the functional form of the treatment effect. Finally, we discuss which methods perform well in each setting and then apply them to four randomized controlled trials to examine effect heterogeneity of treatments for major depressive disorder.

Estimating treatment effects conditional on observed covariates can improve the ability to tailor treatments to particular individuals. Doing so effectively requires dealing with potential confounding, and also enough data to adequately estimate effect moderation. A recent influx of work has looked into estimating treatment effect heterogeneity using data from multiple randomized controlled trials and/or observational datasets. With many new methods available for assessing treatment effect heterogeneity using multiple studies, it is important to understand which methods are best used in which setting, how the methods compare to one another, and what needs to be done to continue progress in this field. This paper reviews these methods broken down by data setting: aggregate-level data, federated learning, and individual participant-level data. We define the conditional average treatment effect and discuss differences between parametric and nonparametric estimators, and we list key assumptions, both those that are required within a single study and those that are necessary for data combination. After describing existing approaches, we compare and contrast them and reveal open areas for future research. This review demonstrates that there are many possible approaches for estimating treatment effect heterogeneity through the combination of datasets, but that there is substantial work to be done to compare these methods through case studies and simulations, extend them to different settings, and refine them to account for various challenges present in real data.

At the end of October 2022, Elon Musk concluded his acquisition of Twitter. In the weeks and months before that, several questions were publicly discussed that were not only of interest to the platform's future buyers, but also of high relevance to the Computational Social Science research community. For example, how many active users does the platform have? What percentage of accounts on the site are bots? And, what are the dominating topics and sub-topical spheres on the platform? In a globally coordinated effort of 80 scholars to shed light on these questions, and to offer a dataset that will equip other researchers to do the same, we have collected all 375 million tweets published within a 24-hour time period starting on September 21, 2022. To the best of our knowledge, this is the first complete 24-hour Twitter dataset that is available for the research community. With it, the present work aims to accomplish two goals. First, we seek to answer the aforementioned questions and provide descriptive metrics about Twitter that can serve as references for other researchers. Second, we create a baseline dataset for future research that can be used to study the potential impact of the platform's ownership change.

Infinitesimal deformations are governed by partition Lie algebras. In characteristic $0$, these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic $p$, they are more subtle. We give explicit models for partition Lie algebras over general coherent rings, both in the setting of spectral and derived algebraic geometry. For the spectral case, we refine operadic Koszul duality to a functor from operads to divided power operads, by taking refined linear duals of $\Sigma_n$-representations. The derived case requires a further refinement of Koszul duality to a more genuine setting., Comment: To appear in Memoirs of the AMS. 106 pages, 1 figure

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$)., Comment: 36 pages. Improved presentation

We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $E$-theory of the weight $p$ summands of iterated loop spaces of spheres (parametrising the weight $p$ operations on $\mathbb{E}_n$-algebras), as well as the $E$-theory of the configuration spaces of $p$ points on a punctured surface. We read off the corresponding Morava $K$-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the $\mathbb{F}_p$-homology of the space of unordered configurations of $p$ particles on a punctured surface., Comment: Final version to appear in the Journal of Topology. 61 pages, 1 figure

A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base., Comment: To appear in Acta Mathematica. 90 pages

We determine the set of catenary degrees, the set of distances, and the unions of sets of lengths of the monoid of nonzero ideals and of the monoid of invertible ideals of orders in quadratic number fields.

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_n|$, which is the $\Sigma_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|\Pi_n|^G$ in terms of subgroup posets for general $G\subset \Sigma_n$ and prove a formula for the restriction of $|\Pi_n|$ to Young subgroups $\Sigma_{n_1}\times \dots\times \Sigma_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover surprising links between strict Young quotients of $|\Pi_n|$, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_n|^\diamond\wedge_{\Sigma_n} (S^\ell)^{\wedge n}$ and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of $|\Pi_n|$ in terms of "atoms" $|\Pi_d|^\diamond\wedge_{\Sigma_d} (S^\ell)^{\wedge d}$ for $\ell$ odd and compute their homology. We thereby also generalise Goerss' computation of the algebraic Andr\'e-Quillen homology of trivial square-zero extensions from $\mathbb{F}_2$ to $\mathbb{F}_p$ for $p$ an odd prime., Comment: Final version to appear in the Publications Math\'ematiques de l'IH\'ES. 79 pages, 15 figures

We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some consequences for $v_n$-periodic homotopy groups: whereas the Goodwillie tower is finite and converges in periodic homotopy when evaluated on spheres (Arone-Mahowald), we show that neither of these statements remains true for wedges and Moore spaces., Comment: 17 pages. Journal version