The Fundamental Theorem on Symmetric Polynomials: History's First Whiff of Galois Theory

We describe the Fundamental Theorem on Symmetric Polynomials (FTSP), exposit a classical proof, and offer a novel proof that arose out of an informal course on group theory. The paper develops this proof in tandem with the pedagogical context that led to it. We also discuss the role of the FTSP both as a lemma in the original historical development of Galois theory and as an early example of the connection between symmetry and expressibility that is described by the theory.Comment: 15 pages, 1 figure. Corrected a misattributio

Machine learning and invariant theory

Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group $G$, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that $G$ is a compact Lie group

Estimation under group actions: recovering orbits from invariants

Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group. In particular, we determine that for cryo-EM with noise variance $\sigma^2$ and uniform viewing directions, the number of samples required scales as $\sigma^6$. We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.Comment: 54 pages. This version contains a number of new result

Degree bounds for fields of rational invariants of Z/pZ\mathbb{Z}/p\mathbb{Z} and other finite groups

Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite group, focusing on abelian groups and especially the case of $\mathbb{Z}/p\mathbb{Z}$. The inquiry is motivated by an application to signal processing. We give new lower and upper bounds depending on the number of distinct nontrivial characters in the representation. We obtain additional detailed information in the case of two distinct nontrivial characters. We conjecture a sharper upper bound in the $\mathbb{Z}/p\mathbb{Z}$ case, and pose questions for further investigation.Comment: 39 pages, 1 tabl

Dimensionless machine learning: Imposing exact units equivariance

Units equivariance is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we employ dimensional analysis and ideas from equivariant machine learning to provide a two stage learning procedure for units-equivariant machine learning. For a given learning task, we first construct a dimensionless version of its inputs using classic results from dimensional analysis, and then perform inference in the dimensionless space. Our approach can be used to impose units equivariance across a broad range of machine learning methods which are equivariant to rotations and other groups. We discuss the in-sample and out-of-sample prediction accuracy gains one can obtain in contexts like symbolic regression and emulation, where symmetry is important. We illustrate our approach with simple numerical examples involving dynamical systems in physics and ecology

Massive stars in the giant molecular cloud G23.3−0.3 and W41

Context. Young massive stars and stellar clusters continuously form in the Galactic disk, generating new Hii regions within their natal giant molecular clouds and subsequently enriching the interstellar medium via their winds and supernovae.Aims. Massive stars are among the brightest infrared stars in such regions; their identification permits the characterisation of the star formation history of the associated cloud as well as constraining the location of stellar aggregates and hence their occurrence as a function of global environment.Methods. We present a stellar spectroscopic survey in the direction of the giant molecular cloud G23.3−0.3. This complex is located at a distance of ~4–5 kpc, and consists of several Hii regions and supernova remnants.Results. We discovered 11 OfK+ stars, one candidate luminous blue variable, several OB stars, and candidate red supergiants. Stars with K-band extinction from ~1.3–1.9 mag appear to be associated with the GMC G23.3−0.3; O and B-types satisfying this criterion have spectrophotometric distances consistent with that of the giant molecular cloud. Combining near-IR spectroscopic and photometric data allowed us to characterize the multiple sites of star formation within it. The O-type stars have masses from ~25–45 M⊙, and ages of 5–8 Myr. Two new red supergiants were detected with interstellar extinction typical of the cloud; along with the two RSGs within the cluster GLIMPSE9, they trace an older burst with an age of 20–30 Myr. Massive stars were also detected in the core of three supernova remnants – W41, G22.7−0.2, and G22.7583−0.4917.Conclusions. A large population of massive stars appears associated with the GMC G23.3−0.3, with the properties inferred for them indicative of an extended history of stars formation