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1. An enriched count of nodal orbits in an invariant pencil of conics

   Bethea, Candace (October 2023, arXiv pre-print repository)

   This work gives an equivariantly enriched count of nodal orbits in a general pencil of plane conics that is invariant under a linear action of a finite group on CP2. This can be thought of as spearheading equivariant enumerative enrichments valued in the Burnside Ring, both inspired by and a departure from R(G)-valued enrichments such as Roberts’ equivariant Milnor number and Damon’s equivariant signature formula. Given a G-invariant general pencil of conics, the weighted sum of nodal orbits in the pencil is a formula in terms of the base locus considered as a G-set. We show this is true for all finite groups except Z/2 × Z/2 and D8 and give counterexamples for the two exceptional groups. 

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2. A General Framework for Robust G-Invariance in G-Equivariant Networks

   Sanborn, Sophia; Miolane, Nina (December 2023, NeurIPS 2023)

   We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks (G-CNNs), which we call the G-triple-correlation (G-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also complete. Many commonly used invariant maps--such as the max--are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the G-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max G-Pooling in G-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for G-CNNs defined on both commutative and non-commutative groups--SO(2), O(2), SO(3), and O(3) (discretized as the cyclic C8, dihedral D16, chiral octahedral O and full octahedral Oh groups)--acting on ℝ2 and ℝ3 on both G-MNIST and G-ModelNet10 datasets. 

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3. A Functional Approach to Rotation Equivariant Non-Linearities for Tensor Field Networks

   Poulenard, Adrien; Guibas, Leonidas (January 2021, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition)

   null (Ed.)

   Learning pose invariant representation is a fundamental problem in shape analysis. Most existing deep learning algorithms for 3D shape analysis are not robust to rotations and are often trained on synthetic datasets consisting of pre-aligned shapes, yielding poor generalization to unseen poses. This observation motivates a growing interest in rotation invariant and equivariant methods. The field of rotation equivariant deep learning is developing in recent years thanks to a well established theory of Lie group representations and convolutions. A fundamental problem in equivariant deep learning is to design activation functions which are both informative and preserve equivariance. The recently introduced Tensor Field Network (TFN) framework provides a rotation equivariant network design for point cloud analysis. TFN features undergo a rotation in feature space given a rotation of the input pointcloud. TFN and similar designs consider nonlinearities which operate only over rotation invariant features such as the norm of equivariant features to preserve equivariance, making them unable to capture the directional information. In a recent work entitled "Gauge Equivariant Mesh CNNs: Anisotropic Convolutions on Geometric Graphs" Hann et al. interpret 2D rotation equivariant features as Fourier coefficients of functions on the circle. In this work we transpose the idea of Hann et al. to 3D by interpreting TFN features as spherical harmonics coefficients of functions on the sphere. We introduce a new equivariant nonlinearity and pooling for TFN. We show improvments over the original TFN design and other equivariant nonlinearities in classification and segmentation tasks. Furthermore our method is competitive with state of the art rotation invariant methods in some instances. 

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4. The equivariant degree and an enriched count of rational cubics

   Bethea, Candace; Wickelgren, Kirsten (February 2025, arXiv pre-print repository)

   NA (Ed.)

   We define the equivariant degree and local degree of a proper G-equivariant map between smooth G-manifolds when G is a compact Lie group and prove a local to global result. We show the local degree can be used to compute the equivariant Euler characteristic of a smooth, compact G-manifold and the Euler number of a relatively oriented G-equivariant vector bundle when G is finite. As an application, we give an equivariantly enriched count of rational plane cubics through a G-invariant set of 8 general points in ℂℙ2, valued in the representation ring and Burnside ring of a finite group. When ℤ/2 acts by pointwise complex conjugation this recovers a signed count of real rational cubics. 

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5. SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION

   https://doi.org/10.1093/qmathj/haz034  

   Guillou, Bertrand J.; May, J. Peter; Merling, Mona; Osorno, Angélica M. (December 2019, The Quarterly Journal of Mathematics)

   Abstract We give an operadic definition of a genuine symmetric monoidal $$G$$-category, and we prove that its classifying space is a genuine $$E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $$G$$-categories to genuine permutative $$G$$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $$G$$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $$G$$-categories as input to an equivariant infinite loop space machine that gives genuine $$\Omega $$-$$G$-spectra as output. 

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