Showing papers on "Equivariant map published in 2016"
Proceedings Article•
19 Jun 2016TL;DR: Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries and achieves state of the art results on CI- FAR10 and rotated MNIST.
Abstract: We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CI- FAR10 and rotated MNIST.
1,321 citations
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TL;DR: Group equivariant convolutional neural networks (G-CNNs) as mentioned in this paper use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers.
Abstract: We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CIFAR10 and rotated MNIST.
239 citations
Posted Content•
TL;DR: In this article, the authors introduce four operations which can be inserted into neural network models as layers to make these models partially equivariant to rotations, and evaluate the effect of these architectural modifications on three datasets which exhibit rotational symmetry.
Abstract: Many classes of images exhibit rotational symmetry. Convolutional neural networks are sometimes trained using data augmentation to exploit this, but they are still required to learn the rotation equivariance properties from the data. Encoding these properties into the network architecture, as we are already used to doing for translation equivariance by using convolutional layers, could result in a more efficient use of the parameter budget by relieving the model from learning them. We introduce four operations which can be inserted into neural network models as layers, and which can be combined to make these models partially equivariant to rotations. They also enable parameter sharing across different orientations. We evaluate the effect of these architectural modifications on three datasets which exhibit rotational symmetry and demonstrate improved performance with smaller models.
183 citations
TL;DR: In this article, it was shown that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kahler-Einstein metric.
Abstract: We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kahler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kahler–Einstein manifolds. We also give analogous results for twisted Kahler–Einstein metrics and Kahler–Ricci solitons.
129 citations
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TL;DR: In this paper, stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties were constructed, which gave an elliptic generalization of the results of arXiv:1211.1287.
Abstract: We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of $q$-difference equations arising the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik-Zamolodchikov equations.
129 citations
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TL;DR: In this paper, an analog of the quantum dynamical Weyl group acting in its equivariant K-theory was constructed for an arbitrary Nakajima quiver variety.
Abstract: For an arbitrary Nakajima quiver variety $X$, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in $Pic(X)\otimes {\mathbb{C}}$. We identify the lattice part of this groupoid with the operators of quantum difference equation for $X$. The cases of quivers of finite and affine type are illustrated by explicit examples.
110 citations
TL;DR: In this article, the McKay conjecture for characters of odd degree was proved for groups of Lie type and primes, such that a Sylow subgroup or maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction.
Abstract: We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and primes ‘ such that a Sylow ‘-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.
93 citations
01 May 2016
TL;DR: In this article, the authors discuss a conjectural correspondence between the enumerative geometry of curves in Calabi-Yau 5-folds Z and 1-dimensional sheaves on 3folds X that are embedded in Z as fixed points of certain C×-actions.
Abstract: Our goal in this paper is to discuss a conjectural correspondence between the enumerative geometry of curves in Calabi–Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×-actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson–Thomas theory, one sums over all Euler characteristics with a weight (−q)χ, where q is a parameter, informally referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter q becomes in dimension 5 the equivariant parameter for the C×-action that defines X inside Z. The 5-dimensional theory effectively sums up the q-expansion in the Donaldson–Thomas theory. In particular, it gives a natural explanation of the rationality (in q) of the DT partition functions. Other expected as well as unexpected symmetries of the DT counts follow naturally from the 5-dimensional perspective. These involve choosing different C×-actions on the same Z, and thus relating the same 5-dimensional theory to different DT problems. The important special case Z = X×C2 is considered in detail in Sections 7 and 8. If X is a toric Calabi–Yau 3-fold, we compute the theory in terms of a certain index vertex. We show that the refined vertex found combinatorially by Iqbal, Kozcaz, and Vafa is a special case of the index vertex. 1. A brief introduction 1.1 Overview Our goal in this paper is to discuss a conjectural correspondence between the enumerative geometry of curves in Calabi–Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×-actions. In both cases, the enumerative information is taken in equivariantK-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson–Thomas theory, one sums over all Euler characteristics with weight (−q)χ, where q is a parameter. (Note the difference with the traditional weighing by qχ as in [MNOP06]. The change of sign of q fits much better with all correspondences.) Informally, q is referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter q becomes in dimension 5 the equivariant parameter for the C×-action Received 15 April 2014, accepted in final form 9 October 2015. 2010 Mathematics Subject Classification 14N35.
84 citations
TL;DR: In this paper, it was shown that the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined.
Abstract: We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, as well as a rigorous definition of cylindrical/linearized contact homology based on an $S^1$-equivariant construction.
82 citations
TL;DR: In this article, the authors consider a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation and derive a generic normal form for a system of coupled phase oscillators with S N symmetry.
78 citations
TL;DR: The Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network architecture encoding rotation equivariance, invariance and covariance, is proposed and a modified convolution operator relying on this representation to obtain deep architectures is developed.
Abstract: In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated as any other variation, the complexity of the problem is decreased, leading to a reduction in the size of the required model. In this paper, we propose the Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network (CNN) architecture encoding rotation equivariance, invariance and covariance. Each convolutional filter is applied at multiple orientations and returns a vector field representing magnitude and angle of the highest scoring orientation at every spatial location. We develop a modified convolution operator relying on this representation to obtain deep architectures. We test RotEqNet on several problems requiring different responses with respect to the inputs' rotation: image classification, biomedical image segmentation, orientation estimation and patch matching. In all cases, we show that RotEqNet offers extremely compact models in terms of number of parameters and provides results in line to those of networks orders of magnitude larger.
TL;DR: In this paper, the authors give a self-contained account of the Fock-Goncharov and Fenchel-Nielsen coordinate system on moduli spaces of flat SL(K)-connections on surfaces.
Abstract: It is known that spectral networks naturally induce certain coordinate systems on moduli spaces of flat SL(K)-connections on surfaces, previously studied by Fock and Goncharov. We give a self-contained account of this story in the case K = 2 and explain how it can be extended to incorporate the complexified Fenchel–Nielsen coordinates. As we review, the key ingredient in the story is a procedure for passing between moduli of flat SL(2)-connections on C (equipped with a little extra structure) and moduli of equivariant GL(1)-connections over a covering \({\Sigma \to C}\); taking holonomies of the equivariant GL(1)-connections then gives the desired coordinate systems on moduli of SL(2)-connections. There are two special types of spectral network, related to ideal triangulations and pants decompositions of C; these two types of network lead to Fock–Goncharov and complexified Fenchel–Nielsen coordinate systems, respectively.
TL;DR: In this paper, the authors construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A).
Abstract: We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.
TL;DR: In this paper, a contour integral formula for the exact partition function of 2 supersymmetric U(N) gauge theories on compact toric four-manifolds was given.
Abstract: We provide a contour integral formula for the exact partition function of $$ \mathcal{N} $$
= 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) $$ \mathcal{N} $$
= 2∗ theory on $$ {\mathrm{\mathbb{P}}}^2 $$
for all instanton numbers. In the zero mass case, corresponding to the $$ \mathcal{N} $$
= 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.
TL;DR: In this paper, a very short proof of the Bloch-Okounkov theorem on the quasimodularity of certain functions defined by sums over partitions is given, and the authors also show how to make their map (i.e.
Abstract: We give a very short proof of the Bloch–Okounkov theorem on the quasimodularity of certain functions defined by sums over partitions, and also show how to make their map \(\mathfrak {s}\mathfrak {l}_2\)-equivariant.
TL;DR: This article develops methodology to obtain low-rank model-based representations of continuous, discrete and ordinal data arrays using a semiparametric transformation model, and shows how orthogonally equivariant parameter estimates can be obtained from Bayesian procedures under invariant prior distributions.
Abstract: Analyses of array-valued datasets often involve reduced-rank array approximations, typically obtained via least-squares or truncations of array decompositions However, least-squares approximations tend to be noisy in high-dimensional settings, and may not be appropriate for arrays that include discrete or ordinal measurements This article develops methodology to obtain low-rank model-based representations of continuous, discrete and ordinal data arrays The model is based on a parameterization of the mean array as a multilinear product of a reduced-rank core array and a set of index-specific orthogonal eigenvector matrices It is shown how orthogonally equivariant parameter estimates can be obtained from Bayesian procedures under invariant prior distributions Additionally, priors on the core array are developed that act as regularizers, leading to improved inference over the standard least-squares estimator, and providing robustness to misspecification of the array rank This model-based approach is extended to accommodate discrete or ordinal data arrays using a semiparametric transformation model The resulting low-rank representation is scale-free, in the sense that it is invariant to monotonic transformations of the data array In an example analysis of a multivariate discrete network dataset, this scale-free approach provides a more complete description of data patterns
TL;DR: In this article, the authors introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category, and show that the categories of topological presheaves indexed by the orbit category of a fixed topological group and the category of $G$-spaces can be endowed with Quillens equivalent model category structures.
Abstract: We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group $G$, under certain conditions on the fixed point functors of the subgroups of $G$. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theorem a la Kropholler and Wall for the normalized chain complexes of simplicial $G$-sets.
TL;DR: Fan, Jarvis and Ruan as mentioned in this paper gave a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G.
Abstract: We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.
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TL;DR: In this paper, a variational method was used to construct a free boundary minimal surface in the unit 3-ball of the cube, tetrahedron and dodecahedron.
Abstract: For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $\Sigma_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $\Sigma_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $\Sigma_g$ converge as varifolds to the union of the disk and critical catenoid. These examples are the first with genus greater than $1$ and were conjectured to exist by Fraser-Schoen. We also construct several new free boundary minimal surfaces in $B$ with the symmetry groups of the cube, tetrahedron and dodecahedron. Finally, we prove that free boundary minimal surfaces isotopic to those of Fraser-Schoen can be constructed variationally using an equivariant min-max procedure. We also prove an $\epsilon$-regularity theorem for free boundary minimal surfaces in $B$.
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TL;DR: In this article, the authors show that for a finite quiver Q, the compactly supported cohomology of the stack of modules of Pi_Q-modules is pure.
Abstract: Let Q be a finite quiver. Using their analogues from noncommutative Donaldson-Thomas theory, we prove cohomological integrality and wall crossing theorems for moduli stacks of objects in Serre subcategories of the category of modules for the preprojective algebra Pi_Q. Via similar techniques, we show that for a finite quiver Q, the compactly supported cohomology of the stack of Pi_Q-modules is pure, and we also give an explicit description of the compactly supported cohomology of the genus one character stack, and the vanishing cycle cohomology of Hilb_n(C^3). We recover as a special case of the wall crossing result a categorification of Hausel's formula for the Hodge polynomials of Nakajima quiver varieties, and as a particular corollary of the integrality result the positivity conjecture for all of the variants of the Kac polynomials introduced in the work of Bozec, Schiffmann and Vasserot. Finally, using the purity result, we prove a degeneration result for Kontsevich-Soibelman cohomological Hall algebras with extra equivariant parameters, and as a corollary we prove the torsion freeness conjecture arising in the work of Schiffmann and Vasserot on the AGT conjectures, proving that the (fully equivariant) cohomological Hall algebra of Pi_Q-modules naturally embeds as a subalgebra of a shuffle algebra.
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TL;DR: In this paper it was shown that Coulomb branches of quiver gauge theories of affine type $A$ are Cherkis bow varieties, which have been introduced as ADHM type description of moduli space of instantons on the Taub-NUT space equivariant under a cyclic group action.
Abstract: We show that Coulomb branches of quiver gauge theories of affine type $A$ are Cherkis bow varieties, which have been introduced as ADHM type description of moduli space of instantons on the Taub-NUT space equivariant under a cyclic group action.
TL;DR: In this article, the authors studied proper and proper invariant biconservative hypersurfaces into the Euclidean space and showed that there exists no proper biharmonic immersion for these invariant families.
Abstract: In this paper, using the framework of equivariant differential geometry, we study proper $$\hbox {SO}(p+1) \times \hbox {SO}(q+1)$$
-invariant biconservative hypersurfaces into the Euclidean space $${\mathbb {R}}^n$$
(
$$n=p+q+2$$
) and proper $$\hbox {SO}(p+1)$$
-invariant biconservative hypersurfaces into the Euclidean space $${\mathbb {R}}^n$$
(
$$n=p+2$$
). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.
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TL;DR: In this article, the authors define an equivariant generalization of a symmetric monoidal category, called a $G$-symmetric monoid category, which is a natural extension of ordinary commutative monoids to this new context.
Abstract: Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also symmetric monoidal powers indexed by arbitrary finite $G$-sets. We then define $G$-commutative monoids to be the natural extension of ordinary commutative monoids to this new context. Using this machinery, we then describe when Bousfield localization in equivariant spectra preserves certain operadic algebra structures, and we explore the consequences of our definitions for categories of modules over a $G$-commutative monoid.
01 Jan 2016
TL;DR: A survey of equivariant stable homotopy theory in the context of orthogonal Gspectra can be found in this paper, where the main reference is the AMS memoir by Mandell and May; the appendices of the paper [10] by Hill, Hopkins and Ravenel contain further material, in particular on the norm construction.
Abstract: We review some foundations for equivariant stable homotopy theory in the context of orthogonal Gspectra. The main reference for this theory is the AMS memoir [17] by Mandell and May; the appendices of the paper [10] by Hill, Hopkins and Ravenel contain further material, in particular on the norm construction. At many places, however, our exposition is substantially different from these two sources, compare Remark 2.7. We do not develop model category aspects of the theory; the relevant references here are again Mandell-May [17], Hill-Hopkins-Ravenel [10] and the thesis of Stolz [25]. For a general, framework independent, introduction to equivariant stable homotopy theory, one may consult the survey articles by Adams [1] and Greenlees-May [9]. We restrict our attention to finite groups (as opposed to compact Lie groups) throughout, which allows to simplify the treatment at various points. Also, we implicitly only deal with the ‘complete universe’ (which can be seen from the fact that we stabilize with respect to multiples of the regular representation). These notes were originally assembled on the occasion of a series of lectures at the Universitat Autonoma de Barcelona in October, 2010, and then subsequently expanded. They are still incomplete and certainly contain typos, but hopefully not too many mathematical errors. At some places, proper credit is also still missing, and will be added later. This survey paper makes no claim to originality. If there is anything new it may be the particular model for the real bordism spectrum MR as a commutative equivariant orthogonal ring spectrum in Example 2.14. Before we start, let us fix some notation and conventions. By a ‘space’ we mean a compactly generated space in the sense of McCord [18], i.e., a k-space (also called a Kelley space) that is also weakly Hausdorff. For a finite dimensional R-vector space V we denote by S the one-point compactification; we consider S as a based space with basepoint at infinity. If V is endowed with a scalar product, we denote by D(V ) the
TL;DR: In this article, the authors study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient and describe how they result from mutations of semiorthogonal decompositions.
TL;DR: In this article, the authors considered the initial value problem for the m-equivariant Chern-Simons-Schr\\\"odinger model in two spatial dimensions with real-valued coupling parameter g. This is a covariant NLS type problem that is L 2-critical.
Abstract: In this article we consider the initial value problem for the m-equivariant Chern-Simons-Schr\\\"odinger model in two spatial dimensions with real-valued coupling parameter g. This is a covariant NLS type problem that is L^2-critical. We prove that at the critical regularity, for any integer-valued equivariance index m, the initial value problem in the defocusing case (g = 1, and in this case we prove that for nonnegative integer-valued equivariance indices m there exist constants c = c_{m, g} such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the L^2 initial data phi_0 is m-equivariant and has L^2-norm less than the square root of c_{m, g}. We also show that c_{m, g}^{1/2} is equal to the minimum L^2 norm of a nontrivial m-equivariant standing wave solution. In the self-dual g = 1 case, we have the exact numerical values c_{m, 1} = 8*pi*(m + 1).
Book•
18 Aug 2016
TL;DR: Physical Motivation Geometrical Description Energy-Type Estimates Analytical Tools General Well-Posedness Theory The Equivariant Problem Singularity Formation as discussed by the authors, which is a special case of the singularity formation problem.
Abstract: Physical Motivation Geometrical Description Energy-Type Estimates Analytical Tools General Well-Posedness Theory The Equivariant Problem Singularity Formation
TL;DR: In this paper, the authors give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences.
Abstract: Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsvath-Szabo, Lipshitz-Treumann, Szabo, Sarkar-Seed-Szabo, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith-type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar-Seed-Szabo's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.
TL;DR: In this paper, the equivariant motivic spectrum representing the homotopy K-theory of G-schemes is constructed as an E-infinity ring.
Abstract: We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
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TL;DR: In this article, an equivariant min-max theory was developed and it can produce many of the known minimal surfaces in the genus and symmetry group. But it cannot produce all of the infinite families of minimal surfaces proposed by Pitts-Rubinstein.
Abstract: We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite families of minimal surfaces in $\mathbb{S}^3$ proposed by Pitts-Rubinstein. These examples are doublings and desingularizations of stationary integral varifolds in $\mathbb{S}^3$.