Showing papers on "Equivariant map published in 2017"

Rotation Equivariant Vector Field Networks

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.

K-semistability is equivariant volume minimization

Abstract: This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over Q-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kahler–Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V,−KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all C∗-invariant valuations on the cone associated to any positive Cartier multiple of −KV. In this case, we show that ordV is the unique minimizer among all C∗-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.

Learning Steerable Filters for Rotation Equivariant CNNs

Abstract: In many machine learning tasks it is desirable that a model's prediction transforms in an equivariant way under transformations of its input. Convolutional neural networks (CNNs) implement translational equivariance by construction; for other transformations, however, they are compelled to learn the proper mapping. In this work, we develop Steerable Filter CNNs (SFCNNs) which achieve joint equivariance under translations and rotations by design. The proposed architecture employs steerable filters to efficiently compute orientation dependent responses for many orientations without suffering interpolation artifacts from filter rotation. We utilize group convolutions which guarantee an equivariant mapping. In addition, we generalize He's weight initialization scheme to filters which are defined as a linear combination of a system of atomic filters. Numerical experiments show a substantial enhancement of the sample complexity with a growing number of sampled filter orientations and confirm that the network generalizes learned patterns over orientations. The proposed approach achieves state-of-the-art on the rotated MNIST benchmark and on the ISBI 2012 2D EM segmentation challenge.

Topological crystalline materials: General formulation, module structure, and wallpaper groups

Abstract: Topological crystalline materials are emergent topological phases due to crystalline space group symmetry. They are either gapful or gapless in the bulk, while hosting topological states at the boundary. Here, the authors define topological crystalline materials rigorously on the basis of a mathematical theory, known as twisted equivariant K-theory. Abstract mathematical ideas, such as the Mayer-Vietoris sequence and module structure, are explained in terms of band theory. The formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states as well as to bulk gapless topological materials, such as Weyl and Dirac semimetals or nodal superconductors. The authors present a complete classification of topological crystalline surface states and band insulators protected by 17 wallpaper groups in the absence of time-reversal invariance, which may support topological states beyond simple Dirac fermions.

Equivariance through parameter-sharing

Abstract: We propose to study equivariance in deep neural networks through parameter symmetries. In particular, given a group G that acts discretely on the input and output of a standard neural network layer ϕW : ℝM → ℝN, we show that ϕW is equivariant with respect to G-action iff G explains the symmetries of the network parameters W. Inspired by this observation, we then propose two parameter-sharing schemes to induce the desirable symmetry on W. Our procedure for tying the parameters achieves G-equivariance and, under some conditions on the action of G, it guarantees sensitivity to all other permutation groups outside G.

Controlled Topological Phases and Bulk-edge Correspondence

Abstract: In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev’s periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane–Mele $${\mathbb{Z}_2}$$ -invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse Mayer-Vietoris exact sequence. As a new example, we study the index of reflection-invariant systems.

Equivariant Verlinde Formula from Fivebranes and Vortices

Abstract: We study complex Chern–Simons theory on a Seifert manifold M_3 by embedding it into string theory. We show that complex Chern–Simons theory on M_3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on Σ×S^1 and (4) index of a spin^c Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on Σ×S^1 and (4) the equivariant index of a spin^c Dirac operator on the moduli space of Higgs bundles.

Cyclic surfaces and Hitchin components in rank 2

Abstract: We prove that given a Hitchin representation in a real split rank 2 group G 0 , there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermit-ian bundle over Teichmuller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of G 0. Some partial extensions of the construction hold for cyclic bundles in higher rank.

A class of non-holomorphic modular forms I

Abstract: This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.

Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

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.

Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells

Abstract: Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex projective manifold $X$, the homogenized, torus equivariant CSM class of a constructible function $\varphi$ is the restriction of the characteristic cycle of $\varphi$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize $X$ to be a (generalized) flag manifold $G/B$. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke orthogonality' of CSM classes, determined by the DL operator and its Poincar{\'e} adjoint. We further use the theory of holonomic $\mathcal{D}_X$-modules to show that the characteristic cycle of a Verma module, restricted to the zero section, gives the CSM class of the corresponding Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rim{\'a}nyi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds $G/P$.

A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

Abstract: We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eisenstein series.

Equivariant character correspondences and inductive McKay condition for type A

Abstract: As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.

The spectrum of the equivariant stable homotopy category of a finite group

Abstract: We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

Quantum K-theory of Quiver Varieties and Many-Body Systems

Abstract: We define quantum equivariant K-theory of Nakajima quiver varieties We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice

Equivariant iterated loop space theory and permutative G–categories

Abstract: We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V–fold loop G–spaces to several avatars of a recognition principle for infinite loop G–spaces. We then explain what genuine permutative G–categories are and, more generally, what E∞–G–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine G–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension G–spectra. Other examples are geared towards equivariant algebraic K–theory.

Asymptotic decomposition for semilinear wave and equivariant wave map equations

Abstract: In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times $t_n\to T_+$ (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities.

Hodge numbers for all CICY quotients

Abstract: We present a general method for computing Hodge numbers for Calabi-Yau manifolds realised as discrete quotients of complete intersections in products of projective spaces. The method relies on the computation of equivariant cohomologies and is illustrated for several explicit examples. In this way, we compute the Hodge numbers for all discrete quotients obtained in Braun’s classification [1].

The positive equivariant symplectic homology as an invariant for some contact manifolds

Abstract: We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a "Viterbo functoriality" property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.

Supersymmetric Renyi Entropy and Anomalies in Six-Dimensional (1,0) Superconformal Theories

Abstract: A closed formula of the universal part of supersymmetric Renyi entropy $S_q$ for six-dimensional $(1,0)$ superconformal theories is proposed. Within our arguments, $S_q$ across a spherical entangling surface is a cubic polynomial of $ u=1/q$, with $4$ coefficients expressed as linear combinations of the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies. As an application, we establish linear relations between the $c$-type Weyl anomalies and the 't Hooft anomaly coefficients. We make a conjecture relating the supersymmetric Renyi entropy to an equivariant integral of the anomaly polynomial in even dimensions and check it against known data in four dimensions and six dimensions.

Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

Abstract: Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases — symmetry-protected topological (SPT) phases in particular — defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry G, we bridge their descriptions in terms of MPSs, and those in terms of G-equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and “open” TFTs, which are TFTs defined on spacetimes with boundaries.

Segre classes and Hilbert schemes of points

Abstract: We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of K3 sufaces, we produce relations intertwining the kappa classes and the Noether-Lefschetz loci. Conjectures are proposed.