Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Semisimple Frobenius structures at higher genus

Abstract: We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product. The conjecture is supported by the corresponding theorem about equivariant GW-invariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.

Quantum cohomology of flag manifolds and Toda lattices

Abstract: We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.

Exploiting Cyclic Symmetry in Convolutional Neural Networks

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.

Fixed point free involutions and equivariant maps

Abstract: was defined to be the least integer n for which there is an equivariant map X -+s n. We abbreviate this invariant to co-ind X. In this terminology the classical Borsuk theorem states that co-ind Sn = n. There are also numerous results (for references, see [2]) which among other things relate co-index to the homology of the quotient space X/T. The main purpose of the present note is the computation of the coindex in several examples in which homotopy, rather than homology, considerations are of primary importance. It should be mentioned that A. S. Svarc has also recently studied the application of homotopy theory to equivariant maps [5]; there is a considerable overlap between his work and our previous paper [2]. We consider as in our previous paper the space p(Sn) of paths on Sn which join a given point x to its antipode A(x) = -x together with the natural involution of p(Sn). It is shown that co-ind P(Sn) = n for n : 1, 2, 4 or 8. Next we consider the space V(Sn) of unit tangent vectors to sn, with its involution (the antipodal map on each fibre), and show that co-ind V(Sn) = n for n : 1, 3, or 7 and co-ind V(Sn) = n - 1 for n = 1, 3 or 7. We also compute the co-index of involutions on low dimensional projective spaces. The arguments rely on suspension and Hopf invariant theorems, using particularly the results of J. F. Adams [1] on maps of Hopf invariant one. 2. The space of paths P(Sn). We choose a base point x e Sn and we let P(Sn) denote the space of all paths in Sn which join x to its antipode - x. A fixed point free involution on P(Sn) is given by 17(p)(t) =-p(l - t), where p(t) is a point in P(S"). In this section we show

Topology of closed one-forms

Abstract: The Novikov numbers The Novikov inequalities The universal complex Construction of the universal complex Bott-type inequalities Inequalities with von Neumann Betti numbers Equivariant theory Exactness of the Novikov inequalities Morse theory of harmonic forms Lusternik-Schnirelman theory, closed 1-forms, and dynamics Appendix A. Manifolds with corners Appendix B. Morse-Bott functions on manifolds with corners Appendix C. Morse-Bott inequalities Appendix D. Relative Morse theory Bibliography Index.