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We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GL n case, we classify the type (1 , ... , 1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4 n -spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators. [ABSTRACT FROM AUTHOR]

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian G (n , m) are the same as multiplication rules for the basis of Schur polynomials in the ring of symmetric polynomials. In this paper, we explain how to recover this somewhat mysterious connection by using the geometric Satake correspondence to put the structure of a representation of G L m on H • (G (n , m)) and comparing it to the Fock space representation on symmetric polynomials. This proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials. [ABSTRACT FROM AUTHOR]

Peterson varieties are subvarieties of flag varieties and their (equivariant) cohomology rings are given by Fukukawa–Harada–Masuda in type A and soon later the author with Harada and Masuda gives an explicit presentation of the (equivariant) cohomology rings of Peterson varieties for arbitrary Lie types. In this note we study the (equivariant) cohomology ring of the intersections of Peterson variety with Schubert, opposite Schubert, and Richardson varieties in more general. By the work of Goldin–Mihalcea–Singh, the intersections of Peterson variety with Schubert varieties are naturally identified with smaller Peterson varieties, so the problem reduces to the problem for opposite Schubert intersections. In this note we provide a technical statement for (equivariant) cohomology ring of a subvariety with some conditions of Peterson variety. By using the statement, we calculate the (equivariant) cohomology rings for some intersections of Peterson varieties with opposite Schubert varieties in type A. We also explicitly present the (equivariant) cohomology rings for some intersections of Peterson varieties with Richardson varieties in type A. [ABSTRACT FROM AUTHOR]

We introduce the topologically twisted index for four-dimensional N = 1 gauge theories quantized on AdS 2 × S 1 . We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on AdS 2 × T 2 , with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a R-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization, the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the R-charge. [ABSTRACT FROM AUTHOR]

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context. [ABSTRACT FROM AUTHOR]

We introduce equivariant formal deformation theory of associative algebra morphisms. We also present an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms. We discuss some examples of equivariant deformations and use the Maurer-Cartan equation to characterize equivariant deformations. [ABSTRACT FROM AUTHOR]

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold (M , ω) of dimension 2n. Then the S 1 -action has at least n + 1 fixed points. We study the case when the fixed point set consists of precisely n + 1 isolated points. We first show certain equivalence on the first Chern class of M and some particular weight of the S 1 -action at some fixed point. Then we show that the particular weight can completely determine the integral cohomology ring of M, the total Chern class of M, and the sets of weights of the S 1 -action at all the fixed points. We will see that all these data are isomorphic to those of known examples, C P n , or G ~ 2 (R n + 2) with n ≥ 3 odd, equipped with standard circle actions. [ABSTRACT FROM AUTHOR]

Let C 2 denote the cyclic group of order 2. We compute the R O (C 2) -graded cohomology of all C 2 -surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether or not the action preserves the orientation, and one other invariant. [ABSTRACT FROM AUTHOR]

We give a Type B analog of Whitehouse’s lifts of the Eulerian representations from S n to S n + 1 by introducing a family of B n -representations that lift to B n + 1 . As in Type A, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain Z 2 -orbit configuration space of R 3 . We show that the lifted B n + 1 -representations also have a configuration space interpretation, and further parallel the Type A story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring. [ABSTRACT FROM AUTHOR]

We relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex symplectic manifolds that are either monotone or exact, using an equivariant version of the PSS isomorphism. We define continuation maps between different Hamiltonians in ℤ / 2 -equivariant Floer cohomology, and prove expected properties of them. We prove a symplectic Cartan relation for the equivariant pair-of-pants product, pointing out the difficulties in stating it. We give a nonvanishing result for the equivariant pair-of-pants product for some elements of S H ∗ (T ∗ S n). We finish by calculating the symplectic square for the negative line bundles M = Tot ((− 1) → ℂ ℙ m) , proving an equivariant version of a result due to Ritter. [ABSTRACT FROM AUTHOR]

In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie–Yamaguti algebras. Next, we study finite group actions on Lie–Yamaguti algebras and introduce equivariant cohomology for Lie–Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology. [ABSTRACT FROM AUTHOR]

A generalisation of the equivariant Dixmier-Douady invariant is constructed as a second-degree cohomology class within a new semi-equivariant Cech cohomology theory. This invariant obstructs liftings of semi-equivariant principal bundles that are associated to central exact sequences of structure groups in which each structure group is acted on by the equivariance group. The results and methods described can be applied to the study of complex vector bundles equipped with linear/anti-linear actions, such as Atiyah's Real vector bundles. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism. [ABSTRACT FROM AUTHOR]

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold. [ABSTRACT FROM AUTHOR]

A theorem of O. Forster says that if R is a noetherian ring of Krull dimension d, then every projective R-module of rank n can be generated by d+n elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d+n elements. We view rank-n projective R-modules as R-forms of the non-unital R-algebra R^n where the product of any two elements is 0. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an R-form of a k-algebra, where k is an infinite field and R is a finitely generated k-ring of Krull dimension d. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack BG, where G is the automorphism group of the algebra in question, by algebraic spaces of a certain type. [ABSTRACT FROM AUTHOR]

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product J C P ∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for J C P ∞ in both cohomology and K -theory. Here, we study the torus-equivariant cohomology of J C P ∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis. Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety. [ABSTRACT FROM AUTHOR]

We consider Bott-Samelson varieties BS c (s) for a semisimple compact Lie group C corresponding to sequences of (not necessarily simple) reflections s. Let n be the length of s , K be a maximal torus in C and W be the Weyl group of C. For any set R of not overlapping integer pairs (i , j) such that 1 ⩽ i ⩽ j ⩽ n and a function v : R → W , we consider the subspace BS c (s , v) ⊂ BS c (s) of solutions of the equations in C / K requiring that the K -orbit of the product of coordinates counted from i to j be equal to the K -orbit of v evaluated at (i , j) ∈ R. We decompose BS c (s , v) into a twisted product (in the sense of iterated fibre bundles) of smaller Bott-Samelson varieties BS c (t) and the fibres of the canonical projections from BS c (t) to the flag variety. Finally, we prove the tensor product decomposition for the K -equivariant cohomology of BS c (s , v). [ABSTRACT FROM AUTHOR]

We compute the equivariant cohomology of moment–angle complexes with respect to the action of coordinate subtori . We give a criterion for to be equivariantly formal, and obtain specifications for the cases of flag complexes and graphs. [ABSTRACT FROM AUTHOR]

We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an -module. Our formula is not exactly the same as the known formula by Chow or Cho, Hong, and Lee, but they are equivalent. We also discuss its higher degree generalization. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras. [ABSTRACT FROM AUTHOR]

The Peterson variety is a subvariety of the flag manifold G / B equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersection multiplicities yields a Z -basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find formulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt. [ABSTRACT FROM AUTHOR]

Let P be an operad with a multiplication, and later with a (higher) derivation. This paper introduces and develops a notion of (partial) group action and corresponding equivariant cohomology and deformation theory on P. It generalizes existing results for equivariant cohomology and deformation for associative algebras, dialgebras, and dendriform algebras, among others. At the end, we discuss compatibilities of group action and some operadic constructions. [ABSTRACT FROM AUTHOR]

Let M be an even-dimensional spin manifold with action of a compact Lie group G. We review the construction of bouquets of analytic equivariant cohomology classes and of K -integration. Given an elliptic curve E , we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on M. Following Bott–Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten's rigidity theorem. Our systematic use of the Chern–Weil construction of equivariant classes allows us to reduce proofs to algebraic identities. [ABSTRACT FROM AUTHOR]

We study finite group actions on Hom-Leibniz algebras. We introduce equivariant cohomology of Hom-Leibniz algebras with coefficients in O G -Hom-algebras. We show that there is a cup-product operation in the graded equivariant Hom-Leibniz cohomology, which satisfies the graded Hom-Zinbiel relation. [ABSTRACT FROM AUTHOR]

We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone. [ABSTRACT FROM AUTHOR]

For finite dimensional free Cp-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of S0 is used to prove the non-existence of certain Cp-maps. These are related to Borsuk-Ulam type theorems, and equivariant maps related to the topological Tverberg conjecture. For certain finite dimensional Cp-spaces which are formed out of representations, it is proved that the cohomology is a free module over the cohomology of a point. All the calculations are done for the cohomology with constant coefficients Z/p. [ABSTRACT FROM AUTHOR]

In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and push-forwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for the equivariant K-theory of some Kac-Moody flag varieties. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the topological equivariant K -ring, in terms of generators and relations, of a Springer variety F λ of type A associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence λ = (λ 1 , ... , λ l). This parallels the description of the equivariant cohomology ring of F λ due to Abe and Horiguchi and generalizes the description of ordinary topological K -ring of F λ due to Sankaran and Uma [13]. [ABSTRACT FROM AUTHOR]

A locally compact contraction group is a pair (G , α) , where G is a locally compact group and α : G → G an automorphism such that α n (x) → e pointwise as n → ∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p , we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G , α) which are central extensions { 0 } → F p ((t)) → G → F p ((t)) → { 0 } of the additive group of the field of formal Laurent series over F p = Z / p Z by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian , as follows from a classification of the abelian locally compact contraction groups. [ABSTRACT FROM AUTHOR]

In this survey article we exemplarily illustrate implications of curvature assumptions on the topology of the underlying manifold. We shall mainly focus on sectional curvature and three different kinds of restrictions, namely on non-negative respectively on positive sectional curvature, as well as on two-sided curvature bounds. We shall see that there are various implications on the side of topology, namely, for example, geometry having an impact on elementary invariants like the Euler characteristic or Betti numbers as well as on concepts from rational homotopy theory or index theory, and that there are connections to K-theory. On our way of making these connections we shall draw on certain simplifications and tools like group actions or metrics with additional properties like geometric formality. [ABSTRACT FROM AUTHOR]

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie triple systems. [ABSTRACT FROM AUTHOR]

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree 1, genus 0 Gromov–Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented. [ABSTRACT FROM AUTHOR]

In this paper, motivated by Chen–Ruan's stringy orbifold theory on almost complex orbifolds, we construct a new equivariant cohomology ring H G ∗ (X) for an equivariant almost complex pair (X, G), where X is a compact connected almost complex manifold, G is a connected compact Lie group which acts on X and preserves the almost complex structure. [ABSTRACT FROM AUTHOR]

We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions. [ABSTRACT FROM AUTHOR]

We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s′), where s is a subsequence of s′. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded H T ∙ (pt) -modules coinciding on the objects with the usual functor H T ∙ of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones). [ABSTRACT FROM AUTHOR]

We describe the torus-equivariant cohomology of weighted partial flag orbifolds w Σ of type A. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to as "Schubert Calculus on w Σ ". For the weighed Schubert classes in w Σ , we give the Chevalley's formula. In addition, we define the weighted analogue of double Schubert polynomials and give the corresponding Chevalley–Monk's formula. [ABSTRACT FROM AUTHOR]

In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type. [ABSTRACT FROM AUTHOR]

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group. [ABSTRACT FROM AUTHOR]

In this paper, we compute the R O (C pq) -graded cohomology of C pq -orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group C p . The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group C p was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem. [ABSTRACT FROM AUTHOR]

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold (M , ω) admitting a Hamiltonian circle action with fixed point set consisting of two connected components X and Y satisfying dim (X) + dim (Y) = dim (M) . Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of M, X and Y, and the total Chern classes of M, X, Y, and of the normal bundles of X and Y. The results show that these data are unique—they are exactly the same as those in the standard example G ~ 2 (R 2 n + 2) , the Grassmannian of oriented 2-planes in R 2 n + 2 , which is of dimension 4n with (any) n ∈ N , equipped with a standard circle action. Moreover, if M is Kähler and the action is holomorphic, we can use a few different criteria to claim that M is S 1 -equivariantly biholomorphic and S 1 -equivariantly symplectomorphic to G ~ 2 (R 2 n + 2) . [ABSTRACT FROM AUTHOR]

We study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked point and strata of the moduli space of Abelian differentials of genus three. [ABSTRACT FROM AUTHOR]

We calculate the ordinary C 2 -cohomology, with Burnside ring coefficients, of C P C 2 ∞ = B C 2 U (1) , the complex projective space, a model for the classifying space for C 2 -equivariant complex line bundles. The R O (C 2) -graded Bredon ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators. These generators include the Euler class of the tautological bundle, which lies outside of the R O (C 2) -graded theory. [ABSTRACT FROM AUTHOR]

We consider the derived category DbG(V) of coherent sheaves on a complex vector space V equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G2, F4, as well as the groups G(m, 1, n) = (μm)n × Sn, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces Vg/C(g), where C(g) is the centralizer subgroup of g ε G. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne [23]. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on Cn, where C is a smooth curve. [ABSTRACT FROM AUTHOR]

This paper is dedicated to introducing and studying a cohomology theory for monoids enriched with the action by endomorphisms of a fixed monoid of operators. This equivariant cohomology theory extends both Whitehead's cohomology for groups with operators and Leech's cohomology for monoids. [ABSTRACT FROM AUTHOR]

We study finite group actions on Leibniz algebras, and define equivariant cohomology groups associated to such actions. We show that there exists a cup-product operation on this graded cohomology, which makes it a graded Zinbiel algebra. [ABSTRACT FROM AUTHOR]

J. De Loera & T. McAllister and K. D. Mulmuley & H. Narayanan & M. Sohoni independently proved that determining the vanishing of Littlewood–Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood–Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas, A. Yong. Our proof then combines a saturation theorem of D. Anderson, E. Richmond, A. Yong, a reading order independence property, and É. Tardos' algorithm for combinatorial linear programming. [ABSTRACT FROM AUTHOR]

We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof. [ABSTRACT FROM AUTHOR]