Equivariant cohomology
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Functors and Computations in Floer homology with Applications Part II Open
The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part pr…
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Universal Invariant and Equivariant Graph Neural Networks Open
Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, …
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Quantum groups and quantum cohomology Open
© Astérisque 408, SMF 2019 — In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra YQ, the Yangian of…
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Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology Open
Causally ordered correlation functions of local operators in near-thermal quantum systems computed using the Schwinger-Keldysh formalism obey a set of Ward identities. These can be understood rather simply as the consequence of a topologic…
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The affirmative answer to Singer's conjecture on the algebraic transfer of rank four Open
During the last decades, the structure of mod-2 cohomology of the Steenrod ring $\mathscr {A}$ became a major subject in Algebraic topology. One of the most direct attempt in studying this cohomology by means of modular representations of …
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Machine learning line bundle cohomology Open
We investigate different approaches to machine learning of line bundle cohomology on complex surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully connected networks with logistic sigmoids is revie…
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Integral p-adic Hodge theory Open
I will describe joint work with M. Morrow and P. Scholze on the construction of a new integral cohomology theory for smooth projective schemes over the ring of integers of a p-adic field. The new theory realizes de Rham cohomology as a spe…
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A spectral sequence for stratified spaces and configuration spaces of points Open
We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomolog…
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Purity for graded potentials and quantum cluster positivity Open
Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$ -action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$ -equivariant with respect to a positive weight action on the base…
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On the lambda algebra and Singer's cohomological transfer Open
Write $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an importa…
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Equivariant oriented cohomology of flag varieties Open
Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identi…
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On the center of quiver Hecke algebras Open
We compute the equivariant cohomology ring of the moduli space of framed\ninstantons over the affine plane. It is a Rees algebra associated with the\ncenter of cyclotomic degenerate affine Hecke algebras of type A. We also give\nsome relat…
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Cohomology of abelian arrangements Open
An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian a…
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On the symplectic cohomology of logCalabi–Yau surfaces Open
We study the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symp…
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Elliptic Dynamical Quantum Groups and Equivariant Elliptic Cohomology Open
We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based o…
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The symplectic arc algebra is formal Open
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology c…
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Blown-up intersection cohomology Open
In previous works, we have introduced the blown-up intersection cohomology\nand used it to extend Sullivan's minimal models theory to the framework of\npseudomanifolds, and to give a positive answer to a conjecture of M. Goresky\nand W. Pa…
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Relations on Mgn via equivariant Gromov-Witten theory of P1 Open
We give a proof of Pixton's generalized Faber-Zagier relations in the\ntautological Chow ring of $\\overline M_{g,n}$. The strategy is very similar to\nthe work of Pandharipande-Pixton-Zvonkine, who have given a proof of the same\nresult i…
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Bredon cohomology and robot motion planning Open
We study the topological invariant [math] reflecting the complexity of algorithms for autonomous robot motion. Here, [math] stands for the configuration space of a system and [math] is, roughly, the minimal number of continuous rules which…
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SYNTOMIC COHOMOLOGY AND p-ADIC REGULATORS FOR VARIETIES OVER p-ADIC FIELDS Open
We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new …
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On two cohomological Hall algebras Open
We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced in [19] associated with each quiver Q , and each algebraic oriented cohomology theory A . It is defined as the A -homology of the moduli …
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3d $$ \mathcal{N} $$ = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians Open
A bstract We study a correspondence between 3d $$ \mathcal{N} $$ = 2 topologically twisted Chern-Simons-matter theories on S 1 × Σ g and quantum K -theory of Grassmannians. Our starting point is a Frobenius algebra depending on a paramet…
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Hochschild cohomology versus the Jacobian ring and the Torelli theorem for cubic fourfolds Open
The Jacobian ring J(X) of a smooth hypersurface X ⊂ P n+1 determines the isomorphism type of X.This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases.However, in Voisin's origina…
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Canonical -deformations in arithmetic geometry Open
In recent work with Bhatt and Morrow, we defined a new integral -adic cohomology theory interpolating between étale and de Rham cohomology. An unexpected feature of this cohomology is that in coordinates, it can be computed by a -deformati…
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Cohomology jump loci of differential graded Lie algebras Open
To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and r…
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Nonsemisimple quantum invariants and TQFTs from small and unrolled quantum groups Open
We show that unrolled quantum groups at odd roots of unity give rise to\nrelative modular categories. These are the main building blocks for the\nconstruction of 1+1+1-TQFTs extending CGP invariants, which are non-semisimple\nquantum invar…
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Motivic homotopy theory in derived algebraic geometry Open
In topology, generalized cohomology theories are representable in the stable homotopy category. The analogue in algebraic geometry is the stable motivic homotopy category, constructed by Morel–Voevodsky, where generalized motivic cohomolog…
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Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes Open
We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme \mathcal{X} at any integer n in terms of Weil-étale cohomology complexes. This extends work…
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Poset pinball, GKM-compatible subspaces, and Hessenberg varieties Open
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compa…
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EQUIVARIANT -THEORY OF GRASSMANNIANS Open
We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$ -theory of Grassmannians, Acta Math . 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grass…