arXiv research

This paper studies ring objects in equivariant derived Satake category from Coulomb branches.

The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the "modern" defi…

Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.

Formulates a new connection between topological and geometric categories.

New homological results for bordered Floer algebras derived from hypertoric categories.

We consider orbifolds as diffeological spaces. This gives rise to a natural notion of differentiable maps between orbifolds, making them into a subcategory of diffeology. We prove that the diffeological approach to orbifolds is equivalent to Satake's notion of a V-manifold and to Haefliger's notion of an orbifold. This…

Defines an equivariant cobordism category for finite groups.

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …

We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we repro…

Category theory enhances understanding of group-equivariant neural networks.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

Study equivariant vector fields near relative equilibria using isomorphic categories.

Lie groupoid equivariant neural networks are a new type of neural network.

Study shows Satake compactifications as horofunctions with polyhedral metrics.

Study shows equivariant Khovanov homotopy types are equivalent.

Formula calculates equivariant LS-category of symplectic toric manifolds.

Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant coho…

In this note, we consider a Lie group G acting on a manifold M. We prove that the category of bundles with connection on the differential quotient stack is equivalent to the category of G-equivariant bundles on M with G-invariant connection.

We determine the extent to which the collection of $Γ$-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of $Γ$-Euler-Satake characteristics corresponding to free or free abelian $Γ$ and are not…

This expository article is an expanded version of talks given at the "Current Developments in Mathematics, 2002" conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally …

Monoidal categorifies genus zero skein algebra using K-theory.

Functor connects symplectic and contact structures via cutting and blowups.

The study proves no $L^2$-eigenvalues for higher rank locally symmetric spaces.

Constructs a functor for equivariant smooth h-cobordisms.

We symmetrise neural networks for groups using Markov categories.

The paper studies equivariant sheaves on toric varieties and their quotients.

Develops parametrised Poincaré duality for equivariant fixed points.

Constructs 2-vector bundles and 2K-theory for Lie groupoids and 2-equivariant settings.

Embeds cohomology of hyperkahler manifolds into torus cohomology.

In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms of objects known as stacks (essentially, sheaves of categories), and develop mu…

We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a …

Geometrically describes Satake compactifications without root data.

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map from the Satake compactification of G/K (associated to V) to the Lie algebra of …

An algorithm for efficient computation of equivariant neural network layers.

The paper classifies equivariant gerbe connections on Lie groups.

Study Gromov-Hausdorff limits of K3 surface metrics via moduli compactification.

Unified mathematical framework for non-compact symmetric spaces with negative curvature.

We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…

Defines a new field theory in 1+1+1 dimensions.

New smooth models for string groups defined in ∞-categories.

Refines quantum annular homology using stable homotopy methods.

Constructs 2-representations and 2-vector bundles for Lie 2-groups.

Equivalent bicategories constructed from action Lie groupoids.

New algorithm speeds up group equivariant neural networks computations.

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…

We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor c…