arXiv research

Develops quantum character theory for complex reductive groups.

We define a functor $\mathcal{Q}$ from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle $X$, there is a one-to-one correspondence between the set of $X$-colorings and that of $\mathcal{Q}(X)$-colorings diagrammatically for any …

Let $G$ be a compact, simply connected Lie group. If $\mathcal{C}_1,\mathcal{C}_2$ are two $G$-conjugacy classes, then the set of elements in $G$ that can be written as products $g=g_1g_2$ of elements $g_i\in \mathcal{C}_i$ is invariant under conjugation, and its image under the quotient map $G\to G/\operatorname{Ad}(G…

We consider the space of differential operators $\mathcal{D}_{λμ}$ acting between $λ$- and $μ$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{λμ}$ which is finer than the classical one, obtained by writting a differen…

We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to $\mathcal{D}$-modules for the homogeneo…

Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consis…

New sl(2) action defined on a mathematical module.

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…

Study of 2d gauged linear sigma models to derive difference equations and spectral data.

Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern cla…

The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are struc…

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …

Paper presents a novel neural network for MIMO symbol detection.

Study Berry connections for 2d GLSMs, linking to cohomology theories.

Proves an equivariant version of index theorem for geometric families.

We consider an equivariant analogue of a conjecture of Borcherds. Let $Y$ be a real $K3$ surface without real points. Let $g$ be a Ricci-flat Kaehler metric on $Y$ invariant under the complex conjugation. We shall prove that the equivariant determinant of the Laplacian of $(Y,g)$ with respect to the complex conjugation…

Proves an equivariant version of Heegaard Floer link surgery formula.

We derive simple explicit formulas for the character of a cycle in the Connes' (b,B)-bicomplex of cyclic cohomology and give applications to the Fredholm modules and equivariant characteristic classes.

We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \math…

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributi…

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of…

Vanishing of equivariant cohomology groups for proper Lie group actions.

The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $λ$ and the space of functions on $T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on ${\mathbb{R}}^n$. However, these mo…

We propose to study equivariance in deep neural networks through parameter symmetries. In particular, given a group $\mathcal{G}$ that acts discretely on the input and output of a standard neural network layer $φ_{W}: \Re^{M} \to \Re^{N}$, we show that $φ_{W}$ is equivariant with respect to $\mathcal{G}$-action iff $\m…

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

New maps help understand deformations of modules over Lie algebroids.

New Lie groupoid and algebroid constructed for octonionic Hopf foliation.

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

Two Pin(2)-equivariant Floer homologies are shown to be equivalent.

Equivariant trisections for group actions on 4-manifolds are introduced and studied.

Develops a new approach to study nonlinear PDEs and their singularities.

Let $Δ$ be a trivial knot in the three-sphere. For every finite cyclic group $G$ of odd order, we construct a $G$-equivariant Khovanov homology with coefficients in the filed $\F_{2}$. This homology is an invariant of links up to isotopy in $(S^{3},Δ)$. Another interpretation is given using the categorification of the …

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

Establishes a connection between skein modules and algebraic sets.

Let G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism cl…

We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3),…

Constructs a functor for equivariant smooth h-cobordisms.

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,…

New method for Lagrangian Floer homology groups using flow trees.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

Lie-Rinehart algebras over $C^\infty$-rings defined and studied.

New connections found on zero-mean multivariate normal distributions.

Explains a property of algebras related to quantum field theories.

New algebraic structure characterizes vector bundles without module requirement.

Study Kauffman bracket skein modules of Seifert fibered spaces.

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on…