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Tatyana Barron

Publications and source records attributed to Tatyana Barron.

18 recordsLinked to original sources
arXiv2026

The Newlander-Nirenberg theorem for complex $b$-manifolds

Melrose defined the b-tangent bundle of a smooth manifold M with boundary as the vector bundle whose sections are vector fields on M tangent to the boundary. Mendoza defined a complex b-manifold as a manifold with boundary together with an involutive splitting of the complexified b-tangent bundle into complex conjugate factors. We prove complex b-manifolds have a single local model depending only on dimension. This can be thought of as the Newlander-Nirenberg theorem for complex b-manifolds. Our proof uses Mendoza's result that complex b-manifolds have no "formal local invariants" and a singular coordinate change to leverage the classical Newlander-Nirenberg theorem and Catlin's generalization for complex manifolds with pseudoconvex boundary.

math.DG↗
arXiv2026

On vector-valued multisymplectic forms

We obtain a standard local presentation for a vector-valued multisymplectic form on a smooth manifold, generalizing the known proof for polysymplectic forms. We show that vector-valued multisymplectic forms on a finite-dimensional real vector space form a non-unital operad. We prove an entropy inequality for partial compositions.

math.DG↗
arXiv2026

Geometry and biological processes

We suggest a geometric approach to modeling biochemical processes, aiming at those processes that occur in humans with food sensitivities or chemical sensitivities.

math.DG↗
arXiv2025

Dimensionality increase for error correction in the interaction between information space and the physical world

The evolution of human intelligence led to the huge amount of data in the information space. Accessing and processing this data helps in finding solutions to applied problems based on finite-dimensional models. We argue, that formally, such a mathematical model can be embedded into a higher-dimensional model inside of which a desired solution will exist. In our model, the physical world and the information space are submanifolds of infinite-dimensional Hilbert spaces, and the processes, including information transmission, are maps between the submanifolds of the physical world or of the information space. We discuss how our perspective fits in the context of existing literature. Our theorem states that a submanifold in the parameter space of the physical world can be deformed to a target submanifold outside that space, with an appropriate count of the deformation parameters. We interpret this assertion as an existence result for a class of problems and we discuss further steps.

cs.IT↗
arXiv2025

Mathematical models for therapeutic approaches involving electric conductors or shielding

We set up a mathematical model for a DC current in a human tissue that shows an attenuation effect in an extended circuit. We give a positive lower bound on the time duration over which this is guaranteed to happen in terms of the parameters of the model. We also discuss shielding and coupling in the context of electrical aspects of biological processes.

physics.med-ph↗
arXiv2025

Signals as submanifolds, and configurations of points

For the purposes of abstract theory of signal propagation, a signal is a submanifold of a Riemannian manifold. We obtain energy inequalities, or upper bounds, lower bounds on energy in a number of specific cases, including parameter spaces of Gaussians and spaces of configurations of points. We discuss the role of time as well as graph embeddings.

cs.IT↗
arXiv2024

Average entropy and asymptotics

We determine the $N\to \infty$ asymptotics of the expected value of entanglement entropy in $H_{1,N}\otimes H_{2,N}$, where $H_{1,N}$ and $H_{2,N}$ are the spaces of holomorphic sections of the $N$-th tensor powers of hermitian ample line bundles on compact complex manifolds.

math.DG↗
arXiv2024

Geometric signals

In signal processing, a signal is a function. Conceptually, replacing a function by its graph, and extending this approach to a more abstract setting, we define a signal as a submanifold M of a Riemannian manifold (with corners) that satisfies additional conditions. In particular, it is a relative cobordism between two manifolds with boundaries. We define energy as the integral of the distance function to the first of these boundary manifolds. Composition of signals is composition of cobordisms. A "time variable" can appear explicitly if it is explictly given (for example, if the manifold is of the form $Σ\times [0,1]$). Otherwise, there is no designated "time dimension", although the cobordism may implicitly indicate the presence of dynamics. We interpret a local deformation of the metric as noise. The assumptions on M allow to define a map $M\to M$ that we call a Fourier transform. We prove inequalities that illustrate the properties of energy of signals in this setting.

math.DG↗
arXiv2023

Coherent states and entropy

Let $H_k$, $k\in {\mathbb{N}}$, be the Hilbert spaces of geometric quantization on a Kähler manifold $M$. With two points in $M$ we associate a Bell-type state $b_k \in H_k\otimes H_k$. When $M$ is compact or when $M$ is ${\mathbb{C}}^n$, we provide positive lower bounds for the entanglement entropy of $b_k$ (asymptotic in $k$, as $k\to\infty$).

math.DG↗
arXiv2023

Semiclassical asymptotics and entropy

We study the entanglement of quantum states associated with submanifolds of Kaehler manifolds. As a motivating example, we discuss the semiclassical asymptotics of entanglement entropy of pure states on the two dimensional sphere with the standard metric.

math.DG↗
arXiv2023

On Automorphisms of Complex $b^k$-Manifolds

The $b$-calculus of Melrose is a tool for studying structures on a smooth manifold with a first order degeneracy at a given hypersurface. In this framework, Mendoza defined complex $b$-manifolds. In the spirit of work of Scott, we extend Mendoza's definition to the case of higher-order degeneracies, introducing the notion of a complex $b^k$-manifold for $k$ a positive integer. We then investigate the local and global automorphisms of complex $b^k$-manifolds. We also propose $b^k$-analogues for some classical spaces of holomorphic functions.

math.DG↗
arXiv2022

Entanglement and products

We consider a sequence of quantum states, $ρ_N$, associated with a submanifold $Λ$ of product of two integral compact Kähler manifolds. We show that, when $Λ$ is a product submanifold, then, in the semiclassical limit, these states are not entangled. We discuss whether geometry of $Λ$ (specifically $Λ$ being a product submanifold) has any relationship with entanglement properties of $ρ_N$.

math.DG↗
arXiv2018

On vector-valued automorphic forms on bounded symmetric domains

We prove a spanning result for vector-valued Poincaré series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of the norms (as the weight goes to infinity) for a class of totally real submanifolds. We give an example of a CR submanifold of the ball, for which the norms of the associated automorphic forms have a different asymptotic behavior.

math.CV↗
arXiv2018

Kähler quantization and entanglement

For a very ample line bundle L on a compact connected complex manifold X, with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L.

math-ph↗
arXiv2014

On varieties of Lie algebras of maximal class

We study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over C, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on N-graded Lie algebras of maximal class. As shown by A. Fialowski (see also [shalev:97], [millionshchikov:04]) there are only three isomorphism types of N-graded Lie algebras $L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by L_1 and L_2, L= . Vergne described the structure of these algebras with the property L= . In this paper we study those generated by the first and q-th components where q>2, L= . Under some technical condition, there can only be one isomorphism type of such algebras. For q=3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

math.RT↗