arXiv research

Conditions found for linearizing divergence-free fields on invariant tori.

The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.

We develop an axiomatic theory of balance functions (future value functions) in the theory of interest that is derived from financial considerations and which applies to general regulated payment streams, including continuous payment streams. Balance functions exist and are unique up to an initial choice of deposit and…

Study one-dimensional topological theories with linear generating functions.

Survey on learning Boolean functions in computational theory.

The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.

We explain the main concepts of Prospect Theory and Cumulative Prospect Theory within the framework of rational dynamic asset pricing theory. We derive option pricing formulas when asset returns are altered with a generalized Prospect Theory value function or a modified Prelec weighting probability function and introdu…

Bound critical points for minimal Radó functions.

Extends Campanato theory to multi-valued functions for geometric variational problems.

New Morse theory for shapes at distances.

Study wave functions in complex Chern-Simons theory, finding integrality and rational points.

Morse theory connects low energy submanifolds in 3-sphere.

3D gauge theories link knot polynomials to vortex partition functions.

This paper is an exposition of the relationship between Witten's Chern-Simons functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory…

New method for analyzing multiparameter persistence modules from smooth functions.

Derives stress-energy identities in Liouville theory on compact surfaces.

The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geo…

TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.

E-string theory reveals modular properties of 4-manifold invariants.

New theory of distributions on spaces with singular submanifolds.

Developed a theory of ultradifferentiable sheafs with applications.

Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.

The Bregman divergence (Bregman distance, Bregman measure of distance) is a certain useful substitute for a distance, obtained from a well-chosen function (the "Bregman function"). Bregman functions and divergences have been extensively investigated during the last decades and have found applications in optimization, o…

Study of gauge theory blowups and Painlevé VI identity.

Deep, wide ConvResNets can approximate functions and their smoothness.

Study gauged supergravity, M5-branes, and class R theories, constraining supergravity coefficients and calculating partition functions.

A new discretisation of a doubled, i.e. BF, version of the pure abelian Chern-Simons theory is presented. It reproduces the continuum expressions for the topological quantities of interest in the theory, namely the partition function and correlation function of Wilson loops. Similarities with free spinor field theory a…

Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.

Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic s…

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition func…

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…

Game theory helps analyze ESOs/EBIs in production and service sectors.

We propose a new description of 3d $\mathcal{N}=2$ theories which do not admit conventional Lagrangians. Given a quiver $Q$ and a mutation sequence $m$ on it, we define a 3d $\mathcal{N}=2$ theory $\mathcal{T}[(Q,m)]$ in such a way that the $S^3_b$ partition function of the theory coincides with the cluster partition f…

This paper applies AMP theory to improve learning tasks.

Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clari…

Karcher reimagined elliptic functions using geometry.

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation …

This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.

New topological quantum gravity theories linked to Ricci flow.

These notes were inspired by the course ''Quantum Field Theory from a Functional Integral Point of View'' given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynman's path integral approach to quantum mechanics and quantum field theory from a functional integral point of view, where the mai…

Theory developed for complex Hessian measures on Hermitian manifolds.

The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on t…

This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such…

Statistical field theory aids in understanding deep learning complexities.

Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.

New model calculates Wilson surfaces in higher gauge theory.

Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalize…

New proof confirms operations on constructible functions match theory.