Study real GIT and Wick-rotations of pseudo-Riemannian manifolds.
problem Understanding Wick-rotations and their conditions for pseudo-Riemannian manifolds.
method Extending earlier results, providing sufficient and necessary conditions for Wick-rotatability, and deriving an invariance theorem.
result Derived sufficient and necessary conditions for pseudo-Riemannian manifolds to be Wick-rotatable.
Mathematical treatment of path integrals in quantum field theory.
problem Understanding the mathematical meaning of path integrals in quantum field theory.
method Perturbative approach using Wick's theorem and coordinate changes.
result Established invariance properties of Wick expansions under coordinate changes and symmetries.
Study Wick-rotations of Lie groups using GIT results.
problem Understanding Wick-rotations of pseudo-Riemannian Lie groups.
method Using results from real GIT, analyze invariant metrics and Lie algebras.
result Existence and conjugacy of Cartan involutions for pseudo-Riemannian Lie groups.
Wick rotations between pseudo-Riemannian spaces using real GIT.
problem Existence and non-existence of Wick-rotations between different pseudo-Riemannian spaces.
method Using real GIT to study structure groups of pseudo-Riemannian spaces as real forms of complex Lie groups.
result Derivation of new results regarding the existence and non-existence of Wick-rotations.
A new metric allows Wick rotation to Riemannian, proving electric tensor properties.
problem Understanding Wick rotation in arbitrary dimensions and signatures.
method Demonstrated through a new metric allowing Wick rotation to a Riemannian metric.
result The Riemann and Weyl tensors are purely electric.
The paper explores connections between three equations via Wick rotations and symmetries.
problem Investigating relations between solutions to specific equations under Wick rotations.
method Analyzing symmetries and transformations of solutions to the minimal surface, zero mean curvature, and Born-Infeld equations.
result Existence conditions and transformations of real and imaginary solutions under Wick rotations.
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index H∈(0,1). This process has sta…
Quantum method prices options by evolving a state in imaginary time.
problem Pricing options in a quantum setting.
method Prepares an initial state, evolves it using imaginary time algorithms, and maps to quantum state.
result Numerical verification for European options; extension to path-dependent options.
We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective str…
Holomorphic Riemannian geometry connects submanifolds in different spaces.
problem Relating submanifolds in pseudo-Riemannian spaces.
method Rephrasing and extending holomorphic Riemannian theory to apply it to submanifolds.
result Holomorphic Riemannian geometry links submanifolds across different spaces.
Paper introduces a new cosmological volume function and its properties.
problem Introducing a new cosmological volume function.
method Introduces and analyzes the cosmological volume function τ_V, showing it's continuously differentiable.
result τ_V leads to a canonical splitting of the metric tensor and a canonical Wick-rotated Riemannian metric.
"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrica…
The paper shows how Scherk-type surfaces can be decomposed into helicoids.
problem Decomposing Scherk-type zero mean curvature surfaces.
method Using a special Euler-Ramanujan identity and Wick rotation, the paper expresses these surfaces as an infinite superposition of dilated helicoids and provides different finite decompositions.
result Scherk-type zero mean curvature surfaces can be expressed as an infinite superposition of dilated helicoids.
Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of Deligne-Mumford-Knudsen moduli spaces Mˉg,n of stable pointed algebraic curves; hence the…
The paper defines a category of Lagrangian correspondences in super Hilbert spaces and constructs a functorial field theory.
problem Understanding composition of Lagrangian correspondences in Hilbert spaces.
method Study of Lagrangian correspondences, construction of a category, and functorial field theory.
result Well-defined composition law in a category of Lagrangian correspondences.
New gravitational solitons and infinite topological manifolds found.
problem Finding new gravitational solitons and Riemannian manifolds.
method Space-periodic solutions of Einstein equations, quotients, Wick rotation.
result Complete Ricci flat Riemannian manifolds of infinite topological type.
The thesis extends Riemannian submanifold theory to non-real settings, unifying various geometries.
problem Generalizing Riemannian geometry and submanifold theory to non-real settings.
method Developing 'Rinehart spaces' and holomorphic Riemannian submanifolds theory.
result Unified framework for various geometries over different ground rings.
Ricci flow emerges from renormalizing nonlinear Sigma models.
problem Understanding the renormalization group flow in nonlinear Sigma models.
method Using Euclidean algebraic quantum field theory and Wick ordering.
result The first-order renormalization group flow of nonlinear Sigma models equals the Ricci flow.
Quantum mechanics models for financial Black-Scholes model.
problem Modeling financial derivatives using quantum mechanics.
method Noncommutative quantum mechanics applied to specific mechanical systems.
result Generalized noncommutative quantum mechanics of financial models.
Given a closed hyperbolic surface S, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on S×R and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on S×R. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in te…
Let D be a homogeneous bounded domain of Cn and A a set of (anti--Wick) symbols that defines a commutative algebra of Toeplitz operators on every weighted Bergman space of D. We prove that if A is rich enough, then it has an underlying geometric structure given by a Lagrangian fo…
The aim of this survey is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonicla Wick rotation-rescaling theory, previously developed b…
Quantum probability theory constructs Martingales for non-Brownian financial models.
problem Constructing Martingales for financial models using fractional Brownian motion.
method Quantum probability theory and Wick product.
result Quantum probability framework allows for Martingale construction without Brownian integrals.
Researchers find a new way to describe maximal surfaces using a specific method.
problem Finding representations for maximal surfaces in Lorentz-Minkowski space.
method Weierstrass-Enneper representation and Barbishov-Chernikov method applied to hodographic coordinates.
result Established a connection between maximal surface equation and Born-Infeld equation via Wick rotation.
The paper extends graph embedding models to handle multiple relations.
problem Link prediction in multi-relational networks.
method Generalized pseudo-Riemannian embedding models to multi-relational networks, considering relations as submanifolds.
result Validation of the approach in link prediction tasks, including knowledge graph completion and biological domain analysis.
Proves M-theory anomaly cancellation on nonorientable manifolds.
problem Anomaly cancellation in M-theory on nonorientable manifolds.
method Computational techniques for eta-invariants, algebraic theory of cubic forms, Adams spectral sequence techniques.
result No parity anomaly in M-theory in the low-energy field theory approximation.
We derive a closed formula for a star-product on complex projective space and on the domain SU(n+1)/S(U(1)×U(n)) using a completely elementary construction: Starting from the standard star-product of Wick type on Cn+1∖{0} and performing a quantum analogue of Marsden-Weinstein reduction, we ca…
Study conjugacy growth in free groups and products of finite groups.
problem Counting conjugacy classes of commutators in free groups and products of finite groups.
method Classification of commutators in free groups and products by Wicks, building on Rivin and Sharp's work.
result Asymptotic formula for conjugacy classes of commutators with given word length.
Exploring distance functions on spacetime models.
problem No canonical distance function exists for Lorentzian manifolds.
method Comparing Riemannianization and null distance function approaches.
result Concrete comparison of distance functions in GRW setting.
In this letter a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely---taking into account that the conjecture lacks any precise formulation yet---first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically st…
Study large N oscillations in 3D theories related to black hole physics.
problem Understanding large N sign oscillations in 3D theories via holography.
method Holographic computation of on-shell actions for Euclidean supergravity solutions, Wick rotation of magnetically charged AdS4 black holes.
result Proposed a non-trivial mathematical conjecture regarding phase factors of twisted Reidemeister-Ray-Singer torsion.
This paper is dedicated to Oleg Viro on his 60-th birthday. The paper is about Khovanov homology and its relationships with statistical mechanics models such as the Ising model and the Potts model. We give a relatively self-contained introduction to Khovanov homology, and also a reformulation of the Potts model in term…
The study finds obstructions to certain Riemannian metrics using Lorentzian geometry.
problem Finding obstructions to curvature distinguished Riemannian metrics.
method Dual Lorentzian metrics and Penrose's plane wave limit.
result Necessary local conditions for certain Riemannian metrics.
New homogeneous M2 brane duals found for specific dimensions.
problem Finding new gravity duals for M2 branes with enhanced supersymmetry.
method Classifying homogeneous M-theory backgrounds with specific symmetry algebras.
result Several new backgrounds with n=5 found, some of which are supersymmetric. Study of convergence in Lorentzian spacetimes using temporal functions.
problem Non-compactness of spacetime isometries and convergence in semi-Riemannian settings.
method Introduced anchored convergence and used Cauchy temporal functions to define convergence for spacetimes.
result Established local and global regularity of Cauchy temporal functions and their properties.
Power-law spectrum of random feature model is preserved in neural networks.
problem Preserving power-law spectrum in neural networks through random feature model.
method Characterized eigenvalues of population random-feature covariance using dyadic head-tail decomposition and Wick chaos expansions.
result Power-law exponent α is inherited from input covariance, modified by a logarithmic correction. New findings on kernel regression in the quadratic regime, improving understanding of machine learning models.
problem Understanding kernel ridge regression in the quadratic asymptotic regime.
method Extended study of kernel regression to the quadratic regime, establishing approximation bounds and spectral distributions.
result Broad class of inner-product kernels exhibit behavior similar to a quadratic kernel, with precise asymptotic training and test errors characterized.
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.