Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.
problem Formulating and resolving the Hauptvermutung in Causal Set Theory.
method Two mathematically well-defined formulations of the Hauptvermutung, one of which is true.
result The Hauptvermutung is true when finite sets are replaced by countable sets.
New theory uses probability sets for data variability, improving machine learning.
problem Variability in data distribution causes learning issues.
method Uses convex sets of probabilities (credal sets) to model data variability.
result Derives bounds for risk of models learned from multiple training sets.
The paper outlines future work in random sets theory.
problem Developing a theory of statistical reasoning with random sets.
method Generalizing logistic regression, probability laws, and geometric uncertainty.
result A new geometric approach to uncertainty with general random sets.
Extends field theory foundations to infinitesimal spaces, simplifying complex concepts.
problem Develop rigorous foundations for field theory, especially for infinitesimal spaces.
method Formulates local Lagrangian field theory in a new category of thickened smooth sets.
result Establishes a firm foundation for field theory, including tangent bundles and perturbative considerations.
In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.
This paper is devoted to the development and applications of some (new) basic concepts in Lie theory, both from `computational" and "observability" viewpoint. We specify set of all "G-equivariant" maps from a given Lie group G to the underlying manifold M, namely G-set, and also we introduce "conjugacy" in Lie group …
In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
This paper introduces a new homology theory for Yang-Baxter solutions.
problem Defining a homology theory for set-theoretic Yang-Baxter solutions.
method Introducing normalized homology theory and proving its split into parts.
result Set-theoretic Yang-Baxter homology can be split into normalized and degenerated parts.
The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows for quick proofs of basic set-theoretic identities and logical tautologies, so i…
We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result i…
A multisymplectic setting for classical field theories subjected to non-holonomic constraints is presented. The infinite dimensional setting in the space of Cauchy data is also given.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.
Develops BV function and finite perimeter set theory on Riemannian manifolds.
problem Theory of BV functions and finite perimeter sets on arbitrary Riemannian manifolds.
method Localization framework combining Euclidean and metric measure space techniques.
result Recovery of key Euclidean results in Riemannian setting.
Differential K-theory gets a λ-ring structure.
problem Establishing a λ-ring structure in differential K-theory. method Splitting principle for differential K-theory, Adams operations construction.
result Differential K0-ring admits a λ-ring structure. Introduces a new geometric framework for field theories.
problem Developing a rigorous mathematical framework for field theories.
method Introduces supergeometric homotopy theory to physics.
result Classical bosonic field theories fit naturally into smooth sets.
We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension …
Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.
problem Computing multiparameter persistence with new tools and methods.
method Adapting Forman's theory to vectorial setting and using combinatorial topological dynamics.
result Established more general result for sublevel sets and found a way to induce Morse decomposition.
The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are su…
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
Paper defines generalized braids and proves their subgroup status.
problem Understanding the structure of generalized braids and knots.
method Defined generalized braid theories and computed their generating sets.
result Quasitoric normal generalized braids form a subgroup of normal generalized braid group.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
Generalizes expansion and collapse theory to metric spaces.
problem Compactification of arbitrary metric spaces.
method Expands dual notions of expansion and collapse to arbitrary metric spaces and infinitely many moves.
result Proves compactification theorems, particularly for Z-set compactifications.
A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists.…
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
Paper establishes robust no-arbitrage conditions under projective determinacy.
problem Understanding financial models under Knightian uncertainty.
method Adopting a projective framework, treating all model components uniformly in terms of measurability.
result Establishes characterizations of robust no-arbitrage condition under PD.
Simplified proof for approximations of set systems.
problem Approximations of set systems in various fields.
method Modular, self-contained proof using Chernoff's bound.
result Accessible proof for a wider audience.
New method distinguishes knots and knotted surfaces.
problem Distinguishing knots and knotted surfaces.
method Twisted set-theoretic Yang-Baxter solutions and Alexander numbering.
result Distinguished 2-twist spun trefoil from its reverse. The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.
problem Understanding multifiltering functions through discrete Morse theory.
method Applying multiparameter discrete Morse theory to vector-valued multifiltering functions.
result Any multifiltering function can be approximated by a compatible MDM function.
The paper examines how the topology of level sets changes with critical points in Morse theory.
problem Understanding how the topology of level sets changes with critical points in Morse theory.
method Study of sublevel sets and level sets of Morse functions, analysis of critical points and their indices.
result For a general class of functions, the topology of a regular level set changes when passing a single critical point, unless the index is half the dimension of the manifold.
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to students familiar with just the fundamentals of algebraic topology.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
Small sets of systoles fill hyperbolic surfaces of large genus.
problem Constructing hyperbolic surfaces with minimal systole sets.
method Theory of Coxeter groups combined with number theory.
result Cardinality of systole sets is in o(g/ ln g) for large genus.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
Paper solves the minimal generating set problem for singular Reidemeister moves.
problem Determine minimal generating sets of oriented singular Reidemeister moves.
method Introduced new invariant for singular links to detect type IV moves and provide obstructions.
result Proved exactly 96 distinct inclusion-minimal generating sets for singular moves.
This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J3Y as well…
We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebr…
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
This study provides a new mathematical structure for Koopman eigenfunctions.
problem Understanding and representing nonlinear dynamics as linear.
method Theoretical, analytical, and numerical approaches to Koopman eigenfunction space.
result Equivalence of minimal generating set and maximal independent set, defining conditions for independence.
Algorithm constructs confidence sets for deep neural networks with PAC guarantees.
problem Ensuring reliable predictions for deep neural networks with high confidence.
method Combines calibrated prediction and learning theory bounds.
result Constructs PAC confidence sets for various deep models.
New theory for clustering in geometric and adaptive settings.
problem Clustering in non-Euclidean spaces and adaptive parameters.
method Asymptotic theory for k-means and related methods. result Strong consistency and asymptotic limit theorems for various clustering procedures.
A families index theorem in K-theory is given for the setting of Atiyah, Patodi and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred cusp, pseudodifferential operators on the fibres (with …
Following the analogies between 3-dimensional topology and number theory, we study an idèlic form of class field theory for 3-manifolds. For a certain set K of knots in a 3-manifold M, we first present a local theory for each knot in K, which is analogous to local class field theory, and then,…
The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.