Anabelian geometry reformulated using Hodge theory for hyperbolic curves.
problem Determining varieties over number fields using their étale fundamental groups.
method Formulating a Hodge-theoretic version of anabelian conjecture, replacing Galois action with Cimes-action. result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over C. We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the A…
In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine ge…
Geographic diversification is fundamental to risk mitigation among investors and insurers of housing, mortgages, and mortgage-related derivatives. To characterize diversification potential, we provide estimates of integration, spatial correlation, and contagion among US metropolitan housing markets. Results reveal a hi…
Lecture notes on linear neural networks for deep learning optimization and generalization.
problem Understanding optimization and generalization in deep learning models.
method Mathematical tools and dynamical systems theory.
result Potential of mathematical tools to enhance understanding of deep learning.
Bayesian methods often misinterpret data and asymptotic concepts.
problem Misunderstandings in Bayesian predictive inference.
method Discussion of two specific misunderstandings.
result Consequences of misinterpretations illustrated through examples.
For any Lie groupoid we construct an analytic index morphism taking values in a modified K−theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constru…
Tangle machines are a topologically inspired diagrammatic formalism to describe information flow in networks. This paper begins with an expository account of tangle machines motivated by the problem of describing `covariance intersection' fusion of Gaussian estimators in networks. It then gives two examples in which ta…
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the sm…
Researchers expand on best subset selection theory, identifying key complexities.
problem Understanding model selection performance in high-dimensional sparse linear regression.
method Analyzing residualized signals, orthogonality, and spurious projections to establish margin conditions.
result Established necessary and sufficient margin conditions for BSS model consistency.
We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir. Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner …
This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exc…
Feature selection from wide datasets leads to misleading results.
problem Feature selection in wide datasets with few samples can lead to misleading results.
method Derived sample size requirement for declaring features different, used real datasets to illustrate issues.
result Feature selection from very wide datasets may lead to misleading results.
Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
Survey of methods for computing volumes of moduli spaces.
problem Computing volumes of moduli spaces for Riemann surfaces with different metrics.
method Combinatorial enumeration, intersection theory, recursion relations.
result Review of key results and methods in computing both Weil-Petersson and Masur-Veech volumes.
Study calculates fundamental groups of torus knots using algebraic topology.
problem Calculating the fundamental group of torus knots.
method Algebraic topology and group theory.
result Computed fundamental groups of torus knots.
Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.
problem Understanding the fundamental groups of ancient Ricci flows.
method Analyzing the structure of ancient Ricci flows and their tangent flows.
result The fundamental group of non-collapsed ancient Ricci flows is finite and a quotient of the regular part's fundamental group.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
Proves one-relator groups are fundamental groups of Sasakian manifolds.
problem Characterizing groups that can be fundamental groups of Sasakian manifolds.
method Analyzes one-relator groups and classifies groups of deficiency at least two.
result Proves sufficient and necessary conditions for one-relator groups to be fundamental groups of Sasakian manifolds.
For every N > 0 there exists a group of deficiency less than -N that arises as the fundamental group of a smooth homology 4-sphere and also as the fundamental group of the complement of a compact contractible submanifold of the 4-sphere. A group is the fundamental group of the complement of a contractible submanifold o…
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
Study mapping class group actions on surface fundamental groups.
problem Understanding group actions on surface fundamental groups.
method Computing image of mapping class group generators as outer automorphisms.
result Computed images of mapping class group generators on fundamental groups.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
Study of fundamental groups of knotted solenoid complements in 3D sphere.
problem Determining fundamental groups of knotted solenoid complements.
method Using canonical sequence of knot groups and embedding up to mirror reflection.
result Fundamental groups of knotted solenoid complements are solely determined by a sequence of knot groups and embedding up to mirror reflection.
New algebraic fundamental groups identified for fake projective planes.
problem Characterizing algebraic fundamental groups of fake projective planes.
method Analysis of complex conjugate pairs and explicit finite étale covers.
result Forty-six distinct isomorphism classes of algebraic fundamental groups.
New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.
problem Constructing manifolds with positive Ricci curvature and non-abelian fundamental groups.
method Constructing specific 9-dimensional manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
result Examples of manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
Characterizes fundamental groups of disjointly tree-graded spaces.
problem Understanding fundamental groups of complex geometric structures.
method Defines and analyzes disjointly tree-graded spaces, characterizing their fundamental groups.
result Fundamental groups of disjointly tree-graded spaces embed into inverse limits of free products of fundamental groups of pieces.
Study homomorphisms from groups to 3-manifold fundamental groups.
problem Understanding homomorphisms between groups and 3-manifold fundamental groups.
method Proves foundational results and answers specific questions.
result Answers questions posed by Reid-Wang-Zhou and Agol-Liu.
The paper classifies 4-manifolds based on their fundamental groups and orientation characters.
problem Classifying 4-manifolds with specific fundamental groups and orientation characters.
method Developed a criterion for groups and homomorphisms to classify 4-manifolds up to homotopy equivalence.
result Closed 4-manifolds with specified fundamental groups and orientation characters are classified by their quadratic 2-types.
In this paper, we give a fully detailed exposition of computing fundamental groups of complements of line arrangements using the Moishezon-Teicher technique for computing the braid monodromy of a curve and the Van-Kampen theorem which induces a presentation of the fundamental group of the complement from the braid mono…
Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.
problem Confirming a 1997 conjecture about fundamental groups of abelian differentials.
method Algebraic-geometric approach using Shimada's techniques for computing fundamental groups via morphisms of varieties.
result Orbifold fundamental groups of these strata are described as framed mapping class groups.
A simple method to create new 4-manifolds by altering fundamental groups.
problem Creating new 4-manifolds with specific fundamental groups.
method Cut-and-paste mechanism to alter fundamental groups.
result Constructs new 4-manifolds with fixed-point free involutions.
We study fundamental groups of non compact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
In this note we investigate to what extent the fundamental group of a metric space can be described as the inverse limit of its discrete fundamental groups. We show that some mild conditions suffice to imply the existence of an isomorphism and we provide a list of counterexamples to possible weakenings of these hypothe…
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
Paper studies fundamental groups of certain Ricci solitons.
problem Understanding fundamental groups of specific Ricci solitons.
method Analyzes properties of complete steady gradient Ricci solitons with nonnegative sectional curvature.
result Fundamental groups of these solitons are either trivial or infinite.
Study surfaces with free product fundamental groups, proving existence and properties.
problem Understanding fundamental groups of quasi-projective surfaces.
method Prove existence of admissible maps and use addition-deletion Lemmas.
result Existence of admissible maps and properties of fundamental groups.
Study fundamental groups of geometric transformation groups using loop spaces.
problem Understanding fundamental groups of geometric transformation groups.
method Use differential forms on loop spaces to prove infinite fundamental groups.
result Proves infinite fundamental groups for specific geometric transformation groups.
Toroidal 3-manifolds have special group structures that can be shown through specific covers.
problem Characterizing the fundamental groups of toroidal 3-manifolds.
method Proving toroidal 3-manifolds have circularly-orderable fundamental groups by showing they admit finite cyclic covers with left-orderable fundamental groups.
result Toroidal 3-manifolds have circularly-orderable fundamental groups, which can be shown through specific covers.
The geography of minimal symplectic 4-manifolds with arbitrary fundamental group and symplectic 6-manifolds with abelian fundamental group of small rank, and with arbitrary fundamental group are addressed.
Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is…
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Study shows different fundamental groups for manifolds with same limit.
problem Understanding fundamental groups of manifolds with non-negative Ricci curvature.
method Constructed sequences of manifolds with specific properties.
result Found manifolds with same limit but different fundamental groups.
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
The study finds smooth structures on specific 4-manifolds with even fundamental groups.
problem Investigating smooth structures on 4-manifolds with even fundamental groups.
method Analyzing topological, closed, oriented, non-spin 4-manifolds with given constraints.
result Existence of either none or infinitely many distinct smooth structures on specified manifolds.
We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …