In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber ge…
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48 results for “real invariants”
The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant -transformations and classification of structures.
result No -maximal real flag manifolds admit integrable invariant generalized almost complex structures.
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…
Proves a specific knot is not smoothly slice using real invariants.
problem Determining the smooth sliceness of -cables of the figure-eight knot.
method Used real Seiberg-Witten Frøyshov invariant and developed an equivariant lattice homotopy type.
result Proves the -cable of the figure-eight knot is not smoothly slice when is odd.
In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures.…
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.
We investigate Seiberg-Witten theory in the presence of real structures. Certain conditions are obtained so that integer valued real Seiberg-Witten invariants can be defined. In general we study properties of the real Seiberg-Witten projection map from the point of view of Fredholm map degrees.
We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Boch…
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes' results, we prove that on a symplectic almost complex manifold with an antisymple…
The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.
problem Counting pseudo-holomorphic curves in symplectic Calabi-Yau 3-folds.
method Constructs three chambered invariants: , , , defined by counting solutions to ADHM vortex equations and pseudo-holomorphic sections of bundles.
result Conjectures a relationship between and and symplectic invariants.
For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.
problem Understanding spectral multiplicity and nodal sets for generic torus-invariant metrics.
method Analyzing real -eigenspaces and nodal sets for generic -invariant metrics.
result For generic -invariant metrics, real -eigenspaces are irreducible and have dimension at most 2, and nodal sets are connected hypersurfaces with specific properties.
New braid invariant derived from octagon solutions.
problem Constructing invariants of braids in the real projective plane.
method Action of braids on graphs in RP^2 with labels.
result Demonstrates octagon solutions yield braid invariants.
We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, real algebraic groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
The paper explores how to make machine learning models robust to domain shifts.
problem Machine learning models are unreliable in domains different from training.
method Introducing a broad formal notion of invariance and causal structures.
result The true underlying causal structure of the data plays a critical role in robustness.
We investigate the -monopole invariants of symplectic -manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic -manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic -manifolds. Further…
Let be a real hypersurface in complex Grassmannians of rank two. Denote by the quaternionic Kähler structure of the ambient space, the normal bundle over and . The real hypersurface is said to be -invariant if $\mathfrak D^\p…
We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.
New field invariant refines real spectrum and relates to absolute Galois group.
problem Understanding field invariants related to absolute Galois groups.
method Introducing Artin-Schreier quandles and computing their properties for different types of fields.
result Artin-Schreier quandles provide relations between fields and their absolute Galois groups.
Smooth figure-eight knot cables have infinite order.
problem Proving infinite order of figure-eight knot cables.
method Introduced new concordance invariants via branched covers and real Seiberg-Witten Floer K-theory.
result Uniform proof for all -cables of the figure-eight knot.
Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gauß map are classified.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth symplectic classification of Lagrangian fibrations near singularities.
result Action variables form complete symplectic invariants for parabolic orbits and cuspidal tori.
Real Heegaard Floer homology gets a new grading for certain 3-manifolds.
problem Real Heegaard Floer homology groups get an absolute Z/2 grading under specific conditions.
method Analyzes real Heegaard Floer homology groups with an involution and nullhomologous fixed points.
result Defines a new invariant of knots equal to the Alexander polynomial evaluated at i.
Developed a real sutured Heegaard Floer theory.
problem Real sutured manifolds and their Floer homology.
method Real nice diagrams, combinatorial computation.
result New structural properties distinguishing from sutured Floer homology.
We define and calculate signature and nullity invariants for complex schemes for curves in the real projective plane. We use an analog of the Murasugi-Tristram inequality to prohibit certain schemes from being realized by real algebraic curves. We give new formulas for Casson-Gordon invariants of graph manifolds, and s…
New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.
problem Relating symplectic bundle signature to surface group representations in real symplectic group.
method Using Atiyah-Patodi-Singer index theorem.
result Obtained a formula for the signature of a flat symplectic vector bundle over a surface with boundary.
The paper characterizes complex Finsler metrics invariant under U(n) and their properties.
problem Characterizing -invariant strongly convex complex Finsler metrics.
method Analyzing conditions for strong convexity and proving theorems about these metrics.
result A -invariant strongly convex complex Finsler metric is a real Berwald metric if and only if it comes from a Hermitian metric.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
The study defines invariants for time-like surfaces with real asymptotic lines.
problem Characterizing time-like surfaces with real asymptotic lines.
method Fundamental theorem of Bonnet-type, canonical parameters, invariant functions, PDEs.
result Time-like surfaces are determined by four invariant functions, two of which can be Gauss and mean curvature.
Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…
We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading …
A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
The main goal is to classify 4-dimensional real Lie algebras $\g$ which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the …
Proposes a method to create fair, robust predictors that remain consistent across different scenarios.
problem Creating fair and robust machine learning models that behave consistently across different scenarios.
method Graphical criteria and a model-agnostic framework called CIP based on HSCIC.
result Demonstrates the effectiveness of CIP in enforcing counterfactual invariance across various datasets.
Real Heegaard Floer Homologymath.GT
Real Heegaard Floer Homology extends Li's real monopole Floer homology.
problem Computing invariants of three-manifolds with symmetries.
method Real version of Lagrangian Floer homology, combinatorial computation.
result Euler characteristic of real Heegaard Floer homology is a knot invariant.
Real secondary index theorymath.GT
In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to which extend the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by 8) which can be defined in this case contains extra information over the complex index (the index of it…
The paper studies invariant measures for specific actions in algebraic groups.
problem Investigating invariant measures for horospherical actions and Anosov groups.
method Analyzing the space of invariant measures for -actions on .
result The space of invariant measures is homeomorphic to .
Study on -type flag manifolds, focusing on invariant metrics and Ricci flow.
problem Characterizing and analyzing metrics on -type flag manifolds.
method Investigation of invariant metrics, analysis of g.o. metrics, and Ricci flow techniques.
result Characterization of metrics invariant under maximal compact subgroups.
Satellite formula connects knot concordance invariants to surgery.
problem Understanding knot concordance invariants.
method Excision theorem for real Floer homotopy types.
result Concordance invariants depend only on zero-framed surgery.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
Develops differential KO-character to determine real vector bundles in multiples of 8.
problem Determining real vector bundles in multiples of 8.
method Constructs eta-invariants and differential KO-character to determine differential KO-theory.
result Eta-invariants and index invariants completely determine differential KO-theory in degree (0 mod 8).
We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.
Constructs projective plane over octonions, proving no higher real division algebras.
problem Existence of higher-dimensional real division algebras.
method Using Adams' solution of the Hopf invariant 1 problem, constructs projective plane over octonions.
result No higher-dimensional real division algebras exist.
New invariant real rank identifies constant real Lie algebroids.
problem Characterizing complex Lie algebroids with constant real rank.
method Introducing real rank and minimal complex subalgebroid.
result Local splitting and characterization of complex Lie algebroids.