Study non-geometric M-theory backgrounds, proving they are differentiable manifolds.
problem Prove non-geometric M-theory backgrounds are differentiable manifolds.
method Prove non-geometric backgrounds are differentiable manifolds, using Kähler covering and deck transformations.
result Non-geometric backgrounds are differentiable manifolds with unique geometric and topological properties.
Derives fluxes in M-theory compactifications and connects them to threebrane sigma-models.
problem Deriving fluxes in M-theory compactifications and understanding their geometric and topological properties.
method Systematic derivation of fluxes from higher Courant brackets and generalized geometry, relating them to threebrane sigma-models.
result Fluxes in M-theory compactifications are understood as generalized Wess-Zumino terms in threebrane sigma-models, linking higher structure to Lie algebroid homotopy.
Motivated by the quest to understand the analog of non-geometric flux compactification in the context of M-theory, we study higher dimensional analogs of generalized Poisson sigma models and corresponding dual string and p-brane models. We find that higher generalizations of the algebraic structures due to Dorfman, Roy…
A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirr…
We study ten-dimensional supersymmetric vacua with NSNS non-geometric fluxes, in the framework of β-supergravity. We first provide expressions for the fermionic supersymmetry variations. Specifying a compactification ansatz to four dimensions, we deduce internal Killing spinor equations. These supersymmetry condition…
This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
Study on metrics on non-geometric 3-manifolds with rigid topological properties.
problem Characterize metrics on non-geometric 3-manifolds.
method Prove lower bounds for systole and volume, deduce local topological rigidity.
result Closed, irreducible manifolds in the class are diffeomorphic if close enough.
We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.
Computes distortion of surfaces in non-geometric 3-manifolds.
problem Distortion of surfaces in non-geometric 3-manifolds.
method Computes distortion of π1(S) in π1(N) and relates it to separability.
result Distortion is linear, quadratic, exponential, or double exponential.
The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.
problem Approximating functionals of non-geometric rough paths.
method Extending rough paths with time and quadratic variation terms, proving uniform approximation.
result Linear functionals of extended signatures uniformly approximate continuous functionals.
Novel approach to financial derivatives pricing using rough path theory.
problem No-arbitrage conditions in financial markets necessitating precise integration methods.
method Developed a polynomial-based approximation class for rough path functionals, extending to non-geometric rough paths.
result Motivated a hypothesis for payoff functionals in financial markets, facilitating analysis.
In this technical note, we adapt an idea of Gabai to construct non-uniquely ergodic, non-geometric, arational trees.
Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic met…
We show that the superconformal symmetries of the (1,1) sigma model decompose into a set of more refined symmetries when the target space admits projectors P±, and the orthogonal complements Q±, covariantly constant with respect to the two natural torsionful connections ∇(±) that arise in the …
Researchers solved the multiple fibration problem for Seifert 3-orbifolds.
problem Determining all inequivalent fibrations of closed orientable Seifert three-orbifolds.
method Geometric and direct arguments for R3 and S2imesR geometries; computer-assisted for S3. result Complete solution for R3 and S2imesR geometries, recovering previous results. New framework robustly handles outliers in Wasserstein DRO for better decision-making.
problem Non-geometric perturbations like adversarial outliers distort Wasserstein distance.
method Proposes an outlier-robust WDRO framework using a robust Wasserstein ball.
result Derives minimax optimal excess risk bounds for robust WDRO.
Unified description of fluxes and T-duality via symplectic manifolds.
problem Unified description of fluxes and T-duality.
method Using supergeometric methods on QP-manifolds and twist of Courant algebroids.
result Unified expressions of NS, F, Q, and R-fluxes in terms of beta- and B-potentials.
New method to write Dirac equation in curved spacetime.
problem Writing Dirac equation in curved spacetime using geometric concepts.
method Using analysis of partial differential equations instead of geometry.
result A non-geometric representation of the Dirac equation in curved spacetime.
We present a ten-dimensional theory, named β-supergravity, that contains non-geometric fluxes and could uplift some four-dimensional gauged supergravities. Building on earlier work, we study here its NSNS sector, where Q- and R-fluxes are precisely identified. Interestingly, the Q-flux is captured in an analogue of the…
Study shows Satake compactifications as horofunctions with polyhedral metrics.
problem Understanding horofunction compactifications of symmetric spaces.
method Investigated invariant Finsler metrics and polyhedral metrics for compactification.
result Generalized Satake compactifications realized as horofunction compactifications.
We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…
A new compactification of outer space is an absolute retract.
problem Compactification of outer space for FN-actions on R-trees. method Defining a new compactification CVN as a blow-up of the classical compactification CVN, assigning orientations to arcs. result The new compactification CVN is an absolute retract. The paper constructs examples of compactifications for Einstein metrics.
problem Compactifying Einstein metrics with specific properties.
method Constructing projective and c--projective compactifications. result Neutral signature Einstein metrics can be compactified canonically.
Extends harmonic maps compactification to punctured Riemann surfaces.
problem Compactifying Teichmüller spaces for punctured Riemann surfaces.
method Using harmonic maps rays to extend compactification.
result The compactification still coincides with Thurston's compactification.
Paper relates new compactification to classical moduli space.
problem Relating new compactification to classical moduli space.
method Defining a continuous finite-to-one surjection between compactifications.
result Uniform upper bound on cardinalities of fibers.
The study classifies and constructs toroidal compactifications of smooth complex hyperbolic surfaces.
problem Classifying and constructing toroidal compactifications of smooth complex hyperbolic surfaces.
method Explicit construction and classification of minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications.
result There are five such surfaces, all arithmetic and associated with commensurable arithmetic lattices.
The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
problem Characterizing the horoboundary of Teichmüller space.
method Using the relationship between the horofunction and visual compactifications of Teichmüller spaces.
result The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.
Existence of Kähler-Einstein metrics on compactifications of Lie groups.
problem Existence of Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. method Proving existence through compactifications of Lie groups.
result Classification of Q-Fano compactifications of SO4(C) with Kähler-Einstein metrics. Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…
Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
problem Characterizing and applying toroidal and semi-toric compactifications to weak K-moduli.
method Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
result Different proof of a theorem of Alexeev-Engel on weak K-moduli compactifications.
The paper explores ball quotient compactifications and their properties.
problem Smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres.
method Use totally geodesic punctured spheres to prove ampleness of KX+αD for α∈(41,1). result First examples of bielliptic ball quotient compactifications are produced.
Study shows infinitely many ways to compactify surfaces as ball quotients.
problem Realizing smooth projective surfaces as ball quotient compactifications.
method Constructing families of distinct ball quotients with biholomorphic smooth toroidal compactifications.
result Arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
Compactifies semi-simple Lie groups to manifolds with corners.
problem Compactification of semi-simple Lie groups.
method hd-compactification to a manifold with corners, 1-1 correspondence with conjugacy classes of parabolic subgroups.
result Harish-Chandra's Schwartz space identified with conormal functions of rapid-logarithmic decay.
New coordinates for Teichmüller space compactification.
problem Compactification of Teichmüller space.
method Defined new coordinates.
result Thurston compactification is radial compactification of Euclidean space.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Embeds Higson compactification into adelic solenoids.
problem Embedding Higson compactification into solenoids.
method Induces isomorphism of 1-dimensional cohomology.
result Higson compactification embeds into adelic solenoids.
Riemannian spaces are shown to be topologically tame via Anosov representations.
problem Understanding the topological nature of Riemannian locally symmetric spaces.
method Constructing compactifications and analyzing properties of Anosov representations.
result These spaces are homeomorphic to the interior of a compact manifold.
We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…
Researchers create a new compactification of character varieties using geometric and algebraic methods.
problem Compactifying character varieties of finitely generated groups in PSL2(R). method Geometric interpretation of elements of the real spectrum compactification as Γ-actions on R-trees, endowed with an orientation. result Continuous surjection from real spectrum compactification to oriented Gromov equivariant compactification.
Paper studies compactifications of Higgs bundles and self-duality equations.
problem Compactification of Hitchin moduli space and Higgs bundles.
method Analyzes maps between algebraic and analytic compactifications.
result Map between compactifications fails to be continuous at boundary over discriminant locus.
This review discusses G2-Manifolds and their role in M-Theory compactifications.
problem Understanding the mathematical structure of G2-Manifolds for M-Theory compactifications. method Mathematical and physical considerations of G2-Manifolds and their compactifications. result Progress in constructing and understanding G2-Manifolds. We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.
Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.
problem Geometric properties preserved by compactifications in relation to coarse structures and group actions.
method Analyzes compactifications of spaces with coarse structures and group actions, proving preservation of geometric properties.
result Geometric properties are preserved by compactifications when coarse structures and group actions are involved.
No natural topological compactification for Fulton-MacPherson.
problem Compactification of configuration spaces on topological manifolds.
method Using recent work on configuration spaces and compactifications.
result Fulton-MacPherson compactification cannot be extended topologically.
The paper classifies and computes limits of equivariant compactifications of groups.
problem Classifying and computing limits of equivariant compactifications of groups.
method Equivariant normal R-test configurations and semistable limits.
result Semistable limits of K-unstable Fano group compactifications are computed.