We describe the topology of the space of all geometric limits of closed abelian subgroups of PSL2C. Main tools and ideas come from the previous paper [BC12].
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We compactify the classical moduli variety of principally polarized abelian varieties of complex dimension by attaching the moduli of flat tori of real dimensions at most in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. Th…
We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits …
New theory connects non-abelian bundle gerbes to abelian ones.
problem Challenges in extending higher gauge theory beyond fake-flat sector.
method Developed a comprehensive theory of adjusted connections on non-abelian bundle gerbes.
result Established a new coordinate-independent formulation of lifting theorem.
We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having a free action on an R^n-tree can be obtained from free abelian groups and surfac…
Study abelian varieties' Weil-Petersson metric asymptotics.
problem Asymptotic behavior of Weil-Petersson metric on abelian varieties.
method Linking asymptotic with multi-scale collapsing limits of parametrized flat tori.
result Refined description of Weil-Petersson metric on abelian varieties.
We consider the Reidemeister torsion associated with SL(2, C)-representations of a knot group. A bifurcation point in the SL(2, C)-character variety of a knot group is a character which is given by both an abelian SL(2, C)-representation and a non-abelian one. We show that there exist limits of the non-acyclic Reidemei…
Study on vortex sheet formation in Abelian gauge theories.
problem Understanding vortex sheet formation in Abelian gauge theories.
method Inspired by Allard's regularity theory, constructs approximate solutions and analyzes their perturbations.
result Establishes a geometric framework and regularity theory for the limiting defect set.
Study of large mass limits of G2 and Calabi-Yau monopoles on specific manifolds.
problem Understanding the behavior of monopoles in the large mass limit on G2 and Calabi-Yau manifolds.
method Developed a structure theory for the limit of -monopoles and Calabi-Yau monopoles, extracting singular abelian G2-monopoles with Dirac singularities.
result Proved an energy identity for monopole bubbles in the large mass limit.
For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.
problem Bounding the dimension of manifolds with nonnegative Ricci curvature and specific fundamental group properties.
method Dimensional estimates for spaces with large Hausdorff dimension.
result If dimension is less than 12, the fundamental group is almost abelian.
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli …
Many physical theories, including notably string theory, require non-abelian higher gauge fields defining higher holonomy. Previous approaches to such higher connections on categorified principal bundles require these to be fake flat. This condition, however, renders them locally gauge equivalent to connections on abel…
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…
For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between t…
We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show…
When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …
Constructs non-abelian G2-instantons on ALC members of B7 family.
problem Constructing non-abelian G2-instantons on ALC members of B7 family.
method Using co-homogeneity one symmetries, classify and describe the solutions as perturbations of abelian instantons.
result Find a one-parameter family of instantons with polynomial decay.
Study of Veech surfaces and their twist tori on moduli spaces of abelian differentials.
problem Distribution of expanding twist tori on moduli spaces of translation surfaces.
method Analysis of Teichmüller geodesic flow and horocycle flow on Veech surfaces.
result Expanding twist tori become dense in the limiting locus as time goes to infinity.
We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan-Warner equations.…
Constructs instantons on a specific G_2-manifold limit.
problem Finding instantons on a particular G_2-manifold limit.
method Dynamical systems approach involving perturbations of abelian and cone solutions.
result Obtains a 1-parameter family of SU(2) instantons with bounded curvature.
Study on mean field games with singular controls and their applications.
problem Optimal productivity expansion in dynamic oligopolies.
method Existence and uniqueness of mean field equilibria through nonlinear equations, Abelian limit for discounted and ergodic games.
result Valid connection between discounted and ergodic games, approximation of Nash equilibria.
The study shows a way to represent Seifert fiber space groups with a limited field size.
problem Representing Seifert fiber space groups with non-abelian .
method Using a triangulation of Seifert fiber spaces and a quotient to limit the field size.
result The lens space recognition problem is shown to be in coNP for Seifert fiber space input.
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singul…
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Generalizing the theorem of Green--Lazarsfeld and Gromov, we classify Kaehler groups of deficiency at least two. As a consequence we see that there are no Kaehler groups of even and strictly positive deficiency. With the same arguments we prove that Kaehler groups that are non-Abelian and are limit groups in the sense …
The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if th…
Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.
problem Rigidity of invariant Ricci flow blowdown limits on nilpotent bundles with zero curvature.
method Construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature.
result Proves blowdown limit is locally an expanding Ricci soliton for three-dimensional Heisenberg group structure group.
We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We…
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…
The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.
problem Understanding Lagrangian structures in Hitchin moduli space.
method Analyzing semistable and polystable Higgs bundles, focusing on the intersection with Lagrangian sublocus.
result The conformal limit of stable Higgs bundles on a specific Lagrangian sublocus exists under certain conditions.
Classifies SU(2)-abelian graph manifolds with a single JSJ torus.
problem Classifying SU(2)-abelian 3-manifolds.
method Using Seifert coefficients and Heegaard Floer homology.
result SU(2)-abelian graph manifolds are Heegaard Floer homology L-spaces.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
3-manifolds with two abelian-filled spaces.
problem Finding 3-manifolds with multiple abelian-filled spaces.
method Constructing 3-manifolds with specific embedding properties.
result 3-manifolds with at least two abelian-filled spaces.
Abstract: Proves non-abelian group of equivariant concordance.
problem Equivariant concordance group non-abelian
method Infinite family of nontrivial commutators
result Equivariant concordance group is not abelian
This is the first in a planned series of papers giving an alternate approach to Zlil Sela's work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a finitely generated group and F is a non-abelian free group.
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the bas…
Characterizes quandles with abelian inner automorphisms.
problem Understanding quandles with specific automorphism properties.
method Generalizes previous work to construct new quandles.
result Homogeneous quandles with abelian inner automorphisms are abelian extensions of trivial quandles.
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
Study abelianization of Lie algebroids and groupoids, providing conditions for existence.
problem Existence conditions for abelianization of Lie algebroids and groupoids.
method Investigation of abelianization for Lie algebroids and groupoids, providing necessary and sufficient conditions.
result Necessary and sufficient conditions for the existence of abelianization in both Lie algebroids and groupoids.
Groups of "flagged homotopies" are introduced of which the usual (abelian for ) homotopy groups is the limit case for flags contracted to a point . Calculus of exterior forms with values in algebra is developped of which the limit cases are differential forms calculus (for $A=\bb R…
The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.