Study balancing embedding of degenerating Abelian varieties.
problem Balancing embedding of degenerating Abelian varieties.
method Maximal unipotent family over punctured disc, base change, theta functions, Gromov-Hausdorff limit.
result Balanced embedding of degeneration into projective space.
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
Study the asymptotics of holomorphic maps from polydisk to disk.
problem Understanding the asymptotics of holomorphic maps from polydisk to disk.
method Analyzing the conjugation action of a unipotent subgroup of PSL2(R).
result Results in the asymptotics of the translation flow on holomorphic maps.
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in Sp4(R). We study subgroups of PU(2,1) generated by two non-commuting unipotent maps A and B whose product AB is also unipotent. We call U the set of conjugacy classes of such groups. We provide a set of coordinates on U that make it homeomorphic to R2 . By considering the actio…
Identifies filtration in Lagrangian fibrations to monodromy weight filtration in degenerations.
problem Understanding the relationship between Lagrangian fibrations and degenerations of hyper-Kähler manifolds.
method Identifies and compares perverse filtration with monodromy weight filtration.
result Identifies the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a degeneration.
The paper proves orbit closure theorems for hyperbolic manifolds with Fuchsian ends.
problem Analyzing the orbit closures of unipotent flows on hyperbolic manifolds with specific ends.
method Establishing an analogue of Ratner's orbit closure theorem for unipotent subgroups in SO(d,1).
result Proves the closure of k-horospheres and (k+1)-geodesic planes are properly immersed submanifolds. Study unipotent group actions on Kähler manifolds using moment maps.
problem Analyzing unipotent group actions on compact Kähler manifolds.
method Moment map techniques, stability conditions, symplectic reduction.
result Geometric quotients with compactifiable Kähler structures.
Reconfigures Milnor invariants using unipotent Magnus embeddings.
problem Computing and understanding Milnor invariants of links.
method Central group extensions and unipotent Magnus embeddings.
result First non-vanishing invariants and refinements of higher degree invariants computed.
Proves polynomial error rate for equidistribution of unipotent flows.
problem Equidistribution of orbits of unipotent subgroups in arithmetic quotients.
method Uses Margulis function, incidence geometry tools, and spectral gap.
result Polynomial error rate for equidistribution theorems.
Effective Lie-Kolchin theorem for quasi-unipotent matrices with common eigenvector.
problem Establishing conditions for quasi-unipotent matrices to have a common eigenvector.
method Using the Jordan Canonical Form and properties of quasi-unipotent matrices.
result Quasi-unipotent matrices A and B have a common eigenvector under certain conditions. Constructs CAT(0) actions for certain groups without unipotent elements.
problem Understanding actions of certain groups on CAT(0) spaces.
method Constructs an isometric action of a group on a CAT(0) space.
result Fundamental groups of certain 3-manifolds do not admit faithful finite-dimensional unitary representations.
Unipotent automorphisms on complex supermanifolds are studied for classification.
problem Classifying supermanifolds based on unipotent automorphisms.
method Analyzing even global degree increasing vector fields and their kernels.
result Classification of strictly t-nildominated supermanifolds up to degree t. In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds with carefully controlled asymptotics near the compactifying divisor; such a m…
Classifies specific types of Lorentzian manifolds with unipotent holonomy.
problem Classifying closed conformally flat Lorentzian manifolds with unipotent holonomy.
method Analyzing the geometric properties and developing maps of the manifolds.
result Identifies four geometric types and two additional types of manifolds.
New group preserves convex shape without repeating patterns.
problem No virtually abelian unipotent group preserves strictly convex domains.
method Examined Heisenberg group acting on convex shape.
result First example of non-virtually abelian unipotent group preserving strictly convex domain.
Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.
The paper extends McShane identities to higher Teichmüller theory and convex real projective surfaces.
problem Deriving McShane identities for higher Teichmüller theory.
method By studying mapping class group invariant functions and generalizing horocycle lengths.
result Established McShane-type identities for various surface group representations.
We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor…
Linear groups without infinite order unipotents have good properties.
problem Properties of linear groups without unipotent elements of infinite order.
method Analyzing the structure and properties of linear groups without unipotent elements of infinite order.
result Linear groups without unipotent elements of infinite order have good properties, including centralisers virtually splitting and finitely generated abelian subgroups being undistorted.
New CR representations are found and shown to be redundant.
problem Identifying and classifying CR representations of 3-manifolds.
method Experimental computation of limit sets and exact computations of triangle groups.
result Many CR representations are redundant and conjugate.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …
Refines mixed Hodge structures using non-abelian Hodge theory.
problem Mixed Hodge structures on Sullivan's minimal models.
method Non-abelian Hodge theory.
result Explicit representatives of real unipotent variations.
The Ptolemy variety for SL(2,C) is an invariant of a topological ideal triangulation of a compact 3-manifold M. It is closely related to Thurston's gluing equation variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the gluing equation var…
For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces d…
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…
We consider the discrete representations of 3-manifold groups into PU(2,1) that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…
Study formalities on closed surfaces using connections.
problem Formalities of Goldman-Turaev Lie bialgebra on closed surfaces.
method Reformulated Kashiwara-Vergne groups and associators in higher genera using non-commutative connections.
result Determined pro-unipotent automorphism group of associated graded.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.
Solves Deligne-Simpson problem for special connections on Gm.
problem Existence of Fuchsian connections with specific singularities.
method Theory of fundamental and regular strata, lattice chain filtration, quiver varieties.
result Characterization of rigid connections with unipotent monodromy at infinity.
Study counts and parametrizes flag components in SO0(p,q) space.
problem Counting and characterizing flag components in SO0(p,q) space.
method Parametrization and computation of Plücker coordinates.
result Anosov subgroups are virtually isomorphic to surface or free groups.
The paper concerns a compactification of the isospectral varieties of nilpotent Toda lattices for real split simple Lie algebras. The compactification is obtained by taking the closure of unipotent group orbits in the flag manifolds. The unipotent group orbits are called the Peterson varieties and can be used in the co…
New algorithm for Coxeter connections with maximally ramified singularities.
problem Constructing connections on the projective line with a maximally ramified irregular singularity.
method Numerical algorithm for matrix completions to solve the Upper Nilpotent Completion Problem.
result Explicit constructions of Coxeter connections with specified singularities.
A graph connects Specht and web bases; matrix is unipotent with vanishing entries.
problem Comparing two bases of irreducible representations of the symmetric group.
method Graph theory and combinatorial analysis to describe relations between bases and prove properties of the transition matrix.
result The transition matrix between Specht and web bases is unipotent with additional vanishing entries.
New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
Researchers provide coordinates for Lie group configurations, linking to Teichmüller space and knot representations.
problem Finding coordinates for Lie group configurations.
method Explicit Fock-Goncharov coordinates for triples and quadruples of affine flags in rank 2 Lie groups.
result Explicit coordinates on higher Teichmüller space and boundary-unipotent representations of 3-manifold groups.
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
New bounds on maximal linkless graphs with improved edge-to-vertex ratios.
problem Finding maximal linklessly embeddable graphs with improved edge-to-vertex ratios.
method Constructing families of graphs and proving necessary and sufficient conditions for clique sums.
result Improved edge-to-vertex ratios for maximal linklessly embeddable graphs.
Classifies generalized cusps in real projective manifolds.
problem Classifying geometric structures on cusps in real projective manifolds.
method Using affine groups and Bieberbach groups to classify cusps.
result Finite Busemann measure condition for cusp classification.
We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.
problem Generalizing Lusztig's total positivity to the setting of general real semisimple Lie groups.
method Classifying Lie groups admitting a positive structure and establishing key properties of unipotent positive semigroups.
result Establishes key geometric properties of elements in the positive semigroup.
Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.
problem Deriving symplectic structures from Torelli groups without holomorphic structures.
method Symplectization of mapping tori derived from Torelli group representatives.
result Symplectic manifolds with complex structure invariants but no holomorphic structures.
The paper describes decompositions of geometric measures on Anosov homogeneous spaces.
problem Decomposing geometric measures on Anosov homogeneous spaces.
method Ergodic decompositions of Burger-Roblin and Bowen-Margulis-Sullivan measures.
result The space of non-trivial invariant ergodic measures is homeomorphic to a product space.
Let X be a projective manifold, and D be a normal crossing divisor of X. By Jost-Zuo's theorem that if we have a reductive representation ρ of the fundamental group π1(X∗) with unipotent local monodromy, where X∗=X−D, then there exists a tame pluriharmonic metric h on the flat bundle V a…