Study on integrals involving differential forms and convexity properties.
problem Integrals of differential forms and their convexity properties.
method Introduce and analyze ext. convexity, quasiconvexity, and polyconvexity for differential forms.
result Relations and examples of ext. convexity, quasiconvexity, and polyconvexity.
Study quasiconvex risk measures in volatile financial markets.
problem Financial risk measurement in markets with variable volatility.
method Defined quasiconvex risk measures on Lp(⋅) space with p(⋅) as a random variable. result Deduced dual representation for the defined quasiconvex risk measures.
Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.
problem Understanding natural quasiconvexity and its implications in risk measures.
method Relates natural quasiconvexity to decomposable sums, proposes a general treatment of convexity index, and proves equivalence for certain spaces.
result Natural quasiconvexity and convexity are equivalent for conditional risk measures on Lp spaces under mild conditions. Exact sequence results show infinite index quasiconvex subgroups in certain groups.
problem Understanding quasiconvex subgroups in specific group structures.
method Analyzing exact sequences of hyperbolic groups and their subgroups.
result Infinite index quasiconvex subgroups in G1 are quasiconvex in G under mild conditions. Computes cohomology of Steenrod algebra for k ≤ 5.
problem Determining a basis of cohomology for Steenrod algebra.
method Algorithm based on generators and Adams relations.
result Verification of hand-computed results for k = 4.
New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups Q1 and Q2 is relatively quasiconvex and isomorphic to Q1∗Q1∩Q2Q2. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…
We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic group G. The first result establishes equivalent conditions for H to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a questi…
Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general…
Study of quasiconvex subgroups in 3-manifold groups.
problem Characterize quasiconvex subgroups in 3-manifold groups.
method Analyzes strongly quasiconvex subgroups in finitely generated 3-manifold groups.
result Characterizes quasiconvex subgroups in graph manifold groups and 3-manifold groups.
We provide a dual representation of quasiconvex maps between two lattices of random variables in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions and the dual representation of conditional convex maps.
Characterizes strongly quasiconvex subsets in hierarchically hyperbolic spaces.
problem Understanding the structure of strongly quasiconvex subsets in HHSs.
method Characterization through contracting properties, relative divergence, and hierarchical structure.
result Proves characterization of hyperbolically embedded subgroups in hierarchically hyperbolic groups.
Disc graphs are uniformly quasiconvex in curve graphs of surfaces.
problem Characterizing quasiconvexity in curve graphs of surfaces.
method Proof using a universal constant K without train tracks.
result Disc graphs are K-quasiconvex in curve graphs.
New condition controls quasiconvex subgroups in hyperbolic 3-manifolds.
problem Controlling behavior of quasiconvex subgroups in hyperbolic 3-manifolds.
method New condition on relatively hyperbolic Dehn filling.
result Provides new proof of virtual fibering of hyperbolic 3-manifolds.
We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H de…
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.
New techniques reveal subgroup properties in Coxeter groups.
problem Characterizing and understanding subgroups of right-angled Coxeter groups.
method Using cube complexes and Stallings-like techniques to study subgroups.
result Reflection and one-ended subgroups are quasiconvex.
In the conditional setting we provide a complete duality between quasiconvex risk measures defined on L0 modules of the Lp type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.
Study dual representations for quasiconvex systemic risk measures.
problem Finding dual representations for quasiconvex systemic risk measures.
method Abstract infinite-dimensional setting, explicit formula for penalty function, nonstandard minimax inequality.
result Explicit formula for the penalty function of quasiconvex compositions.
The study characterizes subgroup stability in finitely generated groups using the Morse boundary.
problem Stability of subgroups in finitely generated groups.
method Using the Morse boundary, an equivalent characterization of subgroup stability is developed.
result Subgroup stability coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups.
The paper introduces new geometric concepts to study quasiconvexity in hyperbolic groups.
problem Characterizing quasiconvexity in hyperbolic and relatively hyperbolic groups.
method Introducing geometric height and graded relative hyperbolicity, using these to characterize quasiconvexity.
result Characterization of quasiconvexity in hyperbolic and relatively hyperbolic groups.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
It has been shown recently that the geometry of D-branes in general topologically twisted (2,2) sigma-models can be described in the language of generalized complex structures. On general grounds such D-branes (called generalized complex (GC) branes) must form a category. We compute the BRST cohomology of open strings …
A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric sp…
The paper examines conditions for the equator map to be minimizing or unstable for higher order energy functionals.
problem Conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
method Analyzes the extrinsic k-energy functional and the equator map to establish conditions for minimization or instability.
result Establishes necessary and sufficient conditions for the equator map to be minimizing or unstable for extrinsic k-energy functionals.
New homological results for bordered Floer algebras derived from hypertoric categories.
problem Homological properties of bordered Floer algebras.
method Affine quasi hereditary property of equivariant hypertoric convolution algebras and computation of Ext groups.
result Existence of standard modules and isomorphism of Ext groups to bordered strands dg algebras.
Groups grow faster than their subgroups, proven using special tools.
problem Growth rates of subgroups within hyperbolic groups.
method Automatic structures, Perron-Frobenius theory, growth tightness, rotating families.
result Non-elementary hyperbolic groups grow exponentially faster than their quasiconvex subgroups.
Classifies hyperbolic groups with surface-like boundaries.
problem Classifying hyperbolic groups with specific surface-like boundaries.
method Analyzing quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
result Identifies hyperbolic groups with surface-like boundaries.
The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.
problem Conditions for amalgamation of subgroups in hierarchically hyperbolic groups.
method Study of amalgamation conditions and preservation of convexity properties.
result Conditions under which amalgamation preserves hierarchical quasiconvexity and strong quasiconvexity.
Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to k-Hessian equations in exterior domains. method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to k-Hessian equations are not quasiconvex in exterior domains. New hierarchy for a special group type.
problem Classifying relatively hyperbolic virtually special groups.
method Constructing a new virtual quasiconvex hierarchy.
result Generalized Malnormal Special Quotient Theorem.
Let Sg,p denote the genus g orientable surface with p punctures. We show that nested train track sequences constitute O((g,p)2)-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus g disk set is O(g2)-quasiconvex. We also show that splitti…
Stable and Morse subgroups coincide in mapping class groups.
problem Understanding subgroup properties in mapping class groups.
method Analyzing stability and Morse properties in mapping class groups.
result Stability and Morse properties coincide for subgroups of infinite index in mapping class groups.
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham coh…
Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the f…
Ridge regression CV loss may have multiple local optima.
problem Can we globally optimize cross-validation loss in ridge regression?
method Analyzing quasiconvexity of CV loss in ridge regression.
result CV loss may fail to be quasiconvex and have multiple local optima.
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…
For any finitely generated, non-elementary, torsion-free group G that is hyperbolic relative to P, we show that there exists a group G∗ containing G such that G∗ is hyperbolic relative to P and G is not relatively quasiconvex in G∗. This generalizes a result of I. Kapovich for hyperbo…
The paper proves conditions for cubulating surface-by-free groups.
problem Proving conditions for cubulating surface-by-free groups.
method Combination theorem for virtually special hyperbolic groups, tracks theory, quasiconvex hierarchy theorem, distance estimates, and model geometry.
result Main result: sufficient conditions for cubulability of G. This paper confirms Singer's conjecture for rank 4 in specific generic degrees.
problem Determining the injectivity of the algebraic transfer for rank 4.
method Using the Singer algebraic transfer and novel algorithms.
result Established Singer's conjecture for rank four in specific generic degrees.
We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompac…
The paper develops algorithms to detect stability and Morse properties in various groups.
problem Detecting stability and Morse properties in finitely generated groups.
method Various detection and decidability algorithms for stability and Morse properties in specific types of groups.
result The algorithms provide a way to determine if a subgroup is stable or Morse in specific group types.
A random group contains many quasiconvex surface subgroups.
The paper studies extPin±-structures on non-oriented 4-manifolds via Lefschetz fibrations.
problem Understanding extPin±-structures on non-oriented 4-manifolds and Lefschetz fibrations. method Extending work on orientable settings, the paper uses Lefschetz fibrations to analyze extPin±-structures on non-orientable 4-manifolds and vector bundles. result The paper provides existence results of extPin+ and extPin−-structures on closed non-orientable 4-manifolds and Lefschetz fibrations over the 2-sphere. Extends portfolio optimization with two quasiconvex risk measures.
problem Optimizing portfolios with dual risk measures for multiple stakeholders.
method Dual problem formulation, bisection algorithm, duality results.
result Approximately optimal solutions can be achieved with prescribed optimality gap.