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.
The paper studies quasifuchsian manifolds and their boundary foliations, providing formulas and extensions.
problem Understanding the boundary behavior of quasifuchsian manifolds and their foliations.
method Variation formula for renormalized volume, upper bound on extremal length, extensions of quadratic differential.
result Upper bound on extremal length of horizontal measured foliation and extensions of quadratic differential.
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.
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 …
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.
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…
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.
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. Study proves existence of solutions for a specific type of parabolic equations.
problem Existence of solutions for second boundary value problem of parabolic equations.
method Established Schnextu¨rer's convergence result and applied it. result Existence of solutions for a family of special Lagrangian equations.
We study the cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of Ext1 as intersection numbers of ta…
Study on mapping class groups of infinite type surfaces.
problem Property Pextnaive for mapping class groups of infinite type surfaces. method Analyzes the existence of elements g and hi satisfying specific group properties. result Establishes the existence of g for any finite collection of non-trivial elements hi. Unified framework connects deformation theory and derived categories for multiparameter persistence.
problem Algebraic complexity of multiparameter persistence modules hinders classification, stability, and interpretability.
method Combines deformation theory and derived categories to study multiparameter persistence geometrically.
result Unified conjecture relating interleaving distance to derived convolution metrics established.
Geometric quantization adapted to polysymplectic manifolds.
problem Quantization of polysymplectic manifolds.
method Adapted geometric quantization framework to polysymplectic setting.
result Polysymplectic Guillemin-Sternberg conjecture is shown to be false with a complex polarization.
New PROP structure defined for automorphism group cohomology.
problem Cohomology of automorphism groups of free groups with specific coefficients.
method Definition of wheeled PROPs and construction of morphisms.
result Construction of morphism φ: E → H with φ(E) = h_1.
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
Introduces a new characteristic class for vector bundles with a connection.
problem Tackles the classification of vector bundles with algebraic connections.
method Defines a new characteristic class using a connection and proves its independence of the choice of connection.
result The class c(E) is an invariant of the vector bundle E and is stronger than the Chern and Euler classes. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as gr…
The paper explores geometric and algebraic structures on Lie groups.
problem Investigating F-manifolds and Fextman-algebras on Lie groups. method Constructing a canonical connection and analyzing curvature and holonomy.
result Established the integrability of a Poisson-algebra distribution.
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…
Degenerations of rank-two bundles on threefolds lead to isolated point singularities, with rigidity and bubbling properties.
problem Degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity.
method Proving a rigidity identity and using it to obtain smoothability obstructions and construct local smoothings.
result Smoothability obstructions and local smoothings are obtained, with a rigidity identity linking algebraic bubbling multiplicity and Ext-length.
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for a…
Develops Morse homology with DG coefficients for manifolds and spaces.
problem Homology with DG coefficients for manifolds and spaces.
method Derived local systems, DG modules, twisting cocycles, Morse trajectories.
result Isomorphic to DG Tor and Ext functors, recovers homology of total space of fibrations.
We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum…
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …
Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).
problem Homology and cohomology of Temperley-Lieb algebra TL_n(a).
method Homological stability and computation of stable homology. Use of chain complex of 'planar injective words'.
result Vanishing of homology and cohomology up to degree (n-2) under certain conditions.
The paper studies quasi-X-convex functions and their applications in optimization.
problem Optimization problems with quasi-X-convex functions. method Definition and study of X-convex, quasi-X-convex, and related functions. result Applications of quasi-X-convex functions in optimization problems. Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. Geodesic convex optimization extends convex optimization to manifolds.
problem Optimizing non-convex functions on manifolds.
method Introducing geodesic convexity on manifolds.
result Certain non-convex problems can be formulated as geodesically convex optimization problems.
2-convex translating solitons are locally strictly convex.
problem Characterizing the convexity of translating solitons in mean curvature flow.
method Analyzing uniformly 2-convex translating solitons in Rn+1. result Locally strictly convex translating solitons are axisymmetric.
The paper introduces geodesic φ-convex functions and their properties.
problem Generalizing geodesic functions to φ-convex functions.
method Introducing geodesic φ-convex functions and investigating their properties.
result Characterization of geodesic φ-convex functions via their φ-epigraphs.
CoRR learns convex relaxations for smooth functions via random point evaluations.
problem Solving non-convex optimization problems efficiently and provably.
method Estimating convex envelopes of smooth functions by fitting a convex function to random point evaluations.
result The solution of CoRR converges to the global optimizer of a function with a specified error rate.
Random convex analysis tackles problems in random environments.
problem Dealing with problems in random environments like conditional convex risk measures.
method Developing random convex analysis over random locally convex modules, establishing inferior limit behavior, continuity, subdifferentiability, and approximating ε-subdifferentials.
result Established relationships among subdifferentiability, Gâteaux-differentiability, and Fréchet-differentiability for proper L0-convex functions. The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. Introduces GG-convex risk measures and derives their dual representations.
problem Defining and studying GG-convex risk measures.
method Introduces GG-convex conjugate, derives dual representations, and studies Orlicz risk measures.
result Derives a general dual representation for GG-convex risk measures.
Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Convex clustering can only learn convex clusters, with significant gaps between clusters.
problem Understanding the limitations and capabilities of convex clustering.
method Analyzing convex clustering solutions, proving properties, and characterizing clusters.
result Convex clustering can only learn convex clusters with significant gaps between clusters.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
New convexity concept applied to sphere yields quermassintegral inequalities.
problem Proving quermassintegral inequalities for horo-convex hypersurfaces on the sphere.
method Smooth convergence of Guan/Li flow for inverse type applied to horo-convex hypersurfaces.
result Full set of quermassintegral inequalities for horo-convex hypersurfaces proved.
First-order methods tackle g-convex optimization on Hadamard manifolds.
problem Geodesically convex optimization on nonlinear metric spaces.
method Iteration complexity analysis for first-order algorithms.
result Upper bounds for global complexity of g-convex optimization.
Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. Geodesic convexity types differ in Riemannian manifolds.
problem Characterizing geodesic convexity types in Riemannian manifolds.
method Reverse engineering to characterize manifolds with coinciding convexity types.
result Characterized complete manifolds with coinciding geodesic convexity types.
Study convex projective manifolds with compact and convex ends.
problem Holonomy of convex projective manifolds.
method Extend Koszul's theorem to non-compact manifolds with boundary.
result Holonomies of properly convex structures form an open subset of the representation variety.
In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n+2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W, which we call \emph{convex open book}, induced b…
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
The paper shows that g-convex functions on manifolds are sparse.
problem Characterizing and understanding the sparseness of g-convex functions.
method Established criteria for g-convexity and used them to prove sparseness results.
result Most g-convex functions on compact manifolds have few critical points.