Complex analytic sets' Lipschitz geometry at infinity characterized.
problem Characterize entire complex analytic sets based on their Lipschitz geometry at infinity.
method Proved a complex non-parametric version of Moser's Bernstein Theorem and characterized algebraicity.
result Entire complex analytic sets at infinity are affine linear subspaces if and only if they are bi-Lipschitz homeomorphic to algebraic sets.
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
Finite rigid sets found in sphere complexes for some but not all cases.
problem Characterizing finite rigid sets in sphere complexes.
method Analyzing locally injective maps and automorphisms.
result Finite rigid sets exist for n≥3 but not for n=2. Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Finite rigid sets found in surface curve complexes.
problem Finding rigid sets in surface curve complexes.
method Incidence-preserving maps to find rigid subcomplexes.
result Finite rigid subcomplexes identified in surface curve complexes.
A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus g with n holes …
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface wit…
Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.
problem Characterizing limit sets of knots in complex hyperbolic geometry.
method Analyzing embeddings of knots as limit sets of discrete subgroups of PU(2, 1).
result Knots are either chains or R-circles as limit sets.
The paper sets sample complexity bounds for identifying LTI systems from a finite set.
problem Identifying an LTI system from a finite set of possible systems using trajectory data.
method Maximum likelihood estimator and information theory tools.
result Upper and lower bounds for sample complexity are derived, independent of stability assumption.
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. Study fixed-point sets of S1-actions on quaternionic manifolds.
problem Characterize fixed-point sets and compatible complex structures on quaternionic manifolds.
method Analyze fixed-point sets and derive equations involving first Chern classes.
result Conditions for the existence of hypercomplex structures on quaternionic manifolds.
New method for identifying best designs in vector optimization with uncertain feedback.
problem Optimizing vector-valued outcomes with uncertain preferences.
method Stochastic bandit feedback, polyhedral ordering cone, (ε,δ)-PAC Pareto set identification. result Sample complexity characterized and matched by the naïve elimination algorithm.
Determines higher smooth surgery structure sets of complex projective spaces.
problem Understanding the higher smooth surgery structure sets of complex projective spaces.
method Analyzes the free subgroup and torsion in low dimensions.
result Obtains information in all dimensions for the free subgroup.
Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We prove that this complex is contractible, which was conjectured by Kakimizu. More ge…
Analyzes the complexity of linear hypothesis sets using Rademacher complexity.
problem Understanding the complexity of linear hypothesis sets for various norms.
method Tight analysis of empirical Rademacher complexity for linear hypothesis classes with bounded weights.
result Improved bounds on Rademacher complexity for linear hypothesis sets, matching or improving existing results.
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold S controls the coisotropic deformation problem of S under both Hamiltoni…
Open problem: fixed-budget best arm identification complexity.
problem Understanding the complexity of identifying the best arm in a fixed budget setting.
method Analyzing existing results and conjectures in the fixed-confidence setting.
result Open questions remain about the fixed-budget setting.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
Fixed point sets of certain group actions are contractible.
problem Fixed point sets of group actions on specific types of complexes.
method Analyzing group actions on diagrammatically reducible complexes with fine 1-skeleton.
result Fixed point sets are contractible under certain conditions.
In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…
We present a study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main result is a generalization bound for data-dependent hypothesis sets expressed in terms of a notion of hypothe…
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating…
The paper analyzes heat trace asymptotics for de Rham and Dolbeault complexes in both real and complex settings.
problem Examining heat trace asymptotics for de Rham and Dolbeault complexes in different geometric settings.
method Analyzing the derived heat trace asymptotics for generalized Witten perturbations in both real and complex settings.
result The integral of the local density for the derived heat trace asymptotics is related to the Euler characteristic and characteristic numbers of the tangent and twisting vector bundles.
Fix a finite set of points in Euclidean n-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. …
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have C1 boundary, and have word hyperbolic divid…
We present two sets of theoretical results on the grouped lasso with overlap of Jacob, Obozinski and Vert (2009) in the linear regression setting. This method allows for joint selection of predictors in sparse regression, allowing for complex structured sparsity over the predictors encoded as a set of groups. This flex…
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
A new definition for vector fields extends the Jacobi set concept.
problem Describing interactions between vector fields on complex domains.
method Piecewise linear approach for simplicial complexes.
result Generalizes Jacobi set concept to vector fields.
Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
This paper studies fixed sets in ribbon complexes using descriptive proximity spaces.
problem Understanding fixed sets in ribbon complexes within descriptive proximity spaces.
method Introduces descriptive fixed sets and their properties in ribbon complexes, using descriptive proximally continuous maps.
result Establishes that proximal descriptive conjugacy preserves fixed sets in ribbon complexes.
Examines properties of holomorphic fibrations in complex geometry.
problem Holomorphic fibrations in complex geometry.
method Analyzes properties of holomorphic fibrations.
result Provides insights into the structure of holomorphic fibrations.
Let S be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex C(S) by a sequence of finite rigid sets.
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…
Improved bounds for function approximation in nonlinear sets.
problem Achieving high probability error with limited samples in nonlinear function approximation.
method Restricting model class to a neighbourhood of the best approximation and estimating sample complexity using tangent and normal spaces' complexities and curvature.
result Improved worst-case bounds for sample complexity in more general sets like tensor networks and neural networks.
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the …
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
New framework improves worst-case generalization bounds for stochastic optimization.
problem Challenges in providing generalization guarantees for stochastic optimization algorithms.
method Introduces random set stability and empirically relevant complexity measures to avoid intractable mutual information terms.
result Bounded worst-case generalization error in terms of random set stability and empirically relevant complexity measures.
CantorNet tests geometric and topological complexity in neural networks.
problem Understanding self-similar patterns in neural networks.
method Inspired by Cantor set, CantorNet introduces novel complexity measures.
result CantorNet's decision boundaries are analytically known and can be arbitrarily ragged.
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.
In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk i…
Paper analyzes complexity of solving nonconvex-strongly-concave problems.
problem Finding approximate stationary points of nonconvex-strongly-concave minimax problems.
method Introduces a generic acceleration scheme to solve crafted subproblems.
result Algorithm nearly matches lower complexity bounds in general setting.