The study examines higher-order modern portfolio theory with complex critical points and feasible portfolio variety.
problem Understanding the complex critical points and feasible portfolio variety in higher-order modern portfolio theory.
method Established genericity conditions for utility functions with higher-order cumulants, analyzed discriminant loci, and determined the dimension and degree of the feasible portfolio variety.
result The utility function has a constant number of complex critical points under genericity conditions, and the feasible portfolio variety has a determined dimension and degree.
The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.
problem Establishing lower bounds on the number of critical points of functions using topological complexity.
method Applying Lusternik-Schnirelmann theory to sequential and parametrized topological complexity.
result Established various lower bounds on the number of critical points using sequential and parametrized topological complexity.
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.
Noninjective monodromy found in polynomial critical point tracking.
problem Tracking critical points in polynomials leads to noninjective monodromy.
method Monic squarefree complex polynomials with prescribed critical point multiplicities.
result Monodromy map is noninjective for polynomials with exactly two critical points.
New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex wi…
Study on critical faces convergence in a Poisson point process.
problem Convergence of point processes associated with critical faces in a Čech filtration.
method Established convergence in M0-topology for critical faces above vanishing threshold. result Obtained limit theorems for positive and negative critical faces.
We present a method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents a substantial extension of previous applications of the Kac-Rice method since it allows to analyze the critical points…
This work analyzes actor-critic methods for faster convergence.
problem Finite-time analysis and sample complexity of two-time-scale actor-critic methods.
method Non-asymptotic analysis under non-i.i.d. setting, proving convergence to first-order stationary point.
result Actor-critic method finds a first-order stationary point with ildeO(ε−2.5) sample complexity. Novel Morse theory for mapping cone cohomology.
problem Cohomology of mapping cones varies with closed forms.
method Introduced a Morse complex for mapping cones.
result Cohomology of cone Morse complex is isomorphic to mapping cone cohomology.
We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a …
Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.
problem Complexity of random Gaussian polynomials with deterministic spikes on a sphere.
method Variational formulas, Kac-Rice formula, determinant asymptotics of finite-rank perturbation of Gaussian Wigner matrices.
result Identification of a topological phase transition in the complexity function.
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
New approach finds Kähler metrics on compact complex manifolds.
problem Finding Kähler metrics on compact complex manifolds.
method Defining a new functional whose critical points are Kähler metrics.
result Critical points of the new functional are precisely the Kähler metrics.
Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map θ:Cn→Cn which outputs the critical values of the canonical polynomial constructed from the…
The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the C∞ topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a…
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
Abstract: Characterizes special Kähler manifolds with specific properties.
problem Characterizing Kähler manifolds with special properties.
method Analyzes properties of functions and gradients on manifolds.
result Characterizes manifolds supporting certain functions and gradients.
We study critical points of the Ginzburg-Landau (GL) functional and the abelian Yang-Mills-Higgs (YMH) functional on the sphere and the complex projective space, both equipped with the standard metrics. For the GL functional we prove that on Sn with n≥2 and CPn with n≥1, stable critical…
Study of polynomial strata using braid groups and translation surfaces.
problem Understanding the monodromy of polynomial strata.
method Using infinite-area translation surfaces and braid groups.
result Determine the monodromy of polynomial strata in the braid group.
We consider the configuration space of planar n-gons with fixed perimeter, which is diffeomorphic to the complex projective space CPn−2. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …
Embedding principle explains loss landscape of deep neural networks.
problem Understanding the structure of loss landscapes in deep neural networks.
method Proposed an embedding principle that critical points of narrower DNNs can be embedded to critical points of wider DNNs.
result Wide DNNs are often attracted by highly-degenerate critical points embedded from narrower DNNs.
This paper improves sample complexity for AC and NAC algorithms under Markovian sampling.
problem Improving sample complexity for actor-critic and natural actor-critic algorithms.
method Characterizes convergence rate and sample complexity under Markovian sampling and mini-batch data.
result Improves sample complexity for AC and NAC algorithms by orders of magnitude.
Study irrational pencils on complex manifolds, finding non-finitely generated homology.
problem Understanding the homology of the kernel induced by irrational pencils on complex manifolds.
method Analyzing critical points and homology of fundamental groups of complex manifolds.
result Homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated.
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.
Machine learning detects tipping points in complex systems.
problem Detecting abrupt shifts in complex dynamical systems.
method Equilibrium-informed neural networks (EINNs) trained on candidate equilibrium states.
result EINNs can identify critical thresholds in nonlinear systems.
The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of crit…
Stock markets are complex systems exhibiting collective phenomena and particular features such as synchronization, fluctuations distributed as power-laws, non-random structures and similarity to neural networks. Such specific properties suggest that markets operate at a very special point. Financial markets are believe…
Study the relationship between braids formed by roots and critical points of polynomials.
problem Relationship between braid formed by roots and braid formed by critical points of polynomials.
method Analyzing pseudo-fibrations and fibrations of complex polynomials.
result For T-homogeneous braids, the pseudo-fibration can be a fibration.
This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
Let M be a smooth closed orientable surface and F=Fp,q,r be the space of Morse functions on M having exactly p critical points of local minima, q≥1 saddle critical points, and r critical points of local maxima, moreover all the points are fixed. Let Ff be the connected component of a function $f\in …
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
CNMs detect tipping points in complex systems using causal network markers.
problem Identifying tipping points ahead of critical transitions in complex systems.
method Introducing CNMs that incorporate causality indicators to detect tipping points.
result CNMs show higher predictive power and accuracy than traditional DNB indicators.
Adapted metrics found on complex manifolds.
problem Finding metrics suitable for complex manifolds.
method Characterizing adapted metrics as critical points of a functional.
result Gauduchon metric is adapted on locally conformally product manifolds.
In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical poin…
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M having fixed number of critical points of each index, moreover at least χ(M)+1 critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a …
The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
Study analyzes landscape complexity of empirical loss functions with correlated data.
problem Understanding the complexity of loss landscapes in machine learning with structured data.
method Kac-Rice formula and random matrix theory applied to high-dimensional empirical loss functions.
result Characterizes the average number of critical points in loss functions with structured data.
The study characterizes complex structures using calculus of variations.
problem Variational characterization of complex structures.
method Calculus of variations for real vector bundle valued differential forms.
result Obtains variational characterization of complex structures.
For a fixed smooth map u0 between two Riemann surfaces Σ and S with non-zero degree, we consider the energy function on Teichmüller space $\mc{T}$ of Σ that assigns to a complex structure $t\in \mc{T}$ on Σ the energy of the harmonic map ut:Σt:=(Σ,t)→S homotopic to u0. We prove that the energy fun…
A new method identifies critical transitions in high-dimensional data.
problem Challenges in identifying critical transitions in high-dimensional time-series data.
method Spatial-temporal Principal Component Analysis (stPCA)
result Identifies tipping points before critical transitions reliably.
For a Morse function f on a compact oriented manifold M, we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is large…
Survey on manifold complexities and motion planning in robotics.
problem Understanding topological complexities of manifolds in robotic motion planning.
method Overview of topological complexities, geodesic motion planning, and connections to critical point theory.
result Estimation of motion planning complexity using Riemannian geometry and critical point theory.
We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated clo…