arXiv research

Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.

Lower bound for complexity of finding flex points on cubic curves.

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.

In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provi…

Groups with special properties always have fixed points.

Study fixed-point sets of $S^{1}$-actions on quaternionic manifolds.

Fixed point sets of certain group actions are contractible.

Paper defines and evaluates DR complex for persistent homology.

The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the per…

The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.

The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle act…

The study examines higher-order modern portfolio theory with complex critical points and feasible portfolio variety.

This paper tackles the computational complexity of finding approximate stationary points in non-convex optimization.

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

In this paper we investigate surfaces in $\mathbb C P^2$ without complex points and characterize the minimal surfaces without complex points and the minimal Lagrangian surfaces by Ruh-Vilms type theorems. We also discuss the liftability of an immersion from a surface to $\mathbb C P^2$ into $S^5$ in Appendix A.

This paper provides a new proof of the Lefschetz fixed point formula using groupoids.

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

The paper classifies and constructs 6D GKM manifolds with 4 fixed points.

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for the configuration spaces of three collinear and non-collinear points.

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 $D$. …

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.

The paper tackles efficient change point detection with limited samples.

Transductive learning considers situations when a learner observes $m$ labelled training points and $u$ unlabelled test points with the final goal of giving correct answers for the test points. This paper introduces a new complexity measure for transductive learning called Permutational Rademacher Complexity (PRC) and …

New method finds stationary points in bilevel optimization problems.

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus …

Paper introduces a neural network-based non-stationary influence kernel for complex event data.

Research examines correlations of complex logarithms of lattice points, showing level repulsion and Poissonian behavior.

We describe the fundamental groups of ordered and unordered k point sets in complex projective space of dimension n generating a projective subspace of dimension i. We apply these to study connectivity of more complicated configurations of points.

We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence…

The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regu…

GOCPD detects change points by maximizing the probability of two independent models.

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is co…

The braid group of a complex reflection group is shown to be an index d subgroup.

Inspired by constructions in complex geometry we introduce a thermodynamic framework for Monge-Ampère equations on real tori. We show convergence in law of the associated point processes and explain connections to complex Monge-Ampère equations and optimal transport.

Consider a circle action on an 8-dimensional compact almost complex manifold with 4 fixed points. To the author's knowledge, $S^2 \times S^6$ is the only known example of such a manifold. In this paper, we prove that if the circle acts on an 8-dimensional compact almost complex manifold $M$ with 4 fixed points, all the…

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…

Improved DP optimization for nonconvex, nonsmooth objectives with reduced sample complexity.

We give nearly matching upper and lower bounds on the oracle complexity of finding $ε$-stationary points ($\| \nabla F(x) \| \leqε$) in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows u…

We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.

New proofs in fixed point theory for manifolds and domains.

We prove that the half-integer valued local index of an isolated umbilic point on a $C^{3+α}$-smooth convex surface in Euclidean 3-space is less than two. The approach is to study the co-kernel of an associated Riemann-Hilbert boundary value problem. The link between the local and global is a semi-local technique that …

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in the complex projective plane. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular p…

For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of th…

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…

Paper introduces Chern minimal surfaces in Hermitian surfaces and establishes identities related to their points and bundles.