arXiv research

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing…

Simple branched coverings established between certain 4-manifolds.

Safe reinforcement learning for autonomous vehicles using prediction constraints.

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with …

Develops a method to compute Morse homology for clean but not necessarily transverse intersections.

A new approach switches between simple and complex models to handle concept drifts in regression tasks.

Let $M$ be a simple 3-manifold, and $F$ be a component of $\partial M$ of genus at least 2. Let $α$ and $β$ be separating slopes on $F$. Let $M(α)$ (resp. $M(β)$) be the manifold obtained by adding a 2-handle along $α$ (resp. $β$). If $M(α)$ and $M(β)$ are $\partial$-reducible, then the minimal geometric intersection n…

This paper offers a new algebraic perspective of GCCA using subspace intersection.

The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefsch…

New findings about twists in 4-sphere diffeomorphisms.

We count meromorphic differentials with fixed residues and poles of fixed orders.

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and …

The paper constructs four-manifolds with lens space boundaries and explores sphere configurations in $\#^n \mathbb{C} P^2$.

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surfa…

Algorithms compute topological info about curves in surfaces.

The paper studies invariants of 2-surfaces embedded in 3-space.

A Heegaard diagram for a 3-manifold M is a closed, oriented surface S together with a pair (X, Y) of compact 1-manifolds in S whose components serve as attaching curves for the 2-handles of the two sides of a Heegaard splitting for M. The diagram is positive if X and Y can be oriented so that the intersection number <X…

Query2box embeds complex queries as boxes to handle logical operations in large KGs.

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmueller curves. For the stratum consisting of holomorphic one-forms in genus three with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over…

Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the…

Develops a method to ensure fairness across multiple sensitive attributes in machine learning.

Subspace segmentation or subspace learning is a challenging and complicated task in machine learning. This paper builds a primary frame and solid theoretical bases for the minimal subspace segmentation (MSS) of finite samples. Existence and conditional uniqueness of MSS are discussed with conditions generally satisfied…

We continue to develop an obstruction theory for embedding 2-spheres into 4-manifolds in terms of Whitney towers. The proposed intersection invariants take values in certain graded abelian groups generated by labelled trivalent trees, and with relations well known from the 3-dimensional theory of finite type invariants…

This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurf…

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

3-manifold triangulation can be reconstructed from its intersection matrix.

The study explores deep and shallow slice knots in 4-manifolds, linking them to conjectures and proving existence and nonexistence results.

The paper finds diffeomorphic complex intersections with distinct Hodge numbers.

New polynomials defined for virtual knots, calculated up to crossing 4.

In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…

We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential ste…

Conditions for curves on a torus with specific pairwise intersections.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

Virtual knots with same writhe polynomial have equivalent intersection graphs.

Study properties of self-similar continua with finite intersection property.

Estimates intersection pairing in hyperbolic 4-manifolds.

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…

Classifies arcs on a 4-punctured sphere that intersect at most once.

James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold $M$ is a subcomplex of $C_*(M)\otimes C_*(M)$ that is quasi-isomorphic to $C_*(M)\otimes C_*(M)$ and, more generally, that the intersection pairing endows $C_*(M)$ with the structure of a partially-defined commutati…

Novel approach for large genus intersection number asymptotics.

The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.

Generic potential primes have no self-intersections or intersections.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Study intersection polynomials of long virtual knots with supporting genera.

Paper computes Atiyah class for DG manifolds of amplitude +1.

Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.