The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.
problem Estimating intersection numbers of psi-classes on Deligne-Mumford's moduli spaces.
method Approximates intersection numbers by closed-form expressions and proves a uniform lower bound.
result Proves a lower bound for intersection numbers in terms of approximating expressions and an explicit factor.
Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.
problem Calculating volumes of specific strata of quadratic differentials.
method Expressed volumes as a sum over stable graphs, with coefficients as intersection numbers of psi classes with combinatorial classes.
result Formula for volumes of odd strata of quadratic differentials.
Study on lengths of random multicurves on hyperbolic surfaces.
problem Distribution of lengths of random multicurves on closed hyperbolic surfaces.
method Using Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres.
result Distribution of lengths admits a polynomial density, with coefficients expressible in terms of intersection numbers of psi-classes.
This paper proves that lattice point enumeration in moduli spaces satisfies topological recursion.
problem Enumeration of lattice points in moduli spaces of curves.
method Proves topological recursion for lattice point enumeration in moduli spaces.
result The enumeration satisfies local topological recursion.
Formulae for Masur-Veech volumes derived from intersection numbers of curves.
problem Computing volumes and densities of geodesics and surfaces.
method Intersection numbers of psi-classes and lattice point count.
result Formulae for Masur-Veech volumes as polynomials in intersection numbers.
Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r-spin and Theta-class intersection numbers. The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.
problem Computing the number of self-intersections of closed geodesics on a pair of pants.
method Used Bowen and Series' coding to compute self-intersections.
result Proved a conjecture and provided bounds for self-intersection numbers.
Method calculates loop intersections on surfaces.
problem Computing intersections of loops on surfaces.
method Nielsen fixed point theory and Gröbner-Shirshov basis.
result Simple method to compute intersection numbers.
The study limits how many parts regular simplicial partitions can overlap.
problem Bounding the intersection number of regular simplicial partitions.
method Analyzing the properties of regular simplicial partitions.
result Established a maximum limit for the intersection number.
Finite intersection numbers between horizontal foliations of quadratic differentials.
problem Intersection properties of horizontal foliations in quadratic differentials.
method Joint continuity of intersection number in L1-norm. result Intersection number is finite and jointly continuous.
We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S2×S1.
The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.
problem Bounding the entropy of geodesic currents on hyperbolic surfaces.
method Established a quantitative upper bound on entropy in terms of self-intersection number and systole.
result Small self-intersection number forces small entropy.
Improved bounds on geodesic intersections on hyperbolic surfaces.
problem Finding the shortest geodesic with a specific number of intersections.
method Proved a new formula for minimal length of geodesics with self-intersection number k.
result Improved the threshold for the existence of geodesics with self-intersection number k.
We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…
Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller space…
Sharp lower bound on fold singularities self-intersections.
problem Finding a lower bound on the number of self-intersections of fold singularities.
method Established a sharp lower bound on the number of self-intersections of the boundary of an immersed surface, then applied this to fold singularities.
result Sharp lower bound on the number of self-intersections of fold singularities.
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…
For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.
We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the …
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. The study finds an upper limit for the number of minimal origami pairs on a surface.
problem Counting the minimal origami pairs on a surface of genus g.
method Algorithm to count minimal origami pairs and using Ménage Problem to establish an upper bound.
result Established a new upper bound for the count of minimal origami pairs.
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most ℓ on a combinatorial surface of complexity n we describe simple algo…
The paper proves the exact number of singular points in the intersection of convex shapes.
problem Determining the exact number of singular points in the intersection of convex shapes.
method Analyzing the intersections of n translates of a strictly convex, smooth, convex body in the Euclidean plane.
result The intersection of n translates of a convex body has exactly n points of singularity along its boundary.
The paper constructs minimal coherent filling pairs on surfaces.
problem Finding minimal intersecting coherent filling pairs on surfaces.
method Geometric procedure starting from a torus filling pair.
result Construction of minimal intersecting coherent filling pairs on Sg for g≥3. We find the minimal number of self-intersections of the boundary of a surface of genus g generically immersed in the plane.
Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
problem Understanding the relationship between intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
method Analyzing the asymptotic behavior of interaction strength I(X) as X approaches infinity in the moduli space of compact hyperbolic surfaces.
result Determined the asymptotic behavior of interaction strength I(X) in terms of the length of the shortest geodesic sys(X).
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is the number of letters of the corresponding word. The self-intersection of a free…
Study intersection cohomology and Lagrangian fibrations in symplectic varieties.
problem Understanding the intersection cohomology and perverse filtration of Lagrangian fibrations in symplectic varieties.
method Analyzes the deformation equivalence class, computes the border of the perverse diamond, and identifies perverse and Hodge numbers.
result Complete description of intersection cohomology and invariant cohomology classes of fibers.
Study on combinatorial k-systoles on surfaces, showing growth in intersection numbers.
problem Understanding the intersection numbers of closed curves on surfaces.
method Analyzing combinatorial k-systoles on punctured tori and pairs of pants. result The maximal intersection number of combinatorial k-systoles grows like k and approaches infinity as k increases. We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most L grows exponentially in L. We get exponentially tighter bounds given…
The study counts geodesics on curved surfaces with specific intersections.
problem Counting geodesics with exact intersection numbers on curved surfaces.
method Introduced a dynamical scattering operator and used Pollicott-Ruelle resonances.
result Asymptotic growth of geodesics with prescribed intersections.
The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than 1. The set Sigma of matrices of positive corank is an example of such a set. It turns out that the inte…
Suppose a smooth planar curve γ is 2π-periodic in the x direction and the length of one period is ℓ. It is shown that if γ self-intersects, then it has a segment of length ℓ−2π on which it self-intersects and somewhere its curvature is at least 2π/(ℓ−2π). The proof involves the projection Γ …
Defines quantum intersection number on pants decompositions and relates it to hyperbolic geometry.
problem Quantum and geometric intersection numbers on surfaces and 3-manifolds.
method Using asymptotic expansions of curve operators in skein theory, we define quantum intersection numbers and relate them to geometric intersection numbers and Teichmüller geometry.
result The pants graph equipped with a metric derived from quantum intersection numbers is quasi-isometric to the Teichmüller space with the Weil-Petersson metric.
The paper proves uniform stable radius and Milnor number equality for specific mappings.
problem Proving uniform stable radius and Milnor number equality for specific mappings.
method Analytic family construction and Newton polyhedra analysis.
result Milnor numbers of non-degenerate isolated complete intersection singularities are equal.
We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…
In a previous paper, we defined an operation μ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…
The study counts curves on a once-punctured torus with self-intersections.
problem Counting closed curves with self-intersections on a once-punctured torus.
method Combinatorial classification of curves with given word-length and self-intersections.
result Determination of curve counts with zero, one, and arbitrary self-intersections.
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.