Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal ℏ-power series solution and Borel summability. result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
Study on Chern-Simons theory at generic levels, revealing universal resurgent structure.
problem Analyzing Chern-Simons theory at generic levels with small boundary holonomy.
method Examined resurgent structure of state integral models on knot complements with generic discrete level.
result Resurgent structure is universal, independent of the level k. The study shows how certain ODEs and integrals are regular under Borel summation.
problem Analyzing the regularity of solutions to ODEs and integration problems.
method Using geometric perspective on Laplace and Borel transforms, the study examines level 1 ODEs and exponential period integrals over Lefschetz thimbles.
result Solutions of certain ODEs and integration problems are Borel regular.
Study shows how nonstandard hulls can contain metric completions.
problem Characterizing the Heine-Borel property in metric spaces.
method Using nonstandard analysis and metric completions.
result Characterization of the Heine-Borel property in terms of inapproachable finite points.
Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.
problem Non-perturbative quantum geometry of open and closed topological string on the resolved conifold.
method Finite difference equations, resurgence analysis, exact WKB techniques.
result Identify 5d BPS states and relate spectral problems to quantum integrable systems.
Novel mathematical approach using resurgent analysis reveals new structures in complex Chern-Simons theory.
problem Curious bijection in vertex algebras and SCFTs.
method Resurgent analysis, numerical algorithms, singularity elimination.
result New structures and patterns in complex Chern-Simons theory on hyperbolic 3-manifolds.
Quantum dilogarithm function proven from a linear difference equation.
problem Proving Faddeev's quantum dilogarithm from a linear difference equation.
method Proved Faddeev's quantum dilogarithm using Borel summation of a formal power series solution of a linear difference equation.
result Borel summation of a formal power series solution produces Faddeev's quantum dilogarithm.
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…
Uniformizes varieties with log-canonical singularities using ball quotients.
problem Uniformizing complex projective varieties with log-canonical singularities.
method Criteria based on Miyaoka-Yau inequality and log-resolutions of singularities.
result Criteria for isomorphism to Baily-Borel-Mok compactifications.
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
Criteria for sharksfin and deltoid singularities from plane to plane, with applications.
problem Identifying and understanding singularities in plane-to-plane mappings.
method Providing criteria and geometric meanings for singularities.
result Geometric meanings and criteria for sharksfin and deltoid singularities.
In this paper we prove that the zeroth Milnor-Thurston homology group coincides with singular homology for Peano Continua. More- over, we show that the canonical homomorphism between these ho- mology theories may not be injective. However, it is proved that it is injective when a space has Borel path-components.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.
Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.
problem Understanding Poisson cohomology for plane structures with isolated singularities.
method Determined Gerstenhaber algebra structure over Poisson cohomology groups.
result GAGA type phenomenon observed in Poisson cohomology.
Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.
problem Non-uniqueness of formal solutions to the Dubrovin equation at irregular singularity.
method Revisited canonical coordinates, formal solutions analysis, Borel resummation, Stokes matrices computation.
result Infinite-dimensional Stokes matrices computed from resummed formal solutions.
The paper proves inequalities for hyperbolic sets and curves.
problem Proving inequalities for sets and curves in hyperbolic geometry.
method Defining horocyclic Minkowski sums and proving inequalities for hyperbolic areas.
result Horocyclic Brunn-Minkowski inequality holds for hyperbolic sets.
For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…
The study examines vertices in curves with singular points in the Euclidean plane.
problem Investigating vertices in curves with singular points in the Euclidean plane.
method Defining vertices using evolutes of frontals and analyzing conditions for the four vertex theorem.
result Conditions for the four vertex theorem to hold for closed frontals.
Divides help construct fibered links from singularities.
problem Understanding complex isolated plane curve singularities.
method Using divides to topologically construct fibered links.
result Explicitly given monodromy diffeomorphism as a product of Dehn twists.
We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of multiple singularities. The obstruction is formulated in terms of a semicontinuity p…
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
Paper proves conditions for rational homology complex projective planes with singularities.
problem Proving conditions for rational homology complex projective planes with singularities.
method Leveraging results from smooth 4-manifolds, including Donaldson diagonalization theorem and Heegaard Floer correction terms.
result Eliminates the possibility of a rational homology complex projective plane with four singularities and identifies families of singularities obstructed by smooth conditions.
We consider the global symplectic classification problem of plane curves. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections. Then the set of symplectic classes form the symplectic moduli space which we compl…
We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.
We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group G. When n=2, the Borel class is equal to the 3-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, …
Invariants count inflections and vertices in singular plane curves.
problem Counting inflections and vertices in singular plane curves.
method Defining invariants If and Vf to count inflections and vertices, respectively, and analyzing their properties. result The invariants If and Vf are finite and bounded for curves without smooth components. Heat kernel resurgent structure from Picard-Lefschetz theory
problem Short-time heat kernel asymptotics
method Picard-Lefschetz theory
result 1-Gevrey small-time expansion
Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discre…
Study plane curve singularities to determine vanishing cycles and monodromy groups.
problem Understanding vanishing cycles and monodromy groups for plane curve singularities.
method Intrinsic description of geometric monodromy group, easy criterion for vanishing cycles, canonical framing.
result Monodromy groups are injective for singularities with Milnor fiber of genus at least 7.
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
New Stein fillings found for rational surface singularities.
problem Exploring Stein fillings of rational surface singularities.
method Using planar open books and Lefschetz fibrations, describe Stein fillings via symplectic disk arrangements.
result Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.
We prove that a closed immersed plane curve with total curvature 2πm has entropy at least m times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature 2πm whose entropy is less than m …
The paper classifies different types of cusps on plane curves.
problem Investigating various types of cusps on plane curves.
method Examining criteria for (n,n+1) cusps with differential conditions and relations to evolutes of fronts. result Complete classifications for (4,5)-cusps. Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct S1…
Study on cohomology of singular foliations with localization results.
problem Understanding cohomology of singular Riemannian foliations.
method Introduced equivariant basic cohomology and proved its properties.
result Equivariant basic cohomology localizes to closed leaves.
The topology of symplectic 4-manifolds is related to that of singular plane curves via the concept of branched covers. Thus, various classification problems concerning symplectic 4-manifolds can be reformulated as questions about singular plane curves. Moreover, using braid monodromy, these can in turn be reformulated …
New moves for singular knots identified and described.
problem Identifying and describing moves for singular knots.
method Provided 96 generating sets of oriented singular Reidemeister moves and selected moves for Legendrian singular knots.
result Surviving moves for Legendrian singular knots were identified and described.
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …
Extends symmetry and rigidity to surfaces with soap film-like singularities.
problem Symmetry and rigidity of minimal surfaces with singularities.
method Method of moving planes applied to surfaces with Plateau-like singularities.
result Extends classical results to surfaces with singularities.
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
Characterizes hyperbolic links with stable maps to the plane.
problem Understanding hyperbolic links through stable maps.
method Characterization of hyperbolic links via stable maps to the plane.
result Complete characterization of hyperbolic links with specific stable maps.
Solved a conjecture about rational homology projective planes with quotient singularities.
problem A conjecture about rational homology projective planes with quotient singularities.
method Combining Donaldson's diagonalization theorem with a distinguished spin^c structure on the smooth locus.
result Proved that rational homology projective planes with quotient singularities have at most three singular points.