Study on deformations of polynomials and their Milnor fibration topology.
problem Understanding the topology of Milnor fibrations of deformations of polar weighted homogeneous polynomials.
method Round handle decomposition of the Milnor fibration, calculation of characteristic polynomials.
result Description of the number of round handles and a formula for characteristic polynomials.
In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.
Study resolves polynomial germs, proving no mixed critical points and strict transform properties.
problem Resolving mixed critical points and properties of strict transforms of polynomial germs.
method Toric resolutions and modifications of weighted homogeneous polynomials.
result No mixed critical points and strict transform properties as germs.
In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.
Extends Khimshiashvili's degree formula to non-isolated singularities.
problem Finding topological properties of non-isolated real singularities.
method Generalizes Khimshiashvili's topological degree formula to non-isolated singularities of real function germs.
result Algebraic formula for the Euler characteristic of fibres of real weighted-homogeneous polynomials.
Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…
Milnor fibrations have been studied since 1960's. In this paper, we study singular points of differentiable maps, called Milnor fibration product maps, obtained by several Milnor fibrations. We give a characterization of singular points of such product maps, and for the case of certain weighted homogeneous polynomials,…
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …
Flat semigroups can represent normal weighted homogeneous surface singularities.
problem Representability of flat semigroups in normal weighted homogeneous surface singularities.
method Study of numerical semigroups associated with surface singularities and prove representability conditions.
result A numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.
We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…
Let Lf be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial f. In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds Lf for which H3(Lf,Z) is completely determined. Using the Boyer-Galicki construction of links…
We consider the significant class of holomorphically nondegenerate CR manifolds of finite type that are represented by some weighted homogeneous polynomials and we derive some useful features which enable us to set up a fast effective algorithm to compute their Lie algebras of infinitesimal CR-automorphisms. This algor…
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed comple…
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
New Stein fillings found for non-weighted homogeneous singularities.
problem Finding Stein fillings for certain non-weighted homogeneous singularities.
method Using spinal open books and nearly Lefschetz fibrations.
result Stein rational homology disk fillings for non-weighted homogeneous singularities.
Study SL(2,C) connections on Seifert-fibered spaces using gauge theory.
problem Counting SL(2,C) connections on Seifert-fibered spaces. method Introduced perturbations of the SL(2,C) Chern--Simons functional and proved a localisation result. result Formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the SL(2,C) character variety of a Seifert-fibered homology 3-sphere. Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-S1 bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or…
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
The study defines K-stability for Sasakian manifolds and finds obstructions to extremal metrics.
problem Finding obstructions to Sasaki-extremal metrics in Sasakian manifolds.
method Defining K-stability relative to a maximal torus of automorphisms and computing the Sasaki cone.
result The existence of a Sasaki-extremal metric implies K-semistability, and the Lichnerowicz obstruction is extended.
The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.
problem Understanding symplectic fillings of Seifert 3-manifolds.
method Rational blowdown surgery and minimal symplectic fillings.
result A necessary and sufficient condition for minimal symplectic fillings to be obtained by rational blowdowns.
We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ satisfying the following conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the Chern connection f…
We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal wei…
This paper completes the classification of certain surface singularities with rational homology disk smoothings.
problem Classifying surface singularities with rational homology disk smoothings.
method Study of configurations of rational curves on projective rational surfaces.
result There is a unique rational homology disk smoothing component except in the cases of an obvious symmetry of the resolution dual graph.
Generalizes Lefschetz fibrations with rational homology disk smoothings.
problem Understanding rational homology disk smoothings of surface singularities.
method Introduces a genus to generic fibers of Lefschetz fibrations.
result Families of relations in mapping class groups represent smoothings.
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are…
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.
The paper classifies minimal symplectic fillings of small Seifert 3-manifolds.
problem Classifying minimal symplectic fillings of small Seifert 3-manifolds.
method Using rational blowdowns and weighted homogeneous complex surface singularities.
result Every minimal symplectic filling of small Seifert 3-manifolds is obtained by a sequence of rational blowdowns.
New findings on QHD smoothing for graphs with 3 or 4 large nodes.
problem Identifying graphs with QHD smoothing and constraints on large nodes.
method Reduction algorithm and enumeration for graphs with QHD smoothings, using the picture deformation technique.
result No singularity with 3 or 4 large nodes has a QHD smoothing.
Study flat connections with logarithmic singularities on complex plane curves.
problem Modeling flat connections with logarithmic singularities.
method Explicit finite-dimensional model construction and detailed investigation of specific cases.
result Construction of shifted Poisson structure on moduli spaces.
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
Given a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regulari…
The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational RX-bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality. result Bi-Lipschitz triviality of deformations of f on (X,0) when X and f are homogeneous of the same degree. The study connects complex surface singularities to numerical semigroups and their properties.
problem Understanding the geometry and properties of strongly flat semigroups and their generalizations.
method Analyzing complex surface singularities and their associated semigroups, proving properties of Frobenius numbers.
result Strongly flat semigroups associated with negative definite Seifert homology spheres are numerical semigroups.
We prove the "End Curve Theorem," which states that a normal surface singularity (X,o) with rational homology sphere link Σ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An "end-curve function" is an analytic function $(X,o)\to (\C,0)$ whose ze…
Study on homology groups of cDV singularity links, identifying their topology.
problem Identify the topology of links of cDV singularities of types cAn and cDn. method Analyzing the second integral homology group of the links, using results from Smale and Thom-Sebastiani sums.
result The homology groups of the links are determined for cDV singularities of types cAn and cDn. The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C5 has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed G2-structure φ induced by the Calabi-Yau 3-orbifold basic geometry. We disti…
Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.
problem Defining and analyzing analytic lattice cohomology for isolated singularities.
method Using a good resolution of the singularity, proving independence of resolution choice, and relating to Hodge spectral numbers.
result Independence of analytic lattice cohomology from the choice of resolution and connection to Hodge spectral numbers.
The paper resolves fundamental groups for three exceptional surface singularity families.
problem Determining the fundamental groups for three exceptional families of surface singularities.
method New explicit constructions and the Pinkham method for some families.
result Fundamental groups for three exceptional families are determined.
New F-polynomial distinguishes knotoid diagrams not previously possible.
problem Polynomial invariants for knotoids.
method Introduced F-polynomial and constructed distinguishing examples. result New invariant F distinguishes knotoids not previously possible. The paper studies polynomials and ideals from colored Jones polynomials for links.
problem Understanding the structure of colored Jones polynomials for links.
method Investigates commutative and noncommutative ideals derived from colored Jones polynomials.
result Formulates the link version of the AJ conjecture.
Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.
problem Examining relationships between Affine Index Polynomial and Sawollek Polynomial.
method New approach to extract Affine Index Polynomial from Sawollek Polynomial.
result Constructs a concise proof of Mellor's Theorem.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Extends interior polynomial to signed bipartite graphs and connects to HOMFLY polynomial.
problem Invariants of signed bipartite graphs and their relation to HOMFLY polynomial.
method Extending interior polynomial to signed bipartite graphs and showing equality to HOMFLY polynomial part.
result Interior polynomial of signed bipartite graphs equals part of HOMFLY polynomial for planar case.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
Developed algorithms to compute three polynomial invariants of veering triangulations.
problem Computing polynomial invariants of veering triangulations.
method Introduced and used algorithms for taut, veering, and Teichmüller polynomials based on upper and lower tracks of veering triangulations.
result Proved that the lower and upper taut polynomials are equal but the veering polynomials can differ.
Study links weaving knots with polynomial coefficients and lattice numbers.
problem Understanding polynomial coefficients of weaving knots and their lattice counterparts.
method Established relationships between Jones and Chebyshev polynomials, and derived explicit formulas for Alexander polynomials.
result Proved coefficients of Jones polynomial are Whitney numbers of Lucas lattices and satisfied Fox's trapezoidal conjecture.