The study provides a criterion for solving complex Hessian-type equations on projective manifolds.
problem Solving complex Hessian-type equations on projective manifolds.
method Proving Nakai-Moishezon-type criteria for these equations.
result Uniform criteria for solving these equations, including complex Hessian and Hessian quotient equations.
A new polynomial invariant for strongly involutive links.
problem Characterizing strongly involutive links using polynomial invariants.
method Introducing a two-variable polynomial invariant \(P^e\) with equivariant skein relations.
result Specialisation of \(P^e\) recovers the graded Euler characteristic of a spectral sequence.
Strongly polynomial algorithm for approximate Forster transforms and halfspace learning.
problem Computing approximate Forster transforms and halfspace learning.
method Strongly polynomial time algorithm for approximate Forster transforms and halfspace learning.
result First strongly polynomial time algorithm for distribution-free PAC learning of halfspaces.
Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients…
Quantum polynomials are derived from a specific tribracket structure.
problem Quantum enhancement polynomials for oriented links.
method Defined using a canonical two-element tribracket, proving polynomials can be derived from five specific ones.
result Universal quantum enhancement polynomials are strictly stronger than the Jones polynomial.
A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility a…
Study subharmonic functions in strongly symmetric Riemannian manifolds, proving polynomial growth.
problem Properties of subharmonic functions in Riemannian manifolds with a pole.
method Introduced polynomial growth of subharmonic functions and proved their properties.
result Proved polynomial growth of degree 1 for non-negative subharmonic functions.
We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, an…
New findings on branched covers of quasipositive links and their L-space properties.
problem Understanding the conditions under which branched covers of quasipositive links are L-spaces.
method Analyzing Alexander polynomials and using properties of cyclic covers.
result Conditions for the L-space property of branched covers of quasipositive links, including specific cases for strongly quasipositive and quasipositive links.
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.
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…
The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
problem Extending geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
method Demonstrates dynamical and counting results for geometrically-finite strictly convex projective structures with Hilbert metric.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
New bounds for knot complexity based on Jones polynomial coefficients.
problem Finding bounds for the crosscap number of knots and links.
method Using coefficients from the Jones polynomial, we derive two-sided bounds for Conway sums of strongly alternating tangles.
result Neither linear bound generalizes for all knots and links.
New formulas for colored Jones polynomials of double twist knots generalize series and duality.
problem Calculating colored Jones polynomials for double twist knots.
method Using Takata's result and comparing with cyclotomic expansions.
result Generalizes Kontsevich-Zagier series and duality at roots of unity.
Detecting convexity in polynomials over boxes is NP-hard even for degree 3.
problem Detecting convexity in polynomials over compact regions, especially boxes.
method Proof by reduction to the NP-hard problem of testing global convexity of polynomials of degree four.
result The problem of testing convexity over a box is strongly NP-hard even for polynomials of degree 3.
Study on knots, genera, and algebraic concordance groups.
problem Understanding the equivariant slice genus of strongly invertible knots.
method Using the Blanchfield form to establish lower bounds and formulate an equivariant algebraic concordance group.
result The equivariant slice genus of an equivariant connected sum of a genus one strongly invertible slice knot is at least n/4.
Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.
problem Understanding Ricci curvature on graphs, especially for specific graph types.
method Introduced modified Ricci curvature, established rigidity theorem, derived formulas for strongly regular graphs.
result Rigidity theorem for complete graphs and explicit formulas for strongly regular graphs.
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
This paper shows how symmetry in special links affects a specific polynomial.
problem Understanding the symmetry in extended strongly periodic links.
method Proving the relationship between the symmetry of these links and the HOMFLYPT polynomial.
result The first coefficients of the HOMFLYPT polynomial reflect the symmetry of extended strongly periodic links.
Research on mixed polynomials, extending non-degeneracy concepts to complex variables.
problem Extending non-degeneracy concepts to mixed polynomials in complex variables.
method Generalization of Mondal's partial non-degeneracy to mixed polynomials, introducing new concepts and proving properties.
result Strong partial non-degeneracy implies isolated singularities, and mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition.
3-manifolds with similar completions have matching slopes and polynomials.
problem Matching slopes and polynomials in cusped hyperbolic 3-manifolds.
method Prove similarity of profinite completions leads to matching A-polynomials and boundary slopes. result Strongly detected boundary slopes match between manifolds with similar completions.
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
Study on Monge-Ampère equations with polynomial growth rates.
problem Analyzing solutions to Monge-Ampère equations with polynomial right-hand sides.
method Utilizing polynomial growth analysis to study regularity and growth rates of solutions.
result Translators for sub-affine-critical curvature flows are smooth and convex with specific growth rates.
Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…
Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…
We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on Mn≥3 share the same unparametrized geodesics, and two of them (say, g and gˉ) are strictly nonproportional (i.e., the minimal polynomial of giαgˉαj coincides with …
New SLC distributions enable easier control over diversity.
problem Lack of easy control over diversity in existing models.
method Developed strongly log-concave distributions and two tools for sampling and mode finding.
result Established weak log-submodularity for SLC functions and optimization guarantees for mode finding.
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
We describe a procedure for creating infinite families of knots, each having the maximum degree of their HOMFLY polynomial strictly less than twice their canonical genus. These families build upon examples first found by Stoimenow.
Improved robust regression with clean covariates achieves better rates than Huber's model.
problem Robust regression under adaptive contamination of responses with clean covariates.
method Exploiting clean covariates to construct an estimator achieving better rates than Huber's model.
result Improved estimation rate even with constant contamination, achieving consistency.
Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure progress toward a trivial embedding. In conjunction with a modified form of arc presen…
The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.
problem Investigating properties and equivalence of complex Finsler metrics.
method Using curvature properties of Bergman metrics and Schwarz lemma, the paper analyzes complex Finsler metrics and their equivalence to the Kobayashi metric.
result Uniform equivalences of the Kobayashi metric and Carathéodory metric on bounded strongly convex domains with smooth boundaries are proven.
We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose ele…
New polynomials with specific braids as links of singularities are constructed.
problem Creating polynomials with given braid closures as isolated singularities.
method Parametrizing braids and constructing polynomials with specific properties.
result Polynomials with closures of certain braids as isolated singularities are real algebraic and satisfy the strong Milnor condition.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order p with a circle as the set of fixed points if and only if M is obtained from the three-sphere by surgery along a strongly p−periodic link L. Moreover, if the quotient three-manifold is an integral ho…
Improved dynamic regret analysis for strongly convex and smooth functions.
problem Analyzing dynamic regret for online learning algorithms.
method Improved analysis of the Online Multiple Gradient Descent (OMGD) algorithm.
result Achieved a best-of-three-worlds guarantee for dynamic regret.
Let M2n−1 be the smooth boundary of a bounded strongly pseudo-convex domain Ω in a complete Stein manifold V2n. Then (1) For n≥3, M2n−1 admits a pseudo-Eistein metric; (2) For n≥2, M2n−1 admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR em…
The paper explores Lagrangians with simplified Euler-Lagrange equations.
problem Variational problems with Euler-Lagrange equations of reduced order.
method Geometrical construction to derive a family of Lagrangians.
result Lagrangians with reduced-order Euler-Lagrange equations are polynomials in the highest-order derivatives.
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a rando…
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions. result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental grou…
We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble s…
New stabilization method in graph braid homology yields polynomial growth.
problem Stabilization in graph braid homology.
method Introduced a stabilization map on graph configuration spaces, leading to a polynomial ring action on homology.
result Homology module is finitely generated and shows polynomial growth in Betti numbers.