arXiv research

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a co…

A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the …

The paper calculates the motive of a specific knot's character variety.

Topological 4-dimensional surgery is conjectured to fail, in general, for free fundamental groups. M. Freedman and P. Teichner have shown that surgery problems with an arbitrary fundamental group have a solution, provided they satisfy a certain condition on Dwyer's filtration on second homology. We give a new geometric…

Develops a new framework for large-scale geometry.

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point model and we give the conditions for the existence of f-divergence minimal equival…

We study in this paper, the existence of tree types of filtrations of the space $\mathcal{D}_{λμ}(\R^{2l+1|n})$ of differential operators on the superspaces $\R^{2l+1|n}$ endowed with the standard contact structure $α$. On this space $\mathcal{D}_{λμ}(\R^{2l+1|n})$, we have the first filtration called canonical and bec…

An algorithm preserves topological features in dimensionality reduction.

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue…

The paper proves a conjecture linking Higgs bundles and Lie algebra actions.

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefo…

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any co…

Study weightings from singular Lie filtrations.

A new method for optimal filtration learning in time-series data analysis.

Develops a new filtration for asset pricing models.

In sequential anytime-valid inference, any admissible procedure must be based on e-processes: generalizations of test martingales that quantify the accumulated evidence against a composite null hypothesis at any stopping time. This paper proposes a method for combining e-processes constructed in different filtrations b…

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…

In a recent paper we defined a new filtration of the mapping class group--the "Lagrangian" filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive invariant of finite-type (e.g. the Casson invariant), and any level of the Lagrangian…

Spatial refinement of Bar-Natan homology constructed.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.

Let $Σ$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $Σ$. By considering the action of $\mathcal{M}$ on the fundamental group of $Σ$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each ter…

We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also establish the analogous result for the Grope filtration of the concordance group…

A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-filtration o…

We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's…

We study knots of order 2 in the grope filtration $\{\G_h\}$ and the solvable filtration $\{\F_h\}$ of the knot concordance group. We show that, for any integer $n\ge4$, there are knots generating a $\Z_2^\infty$ subgroup of $\G_n/\G_{n.5}$. Considering the solvable filtration, our knots generate a $\Z_2^\infty$ subgro…

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polaris…

The knot Floer complex and the concordance invariant $\varepsilon$ can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to $\mathbb{N} \times \mathbb{N}$ and consists of topologically slice knots.

The knot Floer complex together with the associated concordance invariant epsilon can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains the natural numbers cross the integers as an ordered subset.

The paper confirms a conjecture about optimal expected utility in markets with insider information.

The paper develops a new theory of double Johnson filtrations for mapping class groups.

Let M denote the mapping class group of S, a compact connected oriented surface with one boundary component. The action of M on the nilpotent quotients of the fundamental group of S allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of M, called the La…

We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making…

We define a filtration of the smooth concordance group based on the genus of representative knots. We use the Heegaard Floer epsilon and Upsilon invariants to prove the quotient groups with respect to this filtration are infinitely generated. Results are applied to three infinite families of topologically slice knots.

The article explores the mapping class group using unicellular maps and provides filtrations.

Smooth maps show Gromoll filtration for spheres.

In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243--270, arXiv:math.GT/0010247], a group H_g of homology cylinders over the oriented surface of genus g is defined. A filtration of H_g is defined, using the Goussarov-Habiro notion of finite-type. It is e…

Defines a filtration on variational bicomplex for concise functional form conditions.

The study connects norms and filtrations on section rings of projective manifolds.

A new method detects small holes in noisy data.

Researchers determine quantum filtration structure of torus links.

We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given t…

We study multiple defaults where the global market information is modelled as progressive enlargement of filtrations. We shall provide a general pricing formula by establishing a relationship between the enlarged filtration and the reference default-free filtration in the random measure framework. On each default scena…

The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.

The paper introduces new KMEs to capture stochastic process filtrations.

For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger-Gromov rho-invariant, we obtain new real-valued homology cobordism invariants rho_n for closed (4k-1)-dimensional manifolds. For 3-dimensional manifo…

The study computes Gromoll filtration groups and fundamental groups for specific dimensions.