arXiv research

The paper calculates the Saito determinant for Coxeter discriminant strata.

Analyzes Saito vanishing theorem using $L^{2}$ methods.

The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.

We present a marked analogue of Carter and Saito's movie theorem. Our definition of marking was chosen to coincide with the markings that arise in link Floer homology. In order to deal with complications arising from certain isotopies, we define three equivalence relations for marked surfaces and work over an equivalen…

The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors' earlier work, much of the structure comes from the compatibility properties of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of m…

This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generati…

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here a so-called enhanced Burnside ring $\widehat{B}(G)$ of a finite group $G$…

Paper introduces skew-symmetric matrices for virtual doodle classification.

Normal forms and moduli stacks for flat connections on complex manifolds.

We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous …

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared Euclidean distance, Kullback-Leibler and Itakura-Saito divergences as special cases. Learning the model order is impo…

Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.

New stability criterion for Fano manifolds using anticanonically balanced metrics.

We classify simple singularities of functions on space curves. We show that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the bifurcations diagram of functions is true, and both this diagram and the discriminant a…

Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreduci…

For each integer $n\geq2$ we describe the space of stability conditions on the derived category of the $n$-dimensional Ginzburg algebra associated to the $A_2$ quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the $A_2$ singul…

Dubrovin duality connects two F-manifolds on the universal curve.

New definitions of rack and quandle modules are introduced, and shown to generalise the definitions previously studied by Andruskiewitsch, Etingof and Grana. This new construct is shown to coincide with Beck's general definition of a module in an arbitrary category. A theory of Abelian extensions of racks and quandles …

Extends Khovanov homology to surfaces with singularities.

For proper subsets U of {1,2,...,31} we define and construct U-regular isotopy invariants of the Carter-Rieger-Saito movies (representing knotted surfaces) and use these to show that there are ambient isotopic knotted surfaces represented by movies M1 and M2 for which movie-moves of type 31 are required to get from M1 …

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spac…

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

Let $E$ be a vector bundle over a suitable differential manifold $M$ and let $\wedge^p E$ denote $p$-exterior product of $E$. Given sections $ω_1,\dots,ω_k$ of $E$ and a section $η$ of $\wedge^p E$, we consider the problem if $η$ can be written in the form $η=\sum ω_i\wedgeγ_i,$ where $γ_i$ are sections of $\wedge^{p…

The simplest non-trivial solutions of WDVV equations are A_n and B_n-potentials, which describe metrics of K.Saito on spaces of versal deformation of A_n and B_n-singularities. These are some polynomials, which were known for $n\leqslant$ 4. We find some recurrence relations, which give a possibility to find all A_n an…

We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of…

Defines invariants for reflection groups and connects them to Frobenius structures.

T. Saito and M. Teragaito asked whether Berge knots of type VII are hyperbolic, and showed that some infinite sequences of the knots are hyperbolic. We show that Berge knots of types VII and VIII are hyperbolic except the known sequence of torus knots. We used the Reidemeister torsions. As a result, the Alexander polyn…

Equivalent categories of groups and risandles for 3-manifolds.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's p…

Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.

This paper generalizes L2 cohomology theory for complex manifolds.

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to defi…

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in w…

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators(we will call it the "algebraic" version in this article). In 2012, Przytycki defined another homology theory for pre-Yang-Baxter operators which has a nice graphic visualization(we will call it the "graphic" version i…

Nonnegative matrix factorization (NMF) is now a common tool for audio source separation. When learning NMF on large audio databases, one major drawback is that the complexity in time is O(FKN) when updating the dictionary (where (F;N) is the dimension of the input power spectrograms, and K the number of basis spectra),…

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property via L2-Hodge theoretical methods when f satisfies an asymptotic condition…

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structu…

Paper constructs solutions to WDVV equations for Frobenius manifolds.

Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are…

In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notio…

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this pape…

Study of equivariant movie moves for involutive links.

Study flat connections with logarithmic singularities on complex plane curves.

Lie groups applied to tech progress in economic growth.

Study of twisted Alexander matrices for certain quandles and their invariants.