Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients.

First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.

It is known that a 2-knot has the triple point number less than two if and only if it is of ribbon-type. We prove that there is no 2-knot of triple point number two or three. Hence the 2-twist-spun trefoil, which is known as a 2-knot of triple point number four, is one of the simplest non-ribbon 2-knots.

The Morse-Novikov number $\mathcal{MN}(L)$ of a smooth link $L$ in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of $L$). Novikov homology provides lower bounds for $\mathcal{MN}(L)$. In the present paper we introduce the notion of twisted Novikov homology, which allows to obtain better lower bounds for $\mathcal{MN}(L)$ than the usual Novikov homology. Our twisted Novikov homology is a module over the Novikov ring $\mathbf{Z}((t))$ but it contains the information coming from the non abelian homological algebra of the group ring of the fundamental group of the link complement. Using this technique we prove that the Morse-Novikov number of the knot $n\mathfrak{C}$ (the connected sum of $n$ copies of the Conway knot) is not less than $2n/5$ for every positive integer $n$. We prove also that $\mathcal{MN}(n\mathfrak{C})$ is not greater than $2n$. The same estimates hold for the Morse-Novikov numbers of the connected sum of $n$ copies of the Kinoshita-Terasaka knot.

We study generic conformally flat hypersurfaces in the Euclidean $4$-space. The conformal flatness condition for the Riemannian metric is given by a set of several differential equations of order three. In this paper, we first define a certain class $\Xi$ of metrics for 3-manifolds which includes, as a large subset, all metrics of generic conformally flat hypersurfaces in the Euclidean 4-space. We obtain a kind of integrability condition on metrics of the class. Restricting our consideration to metrics of conformally flat hypersurfaces, we define a conformal invariant for generic conformally flat hypersurfaces and obtain a differential equation of order three from the integrability condition. The equation is equal to the simplest one in equations of conformal flatness condition. Next, we study some particular solutions of the equation. We will determine all generic conformally flat hyersurfaces corresponding to these particular solutions under an assumption on the first fundamental form, and characterize these hypersurfaces geometrically. The result includes all known examples of generic conformally flat hypersurfaces in the Euclidean $4$-space. All known examples are the following: The hypersurfaces given by Lafontaine ([6]) which are made from constant Gaussian curvature surface in the three dimentional space forms, the hypersurfaces given by Suyama ([7]), and the flat metrics obtained by Hertrich-Jeromin ([3]). Furthermore, we explicitly construct a series of examples of generic conformally flat hypersurface, which have a geometrical property different from all known examples. Then, we have the following case: There exists a pair of hypersurfaces with the same conformal invariant, each of which is constructed from a surface with constant Gaussian curvature in either the Euclidean 3-space or the standard 3-sphere but does not belong to the known examples. Furthermore, no conformal transformation maps diffeomorphically one hypersurface of the pair to the other hypersurface.

In this paper, we mainly prove that principal orbits of an action of Hermann type on a symmetric space of non-compact type are curvature adapted and proper complex equifocal. The proof is performed by showing that principal orbits of the action are partial tubes over a totally geodesic singular orbit and investingating the shape operatores of the partial tubes over a submanifold in a symmetric space of non-compact type.

Assume that two media are laying in a half-space and the interface wall is parallel to the boundary of the half-space. We can directly observe the data near the boundary of the half-space, but we cannot directly observe inside the half-space. In this situation, we try to identify these unknown things by creating an artificial explosion and observing on the boundary the waves generated by the explosion. In the previous works related to this problem, only the speeds of the waves were treated, but we also take into account the impedances of the media in our setting.

In this paper, we study the following fourth order elliptic problem $(E_p)$: \begin{eqnarray*} (E_p) \left \{ \begin{array}{l} \Delta^2 u = u^p \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right. \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and $p >1$ is any positive number.

We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$. Combining the arguments of Ren-Wei [8] and Wei [10], we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains, they have one or two ``peaks'' away form the boundary. If it happens that the only one peak point appears, we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$ under the Navier boundary condition.

We consider linear partial differential equations with several Fuchsian variables in the sense of M.S. Baouendi and C. Goulaouic [1]. For a holomorphic Fuchsian operator with holomorphic Fuchsian principal part, we prove existence and uniqueness of a holomorphic local solution. Our theorem generalizes the results of ([3, 1, 11]), precises the one of [4] and reduces the proof of their theorems to the proof of the fixed-point theorem. For a holomorphic Fuchsian operator with constant Fuchsian principal part, we establish the existence and uniqueness of a holomorphic global solution. Our aim is to simplify its proof. The methods of proof are based on the application of the fixed-point theorem in some Banach spaces defined by majorant functions that are suitable to this kind of equations.

Using heat kernel Gaussian estimates and related properties, we study the intrinsic regularity of the sample paths of the Hunt process associated to a strictly local regular Dirichlet form. We describe how the results specialize to Riemannian Brownian motion and to sub-elliptic symmetric diffusions.