From Homma's PL Gauss-Bonnet theorem applied on a PL-complex in a plane, many generalizations of Pick's theorem on a lattice PL-figure are obtained in a unified geometric way.

We study analytic smoothing effects for solutions to the Cauchy problem for the Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. The only assumption on the Cauchy data is the weight condition of exponential type and no regularity assumption is imposed.

We show that a closed connected surface embedded in $S^{4} = \partial B^{5}$ bounds a handlebody of dimension 3 embedded in $B^{5}$ if and only if the Euler number of its normal bundle vanishes. Using this characterization, we show that two closed connected surfaces embedded in $S^{4}$ are cobordant if and only if they are abstractly diffeomorphic to each other and the Euler numbers of their normal bundles coincide. As an application, we show that a given Heegaard decomposition of a 3-manifold can be realized in $S^{5}$. We also give a new proof of Rohlin's theorem on embeddings of 3-manifolds into $\mathbf{R}^{5}$.

We consider the Cauchy problem of second order hyperbolic equation with time depending coefficients. Time depending singular coefficient can bring some loss of regularity of the solution; for instance ``infinitely many oscillation'', ``infinite order degeneracy'' and ``accumulation of zeros'' crucially influence on the regularity loss. In this paper we make clear the order of regularity loss from the interaction of the singular effects, and also discuss the optimality.

In 1999 Andersson and Driver proved a piecewise geodesic approximation formula for path integrals on compact Riemannian manifolds. Here, geodesics are those of the Levi-Civita connection. In this paper we will generalize their result for $H^{1}$-type metric to the case of general metric connections.

In the first half of this paper, we complement the theory on discrete polymatroids. More precisely, (i) we prove that a discrete polymatroid satisfying the strong exchange property is, up to an affinity, of Veronese type; (ii) we classify all uniform matroids which are level; (iii) we introduce the concept of ideals of fiber type and show that all polymatroidal ideals are of fiber type. On the other hand, in the latter half of this paper, we generalize the result proved by Stefan Blum that the defining ideal of the Rees ring of a base sortable matroid possesses a quadratic Gröbner basis. For this purpose we introduce the concept of ``$l$-exchange property'' and show that a Gröbner basis of the defining ideal of the Rees ring of an ideal $I$ can be determined and that $I$ is of fiber type if $I$ satisfies the $l$-exchange property. Ideals satisfying the $l$-exchange property include strongly stable ideals, polymatroid ideals of base sortable discrete polymatroids, ideals of Segre-Veronese type and certain ideals related to classical root systems.

We prove results on existence and uniqueness of solutions of the Cauchy Problem for $2\times 2$ weakly hyperbolic systems. The results follow from an extension for systems of the work by O. Oleinik done in the scalar case.

We find upper bounds for the regularity of a singular projective variety over an algebraically closed field of characteristic zero. In particular we study the cases when the variety has finitely many isolated singularities and when it has a 1-dimensional singular locus.

Let $X$ be a projective minimal threefold of general type with only $\mathbb{Q}$-factorial terminal singularities. We study the generic finiteness of the $m$-canonial map for such 3-folds. Suppose $P_g(X)\ge 2$ and $q(X)\ge 3$. We prove that the $m$-canonical map is generically finite for $m\ge 3$, which is a supplement to Kollár's result. Suppose $P_g(X)\ge 5$. We prove that the 3-canonical map is generically finite, which improves Meng Chen's result.

Some quotient algebras arising from the quantum toroidal algebra $U_{q}(\mathit{sl}_{2}(\mathcal{C}_{\gamma}))$ are considered. They are related to integrable highest weight representations of the algebra and are shown to be isomorphic to direct sums of tensor products of two algebras of symmetric Laurent polynomials and Macdonald's difference operators.

We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big' open orbit of Gizatullin [19, 20] and Popov [31] or some of the classification results of Danilov-Gizatullin [12], Bertin [6, 7] and others.