Let $p$ be a hyperbolic periodic point of a diffeomorphism $f$ on a compact $C^{\infty}$ Riemannian manifold $M$. In this paper we introduce the notion of $C^{1}$ stably measure expansiveness of closed $f$-invariant sets, and prove that (i) the chain recurrent set $\mathcal{R}(f)$ of $f$ is $C^{1}$ stably measure expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, and (ii) the homoclinic class $H_{f}(p)$ of $f$ associated to $p$ is $C^{1}$ stably measure expansive if and only if $H_{f}(p)$ is hyperbolic.

Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.

We find some necessary conditions for a smooth irreducible curve $C\subset \mathbb{P}^{4}$ to be isolated in a smooth quintic threefold. As an application, we prove that Knutsen's list of examples of smooth isolated curves in general quintic threefolds is complete up to degree 9.

We consider the ill-posedness issue for the drift-diffusion system of bipolar type by showing that the continuous dependence on initial data does not hold generally in the scaling invariant Besov spaces. The scaling invariant Besov spaces are $\dot{B}_{p, \sigma}^{-2+ n/p} (\mathbb{R}^{n})$ with $1 \leq p, \sigma \leq \infty$ and we show the optimality of the case $p = 2n$ to obtain the well-posedness and the ill-posedness for the drift-diffusion system of bipolar type.

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^{3}$-actions on rational spheres and complex projective spaces, and provide some properties of $S^{1}$-actions on a general rational complex.

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_{d}$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_{d} \gg (\log d)^{2}$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^{2} - d Y^{2} = -1$ (or more generally $X^{2} - D Y^{2} = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $\mu$ are principal.

Let $G$ be a finite group. The Smith equivalence for real $G$-modules of finite dimension gives a subset of real representation ring, called the primary Smith set. Since the primary Smith set is not additively closed in general, it is an interesting problem to find a subset which is additively closed in the real representation ring and occupies a large portion of the primary Smith set. In this paper we introduce an additively closed subset of the primary Smith set by means of smooth one-fixed-point $G$-actions on spheres, and we give evidences that the subset occupies a large portion of the primary Smith set if $G$ is an Oliver group.

We define enhancements of the quandle counting invariant for knots and links with a finite labeling quandle $Q$ embedded in the quandle of units of a Lie algebra $\mathfrak{a}$ using Lie ideals. We provide examples demonstrating that the enhancement is stronger than the associated unenhanced counting invariant and image enhancement invariant.

We show that all twist knots and certain double twist knots are minimal elements for a partial ordering on the set of prime knots. The keys to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.

In this note we establish a new discreteness criterion for a non-elementary group $G$ in $\mathit{SL}(2, \mathbb{C})$. Namely, $G$ is discrete if all the two-generator subgroups are discrete, where one generator is a non-trivial element $f$ in $G$, and the other is in the conjugacy class of $f$.

Using $G$-monopole invariants, we produce infinitely many exotic non-free actions of $\mathbb{Z}_{k}\oplus H$ on some connected sums of finite number of $S^{2}\times S^{2}$, $\mathbb{C}P_{2}$, $\overline{\mathbb{C}P}_{2}$, and $K3$ surfaces, where $k\geq 2$, and $H$ is any nontrivial finite group acting freely on $S^{3}$.

Based on the method of Boston and Leedham-Green et al. for computing the Galois groups of tamely ramified $p$-extensions of number fields, this paper gives a large family of triples of odd prime numbers such that the maximal totally real $2$-extension of the rationals unramified outside the three prime numbers has the Galois group of order $512$ and derived length $3$. This family is characterized arithmetically, and the explicit presentation of the Galois group by generators and relations is also determined completely.

Bertolini--Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbb{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of $\mathrm{GL}(2)$-type associated to a Hilbert modular form of weight 2.

We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi--Kuno [4], [5] and Massuyeau--Turaev [7].