We discuss an analogy between the deformations of hyperbolic structures on a knot complement and of certain nearly ordinary Galois representations.

We study pairs of symmetries of a Riemann surface of genus $g \geq 2$, whose product has order $n > 2$, assuming that one of them is fixed point free. We start our considerations by giving some bounds for the number of ovals of a symmetry with fixed points and showing their attainment, later we take into account the number of points fixed by the product of the symmetries and we study some of its properties. Finally we deal the problem of finding the maximal possible power of $2$ which can be realized as the order of their product.

We classify blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type $G(r,p,n)$ by using the ``residue equivalence'' for multi-partitions.

In this paper, we show qualitative results for the solutions of forward-backward parabolic equations, where the forward or backward behaviour depends on the gradient of the solution.

We study the Cauchy problem for the dissipative Benjamin--Ono equations $u_{t} + \mathcal{H} u_{xx} + \lvert D\rvert^{\alpha} u + uu_{x} = 0$ with $0 \leq \alpha \leq 2$. When $0 \leq \alpha < 1$, we show the ill-posedness in $H^{s}(\mathbb{R})$, $s \in \mathbb{R}$, in the sense that the flow map $u_{0} \mapsto u$ (if it exists) fails to be $\mathcal{C}^{2}$ at the origin. For $1 < \alpha \leq 2$, we prove the global well-posedness in $H^{s}(\mathbb{R})$, $s > -\alpha/4$. It turns out that this index is optimal.

This paper mainly concerns with the stochastic logistic equation with Markovian switching and white noise. We represent the unique global positive solution and show some properties such as the stochastic permanence, global attractivity. The main contribution of this paper is to demonstrate that the solution is asymptotically stable in distribution. Moreover, the similar results for the stochastic Gilpin--Ayala equation with Markovian switching and white noise are yielded by the same way. Finally, we give the definition of the stochastic equilibrium solution and prove the existence and uniqueness of such equilibrium solution of stochastic logistic equation.

We consider a fourth-order nonlinear parabolic type equation on a two-dimensional bounded domain $\Omega$. This equation governs the evolution of the height profile of a thin film in an epitaxial growth process. We show that such equation endowed with no-flux boundary conditions generates a dissipative dynamical system under very general assumptions on $\partial\Omega$ on a phase-space of $L^{2}$-type. This system possesses a global as well as an exponential attractor. In addition, if $\partial\Omega$ is smooth enough, we show that every trajectory converges to a single equilibrium by means of a suitable Łojasiewicz--Simon inequality. An estimate of the convergence rate is also obtained.

This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the $L^{2}$-Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the $L^{2}$-Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the $L^{2}$-Wasserstein geometry. In particular, the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.

Let $X^{(k)}(n)$ be the indicator function of the set of $k$-th power free integers. In this paper, we study refinements of the density theorem $S^{(k)}_{N}(m) := (1/N) \sum_{n = 1}^{N} X^{(k)} (m + n) \to 1/\zeta(k)$, $\zeta$ being the Riemann zeta function. The following is one of our results; \begin{equation*} \lim_{M \to \infty} \frac{1}{M} \sum_{m = 1}^{M} \left(N\left(S^{(k)}_{N}(m) - \frac{1}{\zeta(k)}\right)\right)^{2} \asymp N^{1/k}. \end{equation*} The method we take here is a compactification of $\mathbb{Z}$; we extend $S^{(k)}_{N}$ to a random variable on a probability space $(\hat{\mathbb{Z}},\lambda)$ in a natural way, where $\hat{\mathbb{Z}}$ is the ring of finite integral adeles and $\lambda$ is the shift invariant normalized Haar measure. Then we investigate the rate of $L^{2}$-convergence of $S^{(k)}_{N}$, from which the above asymptotic result is derived.

The mapping class group of a genus $g$ surface $\Sigma_{g,1}$ with one boundary component is known to have a simple yet infinite presentation with generators given by elementary Whitehead moves on marked bordered fatgraphs. In this paper, we introduce an algorithm called fatgraph Nielsen reduction which, from the action of a mapping class $\varphi \in MC_{g,1}$ of $\Sigma_{g,1}$ on the fundamental group $\pi_{1}(\Sigma_{g,1})$ of $\Sigma_{g,1}$, determines a sequence of Whitehead moves representing $\varphi$ beginning at any choice of marked bordered fatgraph. The algorithm utilizes a reduction of bordered fatgraphs to linear chord diagrams, where the desired sequence is given in terms of elementary chord slide moves which continuously decrease some energy function. As a consequence, this leads to an algorithm which factors any mapping class given by its action on $\pi(\Sigma_{g,1})$ in terms of any convenient generating set for $MC_{g,1}$.

Let $S$ be a smooth open rational surface with $\bar{\kappa}(S) = \bar{p}_{g}(S) = 0$ and $\bar{P}_{2}(S) > 0$. We construct a certain minimal model of $S$, which is called a strongly minimal model of $S$ in [15], and determine the strongly minimal model in the case where $S$ has non-contractible boundary at infinity. As an application, we classify the log affine surfaces with $\bar{\kappa} = \bar{p}_{g} = 0$ and $\bar{P}_{2} > 0$ under the minimality condition.

Halphen's equations are given by a remarkable polynomial vector field in $\mathbf{C}^{3}$ having only single-valued solutions, defined in domains bounded by a circle or by a line. By generalizing the Lie-theoretic principle behind Halphen's equations and borrowing some facts from the theory of deformations of Fuchsian groups, we exhibit a family of polynomial vector fields in $\mathbf{C}^{3}$ having only single-valued solutions. The solutions of vector fields within this family are defined in domains which had not been previously observed as domains of definition of solutions of polynomial vector fields in $\mathbf{C}^{3}$. For example, we obtain polynomial vector fields having solutions defined in domains that are bounded by a fractal curve.

We introduce a polynomial invariant of virtual magnetic link diagrams. For virtual links, this polynomial equals the polynomial invariant defined in [3]. We show that it gives lower bounds of the classical crossing number and the virtual crossing number of virtual links. Also we give various properties of this polynomial and examples.

We construct moduli spaces of weighted pointed stable rational curves $\bar{M}_{0,n \cdot \epsilon}$ with symmetric weight data by the GIT quotient of moduli spaces of weighted pointed stable maps $\bar{M}_{0,n \cdot \epsilon}(\mathbb{P}^{1},1)$. As a consequence, we prove that the Knudsen--Mumford space $\bar{M}_{0,n}$ of $n$-pointed stable rational curves is obtained by a sequence of explicit blow-ups from the GIT quotient $(\mathbb{P}^{1})^{n}\qquotient \mathit{SL}(2)$ with respect to the symmetric linearization $\mathcal{O}(1, \ldots, 1)$. The intermediate blown-up spaces turn out to be $\bar{M}_{0,n \cdot \epsilon}$ for suitable ranges of $\epsilon$. As an application, we provide a new unconditional proof of M. Simpson's theorem about the log canonical models of $\bar{M}_{0,n}$.