We establish a large deviation principle for the occupation distribution of a symmetric Markov process normalized by Feynman--Kac functional. The obtained theorem means a large deviation from a ground state, not from an invariant measure.

For all but finitely many compact orientable surfaces, we show that any superinjective map from the complex of separating curves into itself is induced by an element of the extended mapping class group. We apply this result to proving that any finite index subgroup of the Johnson kernel is co-Hopfian. Analogous properties are shown for the Torelli complex and the Torelli group.

A finitely generated torsion free nilpotent group is called an $\mathrsfs{F}$-group. To each $\mathrsfs{F}$-group $\Gamma$ there is associated a connected, simply connected nilpotent Lie group $G_{\Gamma}$. Let TUF be the class of all $\mathrsfs{F}$-group $\Gamma$ such that $G_{\Gamma}$ is totally unimodular. A group in TUF is called TUF-group. In this paper, we are interested in finding non-zero Euler characteristic on the class TUF and therefore, on TUFF, the class of groups $K$ having a subgroup $\Gamma$ of finite index in TUF. An immediate consequence we obtain that any two isomorphic finite index subgroups of a TUFF-group have the same index. As applications, we give two results, the first is a generalization of Belegradek's result, in which we prove that every TUFF-group is co-hopfian. The second is a known result due to G.C. Smith, asserting that every TUFF-group is not compressible.

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant $\tau_{L}(R) := -K_{X} \cdot \Gamma/(L \cdot \Gamma)$, where $\Gamma$ is a minimal curve of an extremal ray $R := \mathbb{R}_{+}[\Gamma]$ on $X$ such that $L \cdot R > 0$.

The objective of this note is to prove an existence result for brake orbits in classical Hamiltonian systems (which was first proved by S.V. Bolotin) by using Floer theory. To this end, we compute an open string analogue of symplectic homology (so called wrapped Floer homology) of some domains in cotangent bundles, which appear naturally in the study of classical Hamiltonian systems. The main part of the computations is to show invariance of wrapped Floer homology under certain handle attaching to domains.

We prove the existence of torus invariant almost complex structure on any positively omnioriented four dimensional primitive quasitoric orbifold. We construct pseudo-holomorphic blowdown maps for such orbifolds. We prove a version of McKay correspondence when the blowdowns are crepant.

The uniqueness of invariant measure is one of the most interesting problems in theory of Markov processes. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.

We prove that the asymptotic completion of a developable Möbius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip is generated by a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.

For a Lie group $G = \mathbb{R}^{n}\ltimes_{\phi}\mathbb{R}^{m}$ with the semi-simple action $\phi\colon \mathbb{R}^{n}\to \Aut(\mathbb{R}^{m})$, we show that if $\Gamma$ is a finite extension of a lattice of $G$ then $K(\Gamma, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $\Gamma$ satisfies the hard Lefschetz property. By those results we give many examples of formal solvmanifolds satisfying the hard Lefschetz property but not admitting Kähler structures.

We introduce a family of orientable $3$-manifolds induced by certain cyclically presented groups and show that this family of $3$-manifolds contains all Dunwoody $3$-manifolds by using the planar graphs corresponding to the polyhedral description of the $3$-manifolds. As applications, we consider two families of cyclically presented groups, and show that these are isomorphic to the fundamental groups of the certain Dunwoody $3$-manifolds $D_{n}$ ($n \geq 2$) which are the $n$-fold cyclic coverings of the $3$-sphere branched over the certain two-bridge knots, and that $D_{n}$ is the $(\mathbb{Z}_{n}\oplus \mathbb{Z}_{2})$-fold covering of the $3$-sphere branched over two different $\Theta$-curves.

We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration space.

Recently, Y. Ollivier defined the Ricci curvature of Markov chains on Polish spaces via the contractivity of transition kernels under the $L^{1}$ Wasserstein metric. In this paper, we will discuss further the spectral gap, entropy decay, and logarithmic Sobolev inequality for the $\lambda$-range gradient operator. As an application, given resistance forms (i.e. symmetric Dirichlet forms with finite effective resistance) on fractals, we can construct Markov chains with positive Ricci curvature, which yields the Gaussian-then-exponential concentration of invariant distributions for Lipschitz test functions.

A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.

Recently we generalized Toponogov's comparison theorem to a complete Riemannian manifold with smooth convex boundary, where a geodesic triangle was replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface was replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. The aim of this article is to prove splitting theorems of two types as an application. Moreover, we establish a weaker version of our Toponogov comparison theorem for open triangles, because the weaker version is quite enough to prove one of the splitting theorems.

Let $p$ be a fixed odd prime number and $K_{n}$ the $p^{n+1}$-st cyclotomic field. For a fixed integer $d \in \boldsymbol{Z}$ with $\sqrt{d} \notin K_{0}$, denote by $L_{n}$ the imaginary quadratic subextension of the biquadratic extension $K_{n}(\sqrt{d})/K_{n}^{+}$ with $L_{n} \neq K_{n}$. Let $h_{n}^{*}$ and $h_{n}^{-}$ be the relative class numbers of $K_{n}$ and $L_{n}$, respectively. We give an explicit constant $n_{d}$ depending on $p$ and $d$ such that (i) for any integer $n \geq n_{d}$, the ratio $h_{n}^{-}/h_{n-1}^{-}$ is odd if and only if $h_{n}^{*}/h_{n-1}^{*}$ is odd and (ii) for $1 \leq n < n_{d}$, $h_{n}^{-}/h_{n-1}^{-}$ is even.

We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a surjective morphism to the Thompson group of piecewise linear automorphisms of $\mathbb{Z}^{2}$. We prove that the presentation for this group conjectured by Usnich is correct.