To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetrahedral moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length.

Under K.-T. Sturm's formulation, we obtain a Gaussian upper bound for tail probability of mean value of independent, identically distributed random variables with values in $\mathbb{R}$-trees and Hadamard manifolds.

In [7], Yasushi Mizusawa gives computations which lead to a pro-$2$-presentation of the Galois group of the maximal unramified pro-$2$-extension of the cyclotomic $\mathbb{Z}_{2}$-extension over some imaginary quadratic fields, with low $\lambda$-invariants. We show that these methods can be applied to some maximal tamely ramified pro-$2$-extensions, depending on the quadratic imaginary field, and the condition of ramification.

We prove that a locally Stein open subset of the blow-up of $\mathbb{C}^{n}$ at a point is Stein if and only if it does not contain a subset of the form $U \setminus A$ where $A$ is the exceptional divisor and $U$ is an open neighborhood of $A$. We also study an analogous statement for locally Stein open subsets of line bundles over $\mathbb{P}^{n}$.

The fiber-sum construction gives us many interesting examples of Lefschetz fibrations. Which Lefschetz fibrations can be decomposed as fiber-sums? Stipsicz obtained some results on the fiber-sum decomposition, which state about the relationship between the minimality and the fiber-sum decomposability of Lefschetz fibrations. He proved that every Lefschetz fibration with section of self-intersection number $-1$ cannot be decomposed as any nontrivial fiber-sum. In this paper, we show that the reverse of this theorem does not hold and we characterize genus-2 decomposable Lefschetz fibrations with $b_{2}^{+} = 1$.

For a knot $K$ in $S^{3}$, J. Ma and R. Qiu defined an integral invariant $a(K)$ which is the minimal number of elements that generate normally the commutator subgroup of the knot group, and showed that it is a lower bound of the unknotting number. We prove that it is also a lower bound of the tunnel number. If the invariant were additive under connected sum, then we could deduce something about additivity of both the unknotting numbers and the tunnel numbers. However, we found a sequence of examples that the invariant is not additive under connected sum. Let $T(2, p)$ be the $(2, p)$-torus knot, and $K_{p, q}=T(2, p) \sharp T(2, q)$. Then we have $a(K_{p, q})=1$ if and only if $\gcd(p, q)= 1$.

In his two pioneering articles [9, 10] Jerry Levine introduced and completely determined the algebraic concordance groups of odd dimensional knots. He did so by defining a host of invariants of algebraic concordance which he showed were a complete set of invariants. While being very powerful, these invariants are in practice often hard to determine, especially for knots with Alexander polynomials of high degree. We thus propose the study of a weaker set of invariants of algebraic concordance---the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. Also, we obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference ``Fifty years since Milnor and Fox'' held at Brandeis University on June 2--5, 2008.

The aim of this paper is to develop a theory of decomposition in the weighted modulation spaces $M_{p,q}^{s,W}$ with $0 < p,q \le \infty$, $s \in \mathbb{R}$ and $W \in A_{\infty}$, where $W$ belongs to the class of $A_{\infty}$ defined by Muckenhoupt. For this purpose we shall define molecules for the modulation spaces. As an application we give a simple proof of the boundedness of the pseudo-differential operators with symbols in $M_{\infty,\min(1,p,q)}^{0}$. We shall deal with dual spaces as well.

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen--Ruan cohomology groups of an almost complex quasitoric orbifold.

The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my papers [7] and [8], and for the space $\mathrm{SU}(3)/\mathrm{SO}(3)$ in Section 6 of [9], we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: $\mathrm{SU}(6)/\mathrm{Sp}(3)$, $\mathrm{SO}(10)/\mathrm{U}(5)$, $E_{6}/(\mathrm{U}(1) \cdot \mathrm{Spin}(10))$, $E_{6}/F_{4}$, $G_{2}/\mathrm{SO}(4)$, $\mathrm{SU}(3)$, $\mathrm{Sp}(2)$ and $G_{2}$. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([5], \S9) is incomplete. In particular, in the spaces $\mathrm{Sp}(2)$, $G_{2}/\mathrm{SO}(4)$ and $G_{2}$, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a ``skew'' position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [5].

We shall prove that for any finite $p$-group $G$, there exists an elementary abelian $p$-extension $k/\mathbf{Q}$ and an unramified extension $K/k$ such that the Galois group $\mathrm{Gal}(K/k)$ is isomorphic to $G$.

A normalized holomorphic motion of a closed set in the Riemann sphere, defined over a simply connected complex Banach manifold, can be extended to a normalized quasiconformal motion of the sphere, in the sense of Sullivan and Thurston. In this paper, we show that if the given holomorphic motion, defined over a simply connected complex Banach manifold, has a group equivariance property, then the extended (normalized) quasiconformal motion will have the same property. We then deduce a generalization of a theorem of Bers on holomorphic families of isomorphisms of Möbius groups. We also obtain some new results on extensions of holomorphic motions. The intimate relationship between holomorphic motions and Teichmüller spaces is exploited throughout the paper.