Let $X$ be a CW-complex with basepoint. We obtain a simple description of the Borel construction on the free loopspace of the suspension of $X$ as a wedge of the classifying space of the circle and the homotopy colimit of a diagram consisting of products of a number of copies of $X$ and the standard topological $n$-simplex. This is obtained by filtering the cyclic bar construction on the James model of the based loopspace by word length in order to express the homotopy type of the free loopspace as a colimit of powers of $X$ and standard cyclic sets. It is shown that this colimit is in fact a homotopy colimit and commutativity of homotopy colimits is used to describe the Borel construction.

Let $M^{2n}$ denote a closed ($n-1$)-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\bar{\Theta}_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_{c}(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_{h}(M^{2n})=0$ for $n=4$. On the way, following Wall's approach [16] we present a new proof of the main result in [9], namely, for $n=4$, $8$ and $H^{n}(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong \mathbb{Z}_{2}$. We also show that, up to orientation-preserving diffeomorphism, $M^{8}$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $\pi$-manifold, $M^{14}$ has exactly two distinct smooth structures.

Let $k$ be an algebraic number field of finite degree and $k_{\infty}$ be the maximal cyclotomic extension of $k$. Let $\tilde{L}_{k}$ and $L_{k}$ be the maximal unramified Galois extension and the maximal unramified abelian extension of $k_{\infty}$ respectively. We shall give some remarks on the Galois groups $\mathrm{Gal}(\tilde{L}_{k}/k_{\infty})$, $\mathrm{Gal}(L_{k}/k_{\infty})$ and $\mathrm{Gal}(\tilde{L}_{k}/k)$. One of the remarks is concerned with non-solvable quotients of $\mathrm{Gal}(\tilde{L}_{k}/k_{\infty})$ when $k$ is the rationals, which strengthens our previous result.

We extend an equivariant mountain pass theorem, due to Bartsch, Clapp and Puppe for compact Lie groups to the setting of infinite discrete groups satisfying a maximality condition on their finite subgroups.

It is known that every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. In this paper, we give another proof which improves the result of Korkmaz. In addition, Korkmaz defined the genus of a finitely presented group. We also evaluate upper bounds for genera of some finitely presented groups.

Buchstaber invariant is a numerical characteristic of a simplicial complex (or a polytope), measuring the degree of freeness of the torus action on the corresponding moment-angle complex. Recently an interesting combinatorial theory emerged around this invariant. In this paper we answer two questions, considered as conjectures in [2], [11]. First, Buchstaber invariant of a convex polytope $P$ equals $1$ if and only if $P$ is a pyramid. Second, there exist two simplicial complexes with isomorphic bigraded $\mathrm{Tor}$-algebras, which have different Buchstaber invariants. In the proofs of both statements we essentially use the result of N. Erokhovets, relating Buchstaber invariant of simplicial complex $K$ to the distribution of minimal non-simplices of $K$. Gale duality is used in the proof of the first statement. Taylor resolution of a Stanley--Reisner ring is used for the second.

We derive explicit inequalities for sums of eigenvalues of one-dimensional Schrödinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.

In this note, we consider the three-dimensional (3D) incompressible Boussinesq equations. We obtain the logarithmically improved regularity criterion of smooth solutions in terms of the velocity field. This result improves some previous works.

The concept of a module $M$ being almost $N$-injective, where $N$ is some module, was introduced by Baba (1989). For a given module $M$, the class of modules $N$, for which $M$ is almost $N$-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module $U$ is almost $V$-injective, where $V$ is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of $V$. Let $M$ be a uniform module and $V$ be a finite direct sum of indecomposable modules. Some conditions under which $M$ is almost $V$-injective are determined, thereby Baba's result is generalized. A module $M$ that is almost $M$-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is ($-$)-amphicheiral with odd minimal crossing number. We call the latter knot a Stoimenow knot. We also show that the Stoimenow knot is not invertible by the Alexander polynomials.

We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of type $C_{l}^{(1)}$.

We study a nonlinear elliptic equation with a singular term and a continuous perturbation. We look for positive solutions. We prove three multiplicity theorems producing at least two positive solutions. The first multiplicity theorem concerns equations driven by a nonhomogeneous in general differential operator. Also, two of the theorems have a superlinear perturbation (but without the Ambrosetti--Rabinowitz condition), while the third has a sublinear perturbation. Our approach is variational together with suitable truncation and comparison techniques.

In this paper, we study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of the Cotton--York tensor and the $L^{1}$-norm of it under the Ricci flow. In particular, we investigate the behavior of the $L^{1}$-norm of the Cotton--York tensor under the Ricci flow on three-dimensional simply-connected Riemannian homogeneous spaces which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the $L^{1}$-norm for the product metric of the Rosenau solution for the Ricci flow on $S^{2}$ and the standard metric of $S^{1}$.

We prove the abundance theorem for semi log canonical surfaces in positive characteristic.