There are four groups $G$ fitting into a short exact sequence $1\rightarrow\operatorname{SL}(2,5)\rightarrow G\rightarrow C_2\rightarrow 1,$ where $\operatorname{SL}(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five elements. Except for the direct product of $\operatorname{SL}(2,5)$ and $C_2$, there are two other semidirect products of these two groups and just one non-semidirect product $\operatorname{SL}(2,5).C_2$, considered in this paper. It is known that each finite nonsolvable group can act on spheres with arbitrary positive number of fixed points. Clearly, $\operatorname{SL}(2,5).C_2$ is a nonsolvable group. Moreover, it turns out that $\operatorname{SL}(2,5).C_2$ possesses a free representation and as such, can potentially act pseudofreely with nonempty fixed point set on manifolds of arbitrarily large dimension. We prove that $\operatorname{SL}(2,5).C_2$ cannot act effectively with odd number of fixed points on homology spheres of dimensions less than $14$. In the special case of effective one fixed point actions on homology spheres, we are able to exclude $15$, $16$, and $17$ from the dimension of them. Moreover, we prove that $5$-pseudofree one fixed point actions of $\operatorname{SL}(2,5).C_2$ on spheres do not exist.

Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\mathrm{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.

We study a generalized nonlinear Schrödinger equations with spatial variable coefficients, which models the remarkable inhomogeneous Schrö dinger maps (ISM). A new weighted Sobolev space $\mathcal{W}^{1,q}(\Bbb{R}^{+})$ is introduced and the existence of blow-up solutions of this equations, including the integrable radial ISM, with the initial data in $\mathcal{W}^{1,2}(\Bbb{R}^{+})$ is proved.

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$ for integers $m$. Namely, we first provide a condition on $m$ that gives a bound of the size of Selmer group and then we provide a condition on $m$ that makes $L$-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch-Swinnerton-Dyer conjecture.

We prove that the extended mapping class group, $\mod^{*}(\Sigma_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\mod^{*}(\Sigma_{g,p})$ can be generated by three involutions for $g\geq 10$ and $p\geq 6$ (with the exception that for $g\geq 11$, $p$ should be at least $15$).

For each connected alternating tangle, we provide an infinite family of non-left-orderable $L$-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational homology $3$-sphere is an $L$-space if and only if it is non-left-orderable. These $3$-manifolds are obtained as Dehn fillings of the double branched covering of any alternating encircled tangle. We give a presentation of these non-left-orderable $L$-spaces as double branched coverings of $S^3$, branched over some specified links that turn out to be hyperbolic. We show that the obtained families include many non-Seifert fibered spaces. We also show that these families include many Seifert fibered spaces and give a surgery description for some of them. In the process we give another way to prove that the torus knots $T(2,2m+1)$ are $L$-space-knots as has already been shown by Ozsváth and Szabó in [24].

The restricted Feynman path integrals (RFPIs) have been proposed to study continuous quantum measurements in physics. The RFPIs are heuristically determined in terms of the usual probability amplitude multiplied by weight for each path, which contains information about the results and the resolution of the measuring device. In the present paper we will consider the RFPIs particularly for the position measurements and will prove rigorously that these RFPIs are well defined in the $L^{2}$ space and are the solutions to the non-self-adjoint Schrödinger equations. Our results in the present paper give a generalization of the results on the usual Feynman path integrals for the Schrödinger equations. Furthermore, our results are extended to quantum spin systems.

In this paper, first we introduce a general notion of affine Killing vector fields on the complex quadric $Q^{m}$, which is weaker than usual Killing vector field. Next, we give a complete classification of Hopf real hypersurfaces $M$ with affine Killing Reeb vector field in the complex quadric $Q^{m}$, $m \geq 3$.

A Fuchsian system of rank 8 in 3 variables with 4 parameters is found. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equations, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions of these ordinary differential equations are found. Their coefficients are \textit{sums} of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series $_4F_3$ at 1.

We establish a relative Bertini type theorem for multiplier ideal sheaves. Then we prove a relative version of the Kollár-Nadel type vanishing theorem as an application.

In this paper, we consider the equivariant index of a generalized Bott manifold. We show the multiplicity function of the equivariant index is given by the density function of a generalized twisted cube. In addition, we give a Demazure-type character formula of this representation.