Let $G$ be a finite group. It is known that if a homotopy sphere $X$ has a one-fixed-point smooth $G$-action then the dimension of $X$ is greater than or equal to $6$. It is also known that there is an effective $2$-pseudofree one-fixed-point smooth $G$-action on the sphere $S^n$ of dimension $n$ if and only if $n$ is equal to $6$ and $G$ is isomorphic to the alternating group $A_5$ on five letters. E. Stein proved that for the group $G = {\textrm{SL}}(2, 5) \times Z_m$ such that $m$ is prime to $30$, there is a $3$-pseudofree one-fixed-point smooth $G$-action on $S^7$, where $Z_m$ is a cyclic group of order $m$. In this article, we determine the finite groups $G$ possessing $3$-pseudofree one-fixed-point smooth $G$-actions on $S^6$. In addition, for an arbitrary finite group $G$ isomorphic to $A_5$, $A_5 \times Z_2$, or ${\textrm{SL}}(2, 5) \times Z_m$ such that $m$ is prime to $30$, we prove that there is a $3$-pseudofree one-fixed-point smooth $G$-action on $S^7$.

We study the relationship between the central ideals and the Jacobson radicals of blocks of group algebras. In particular, we characterize the blocks with the property that the square of the radical is a non-zero central ideal. Moreover, we also consider the blocks with the property that the cube of the radical becomes a non-zero central ideal. These are improvements of a result by Külshammer in 2020.

We show pathwise uniqueness for a class of degenerate Itô-SDE among all of its weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Consequently, by the Yamada-Watanabe Theorem and a weak existence result, the pathwise unique solutions can be shown to be strong and to exist. The main tools to show pathwise uniqueness are inequalities associated with maximal functions and a Krylov type estimate derived from elliptic regularity and uniqueness in law.

We prove that a radial Kähler metric $g$ is Kähler-Einstein if and only if one of the following conditions is satisfied: 1. $g$ is extremal and it is associated to a Kähler-Ricci soliton; 2. two different generalized scalar curvatures of $g$ are constant; 3. $g$ is extremal (not cscK) and one of its generalized scalar curvature is constant.

We prove that the blowup of the jet scheme of a singular hypersurface along a certain jet-related module is not an isomorphism. In conjunction with recent developments in the theory of Nash blowups, our result holds over fields of arbitrary characteristic. Our approach is based on explicit presentations given by a higher-order Jacobian matrix combined with a certain jet-related matrix.

In this article, we prove that Buchstaber invariant of 4-dimensional real universal complex is no less than 24 as a follow-up to the work of Ayzenberg and Sun. Moreover, a lower bound for Buchstaber invariants of $n$-dimensional real universal complexes is given as an improvement of result of Erokhovets.

We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields and present some sufficient conditions for this to hold.

A quandle is an algebraic system which excels at describing limited symmetries of a space. We introduce the concept of Schläfli quandles which are defined relating to chosen rotational symmetries of regular tessellations. On the other hand, quandles have a good chemistry with knot theory. Associated with a knot we have its knot quandle. We show that the knot quandle of the $m$-twist-spun trefoil is a central extension of the Schläfli quandle related to the regular tessellation $\{ 3, m \}$ in the sense of the Schläfli symbol if $m \geq 3$.

Extended affine Lie superalgebras are super versions of extended affine Lie algebras and, more generally, invariant affine reflection algebras. By employing a method known as “affinization", we construct several classes of extended affine Lie superalgebras of arbitrary nullity.

It is well-known that the $T$-fixed points of a Schubert variety in the flag variety $GL_n({\mathbb C})/B$ can be characterized purely combinatorially in terms of Bruhat order on the symmetric group ${\mathfrak{S}}_n$. In a recent preprint, Cho, Hong, and Lee give a combinatorial description of the $T$-fixed points of Hessenberg analogues of Schubert varieties (which we call Hessenberg Schubert varieties) in a regular semisimple Hessenberg variety. This note gives an interpretation of their result in terms of Bruhat order by making use of a partition of the symmetric group defined using so-called subsets of Weyl type. The Appendix, written by Michael Zeng, proves a lemma concerning subsets of Weyl type which is required in our arguments.

Given a categorical dynamical system, \textit{i.e.} a triangulated category together with an endofunctor, one can try to understand the complexity of the system by computing the entropy of the endofunctor. Computing the entropy of the composition of two endofunctors is hard, and in general the result doesn't have to be related to the entropy of the single pieces.In this paper we compute the entropy of the composition of two spherical twists around spherical objects, showing that it depends on the dimension of the graded vector space of morphisms between them. As a consequence of these computations we produce new counterexamples to Kikuta-Takahashi's conjecture. In particular, we describe the first counterexamples in odd dimension and examples for the $d$-Calabi-Yau Ginzburg dg algebra associated to the $A_2$ quiver.

A region crossing change is a local transformation on spatial graph diagrams switching the over/under relation at all the crossings on the boundary of a region. In this paper, we show that a spatial graph of a planar graph is unknottable by region crossing changes if and only if the spatial graph is non-Eulerian or is Eulerian and proper.

We construct a new compactification of Teichmüller space. We prove that this new compactification is finer than the Gardiner-Masur compactification of Teichmüller space and the action of the mapping class group on Teichmüller space extends continuously to this new compactification. We also construct some special points in the new boundary. The construction of the new compactification is based on the Hubbard-Masur theorem, which states that there is an one-to-one corresponding between holomorphic differentials and measured foliations.

We construct torsion Selmer pointed sets, which are a discrete analogue of Selmer schemes. Such torsion objects were first defined by K. Sakugawa under a hypothesis, who also established a control theorem for them. We remove the hypothesis and provide the discrete analogue in full generality, to which Sakugawa's control theorem extends as well. As a key ingredient, we use Lazard's theory to build the unipotent completion of a finitely generated group over $p$-adic integers.

Consider the Schrödinger operator with constant magnetic field and smooth potential $V$ : $H(\epsilon)=H+V(\epsilon x,\epsilon y),\,\, H=D_x^2+(D_y+\mu x)^2,\,\, (x,y)\in \Omega_d,$ with Dirichlet boundary conditions. Here $ \Omega_d=\Pi_{j=1}^d]-a_j,a_j[\times \mathbb R_y^d$. The spectral properties of two operators $H$ and $H(\epsilon)$ are investigated. For $\epsilon$ small enough, we study the effect of the slowly varying potential $V(\epsilon x,\epsilon y)$. In particular, we derive asymptotic trace formula and we give an asymptotic expansion in powers of $\epsilon$ of the spectral shift function corresponding to $(H(\epsilon), H)$.