We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups $G(m,d,n)$ from the corresponding partition for $G(m,1,n)$. This confirms, in the case $W=G(m,d,n)$, a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

For a monomial ideal $I$ of a polynomial ring $S$, a polarization of $I$ is a square-free monomial ideal $J$ of a larger polynomial ring $\tilde{S}$ such that $S/I$ is a quotient of $\tilde{S}/J$ by a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sends $x{y}^2\in S$ to $x_1{y}_1{y}_2\in \tilde{S}$, ours sends it to $x_1{y}_2{y}_3$. Using this idea, we recover/refine the results on square-free operation in the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of $p$-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension $n$ and it is not $p$-parabolic for $p>n-1$, then it admits a lot of $p$-harmonic functions with finite Dirichlet sum of order $p$.

The chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the $E_2$-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The $E_1$-term $E_1^{s,t}\left(k\right)$ of the spectral sequence is an Ext group of $B{P}_{\ast }BP$-comodules. There is a sequence of Ext groups $E_1^{s,t}(n-s)$ for nonnegative integers $n$ with $E_1^{s,t}\left(0\right)=E_1^{s,t}$, and there are Bockstein spectral sequences computing a module $E_1^{s,\ast }(n-s)$ from $E_1^{s-1,\ast }(n-s+1)$. So far, a small number of the $E_1$-terms are determined. Here, we determine the $E_1^{1,1}(n-1)={Ext}^1{M}_{n-1}^1$ for $p>2$ and $n>3$ by computing the Bockstein spectral sequence with $E_1$-term $E_1^{0,s}\left(n\right)$ for $s=1,2$. As an application, we study the nontriviality of the action of ${\alpha }_1$ and ${\beta }_1$ in the homotopy groups of the second Smith-Toda spectrum $V\left(2\right)$.