Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X\left[2\right]$ of the Jacobian of $X$ in terms of the Ekedahl–Oort type. The interesting feature is that $J_X\left[2\right]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the $2$-torsion group schemes of Jacobians of hyperelliptic $k$-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur.

Let $\mathcal{V}=Spec\left(R\right)$ and $R$ be a complete discrete valuation ring of mixed characteristic $\left(0,p\right)$. For any flat $R$-scheme $\mathcal{X}$, we prove the compatibility of the de Rham fundamental class of the generic fiber and the rigid fundamental class of the special fiber. We use this result to construct a syntomic regulator map ${reg}_{syn}:{CH}^i\left(\mathcal{X}∕\mathcal{V},2i-n\right)\to H_{syn}^n\left(\mathcal{X},i\right)$ when $\mathcal{X}$ is smooth over $R$ with values in the syntomic cohomology defined by A. Besser. Motivated by the previous result, we also prove some of the Bloch–Ogus axioms for the syntomic cohomology theory but viewed as an absolute cohomology theory.

We prove that if all real-valued irreducible characters of a finite group $G$ with Frobenius–Schur indicator $1$ are nonzero at all $2$-elements of $G$, then $G$ has a normal Sylow $2$-subgroup. This result generalizes the celebrated Ito–Michler theorem (for the prime $2$ and real, absolutely irreducible, representations), as well as several recent results on nonvanishing elements of finite groups.

For any finite Coxeter group $W$, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on $W$. The construction of the biHecke monoid relies on the usual combinatorial model for the $0$-Hecke algebra $H_0\left(W\right)$, that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits $\left|W\right|$ simple and projective modules. In order to construct the simple modules, we introduce for each $w\in W$ a combinatorial module $T_w$ whose support is the interval ${\left[1,w\right]}_R$ in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

Shuffle algebras are monoids for an unconventional monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations.

Let $a\left(\lambda \right),b\left(\lambda \right)\in ℂ\left[\lambda \right]$, and let $f_{\lambda }\left(x\right)\in ℂ\left[x\right]$ be a one-parameter family of polynomials indexed by all $\lambda \in ℂ$. We study whether there exist infinitely many $\lambda \in ℂ$ such that both $a\left(\lambda \right)$ and $b\left(\lambda \right)$ are preperiodic for $f_{\lambda }$.

We study $F$-blowups of non-$F$-regular normal surface singularities. Especially the cases of rational double points and simple elliptic singularities are treated in detail.