We calculate the motivic Donaldson–Thomas invariants for $\left(-2\right)$-curves arising from $3$-fold flopping contractions in the minimal model program. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed earlier by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson–Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.

We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion for silting-discreteness.

By a “generalized Calabi–Yau hypersurface” we mean a hypersurface in ${\mathbb{P}}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function $\mathcal{ℱ}$. That function $\mathcal{ℱ}$ is the $p$-adic analytic continuation of the ratio $F\left(\Lambda \right)∕F\left({\Lambda }^p\right)$, where $F\left(\Lambda \right)$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard–Fuchs equation of the family.

We propose upper bounds for the number of modular constituents of the restriction modulo $p$ of a complex irreducible character of a finite group, and for its decomposition numbers, in certain cases.

Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in ${ℝ}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one “slice” corresponding to monic polynomials; this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser–Vaaler and Barroero.

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a,b\in ℂ\left[x\right]$, there is a polynomial $h$ such that for all $n$, we have

$$gcd\left(a^n-1,b^n-1\right)|h$$

We prove a compositional analog of this theorem, namely that if $f,g\in ℂ\left[x\right]$ are compositionally independent polynomials and $c\left(x\right)\in ℂ\left[x\right]$, then there are at most finitely many $\lambda$ with the property that there is an $n$ such that $\left(x-\lambda \right)$ divides $gcd\left(f^{\circ n}\left(x\right)-c\left(x\right),g^{\circ n}\left(x\right)-c\left(x\right)\right)$.

Suppose that $R$ is a 2-dimensional excellent local domain with quotient field $K$, $K^{\ast }$ is a finite separable extension of $K$ and $S$ is a 2-dimensional local domain with quotient field $K^{\ast }$ such that $S$ dominates $R$. Suppose that ${\nu }^{\ast }$ is a valuation of $K^{\ast }$ such that ${\nu }^{\ast }$ dominates $S$. Let $\nu$ be the restriction of ${\nu }^{\ast }$ to $K$. The associated graded ring ${gr}_{\nu }\left(R\right)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $\left(K,\nu \right)\to \left(K^{\ast },{\nu }^{\ast }\right)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blowup of $R$, $S_1$ is a local ring of a blowup of $S$, ${\nu }^{\ast }$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${gr}_{{\nu }^{\ast }}\left(S_1\right)$ is a finitely generated ${gr}_{\nu }\left(R_1\right)$-algebra.

We also investigate the role of splitting of the valuation $\nu$ in $K^{\ast }$ in finite generation of the extensions of associated graded rings along the valuation. We say that $\nu$ does not split in $S$ if ${\nu }^{\ast }$ is the unique extension of $\nu$ to $K^{\ast }$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, ${\nu }^{\ast }$ has rational rank 1 and is not discrete and ${gr}_{{\nu }^{\ast }}\left(S\right)$ is a finitely generated ${gr}_{\nu }\left(R\right)$-algebra, then $S$ is a localization of the integral closure of $R$ in $K^{\ast }$, the extension $\left(K,\nu \right)\to \left(K^{\ast },{\nu }^{\ast }\right)$ is without defect and $\nu$ does not split in $S$. We give examples showing that such a strong statement is not true when $\nu$ does not satisfy these assumptions. As a consequence, we deduce that if $\nu$ has rational rank 1 and is not discrete and if $R\to R^{\prime }$ is a nontrivial sequence of quadratic transforms along $\nu$, then ${gr}_{\nu }\left(R^{\prime }\right)$ is not a finitely generated ${gr}_{\nu }\left(R\right)$-algebra.