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We prove the relative Grauert-Riemenschneider vanishing, Kawamata-Viehweg vanishing, and Koll\'ar injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses…

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $\beta\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups…

F. Gross conjectured that any meromorphic solution of the Fermat Cubic $F_3\colon\ x^3+y^3=1$ are elliptic functions composed with entire functions. The conjecture was solved affirmatively first by I. N. Baker who found explicit formulas of…

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz…

Let $K$ be a field, and let $f\in K(z)$ be a rational function of degree $d\geq 2$. The Galois group of the field extension generated by the preimages of $x_0\in K$ under all iterates of $f$ naturally embeds in the automorphism group of an…

This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$,…

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category…

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral.…

The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of…

Let G be a group and f be an endomorphism of G. A subgroup H of G is called f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups that are f-inert for all inner automorphisms of G are widely known and studied in…

We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically…

Generalised algebraic theories (GATs) allow multiple sorts indexed over each other. For example, the theories of categories or Martin-L{\"o}f type theories form GATs. Categories have two sorts, objects and morphisms, and the latter are…

This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with…

We study canonical filtrations of finite-dimensional associative algebras and Lie algebras. These filtrations are defined via optimal destabilizing one-parameter subgroups in the sense of geometric invariant theory (GIT), and appear to be a…

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field $F$ and exponents in an additive submonoid $M$ of $\mathbb{Q}_{\ge 0}$ is called a Puiseux algebra and denoted by $F[M]$. Here we study the…

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le…

Let $R$ be the complete local ring of a complex plane curve germ and $S$ its normalization. We propose a "Hilb-vs-Quot" conjecture relating the virtual weight polynomials of the Hilbert schemes of $R$ to those of the Quot schemes that…

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if…

This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras…

Equational reasoning about circuits is central in quantum software for validation, optimisation, and verification. For qubits, the CNOT-dihedral fragment supports efficient rewriting via phase polynomials and layered normal forms, yielding…