We consider two overlapping classes of fields, IAC and VAC, which are defined using valuation theory but which do not involve a distinguished valuation. Rather, each class is defined by a condition that quantifies over all possible valuations on the field. In his thesis, Hong asked whether these two classes are equal. In this paper, we give an example that negatively answers Hong’s question. We also explore several situations in which the equivalence does hold with an additional assumption, including the case where every $K^{\prime }\equiv K$ is IAC.

In this paper, we extend the constructions of Boava and Exel to present the C^∗-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation, we present another way to study such C^∗-algebras, only using tools from partial crossed products.

For each countable ordinal α, let ${\mathcal{S}}_{\mathit{\alpha }}$ be the Schreier set of order α and $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ be the corresponding Schreier space of order α. In this paper, we prove several new properties of these spaces.

The fact that for nonzero α, $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ is polyhedral and has the λ-property implies that each $X_{{\mathcal{S}}_{\mathit{\alpha }}}$ is an example of a space that solves a problem of Lindenstrauss from 1966 about the existence of a polyhedral infinite dimensional Banach space whose unit ball is the closed convex hull of its extreme points. The first example of such a space was given by De Bernardi in 2017 using a renorming of $c_0$.

We show that a compact half-conformally flat manifold of negative type with bounded $L^2$ energy, sufficiently small scalar curvature, and a noncollapsing assumption has all Betti numbers bounded in terms of the $L^2$ curvature norm. We give examples of multi-ended bubbles that disrupt attempts to improve these Betti number bounds. We show that bounded self-dual solutions of $d\mathit{\omega }=0$ on asymptotically locally Euclidian (ALE) manifold ends display a rate-of-decay gap: they are either asymptotically Kähler, or they have a decay rate of $O(r^{-4})$ or better.

The F-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focusing on the number of nonfree direct summands. In this paper, we provide several methods to compute the (generalized) F-signature of a Hibi ring which is a special class of toric rings. In particular, we show that it can be computed by counting the elements in the symmetric group satisfying certain conditions. As an application, we also give the formula of the (generalized) F-signature for some Segre products of polynomial rings.

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted $L^p$ spaces over ${\mathbb{R}}^d$ with weights of the form $exp(-\mathit{\varphi }(x))$, for ϕ a $C^2$ function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on $L^p$ is determined, and its properties are analyzed. This theory is used to calculate an upper bounded on the $H^{\mathrm{\infty }}$ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.

We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen–Series boundary maps, and their natural extensions, associated to co-compact torsion-free Fuchsian groups. If the boundary map parameters are extremal—that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon—then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. Moreover, each extremal parameter choice has a corresponding dual parameter choice such that the “past” of the arithmetic code of a geodesic is the “future” for the code using the dual parameter. This duality was observed for two classical parameter choices by Adler and Flatto; here we show constructively that every extremal parameter choice has a dual.

Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko’s fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.

We show that certain $L^2$ space-time estimates for generalized density matrices which have been used by several authors in recent years to study equations of BBGKY or Hartree-Fock type, do not have non-trivial $L^p{L}^q$ generalizations.

Let ν be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field K. We prove comparison theorems between MacLane–Vaquié key polynomials for valuations $\mathrm{\mu }\le \mathrm{\nu }$ and abstract key polynomials for ν. Also, some results on invariants associated to limit key polynomials are obtained. In particular, if $char(K)=0$, we show that all the limit key polynomials of unbounded continuous families of augmentations have the numerical character equal to one.

We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that, we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.