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For a curve C and a reductive group G in prime characteristic, we relate the de Rham moduli of logarithmic G-connections on C to the Dolbeault moduli of logarithmic G-Higgs bundles on the Frobenius twist of C. We name this result the Log-p-NAHT. It is a logarithmic version of Chen-Zhu's characteristic p Non Abelian Hodge Theorem (p-NAHT). In contrast to the no pole case, the two moduli stacks in the log case are not isomorphic etale locally over the Hitchin base. Instead, they differ by an Artin-Schreier type Galois cover of the base. In contrast to the case over the complex numbers, where some parabolic/parahoric data are needed to specify the boundary behavior of the tame harmonic metrics, no parabolic/parahoric data are needed in Log-p-NAHT. We also establish a semistable version of the Log-p-NAHT, and deduce several geometric and cohomological consequences. In particular, when G=GL_r, the Log-p-NAHT induces an embedding of the intersection cohomology of the degree d Dolbeault moduli to that of the degree pd de Rham moduli, and the embedding is an isomorphism when r is coprime to d and p>r.

We introduce conditions on cones of normal toric varieties under which the polyhedron defining the normalized Nash blowup does not depend on the characteristic of the base field. As a consequence, we deduce several results on the resolution of singularities properties of normalized Nash blowups. In particular, we recover all known results of the families that can be resolved via normalized Nash blowups in positive characteristic. We also provide new families of toric varieties whose 23normalized Nash blowup is non-singular in arbitrary characteristic., Comment: 23 pages,3 figures

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples., Comment: 15 pages, 1 table, 1 figure

In this paper,we extend the definition of the Chern-Simons type characteristic classes in the continuous case to abelian lattice gauge theory. Then, we show that the exterior differential of a k-th Chern-Simons type characteristic class is exactly equal to the coboundary of the cochain of the (k-1)-th Chern-Simons type characteristic classes based upon the noncommutative differential calculus on the lattice., Comment: 12 pages

The classification of finite dimensional Nichols algebras of diagonal type plays an important role in the classification of Hopf algebras by the lifting method of N. Andruskiewitsch and H.-J. Schneider over fields of characteristic zero. In this paper, we obtain the classification theorem of all finite-dimensional rank 5, rank 6 and rank 7 Nichols algebras of diagonal type over fields of positive characteristic. Weyl groupoids and finite arithmetic root systems are crucial tools to the classification theorem., Comment: arXiv admin note: text overlap with arXiv:1911.03555

This paper is concerned with high-order numerical methods for hyperbolic systems of balance laws. Such methods are typically based on high-order piecewise polynomial reconstructions (interpolations) of the computed discrete quantities. However, such reconstructions (interpolations) may be oscillatory unless the reconstruction (interpolation) procedure is applied to the local characteristic variables via the local characteristic decomposition (LCD). Another challenge in designing accurate and stable high-order schemes is related to enforcing a delicate balance between the fluxes, sources, and nonconservative product terms: a good scheme should be well-balanced (WB) in the sense that it should be capable of exactly preserving certain (physically relevant) steady states. One of the ways to ensure that the reconstruction (interpolation) preserves these steady states is to apply the reconstruction (interpolation) to the equilibrium variables, which are supposed to be constant at the steady states. To achieve this goal and to keep the reconstruction (interpolation) non-oscillatory, we introduce a new LCD of equilibrium variables. We apply the developed technique to the fifth-order Ai-WENO-Z interpolation implemented within the WB A-WENO framework recently introduced in [S. Chu, A. Kurganov, and R. Xin, Beijing J. of Pure and Appl. Math., to appear], and illustrate its performance on a variety of numerical examples.

We shall calculate the automorphism group of the Kummer surfaces associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a K3 surface. Moreover we discuss the corresponding complex case and a generalized Kummer surface in characteristic three. The paper has an appendix by the second author., Comment: 33 pages, 18 Figures

This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes a framework and carries out the classification work for arc-transitive maps with solvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the constructions of various new and interesting arc-regular maps.

Coates, Fukaya, Kato, Sujatha and Venjakob come up with a procedure of attaching suitable characteristic element to Selmer groups defined over a non-commutative $p$-adic Lie extension, which is subsequently refined by Burns and Venjakob. By their construction, these characteristic elements are realized as elements in an appropriate localized $K_1$-group. In this paper, we will introduce a notion of modulo $p$ for these elements and study some of their properties. As an application, we study the Greenberg Selmer group of a tensor product of modular forms, where $p$ is an Eisenstein prime for one of these forms.

We compute the $S_n$-equivariant topological Euler characteristic of the Kontsevich moduli space $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$. Letting $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d) \subset \overline{\mathcal{M}}_{1, n}(\P^r, d)$ denote the subspace of maps from curves without rational tails, we solve for the motive of $\overline{\mathcal{M}}_{1, n}(\mathbb{P}^r, d)$ in terms of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)$ and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic $\mathbb{C}^\star$-action on $\mathbb{P}^r$, we derive a closed formula for the Euler characteristic of $\overline{\mathcal{M}}_{1, n}^{\mathrm{nrt}}(\mathbb{P}^r, d)^{\mathbb{C}^\star}$ as an $S_n$-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^r, d)$. Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types $A$ and $B$, as well as the enumeration of graph colourings with prescribed symmetry. We also prove a structural result about the $S_n$-equivariant Euler characteristic of $\overline{\mathcal{M}}_{g, n}(\mathbb{P}^r, d)$ in arbitrary genus.

Using the \texttt{WeylModules} \textsf{GAP} Package, we compute structural information about certain Weyl modules for type $G_2$ in characteristic $2$. This gives counterexamples to two conjectures stated by S.~Donkin in 1990. It also illustrates capabilities of the package, which can in principle be applied to Weyl modules for any simple, simply-connected algebraic group in any characteristic, subject of course to time and space limitations of computational resources., Comment: 35 pages

Supercooled liquid state is a particularly interesting state in that it exhibits several unusual physical properties. To illustrate, the liquid displays a single peak relaxation frequency at high temperatures, which splits into $\alpha$ relaxation and $\beta$ relaxation in the moderately supercooled regime, with relaxation a disappearing at the glass transition temperature. The mechanism underlying these unusual physical properties of liquids has always been one of the important research topics in condensed matter. Here, a new mechanism is proposed. A distinctive physical state is built, and its most salient feature is that its independent variables are difficult or impossible to measure. Theoretical calculations indicate that there exist two sets of measurable variables in this physical state that cannot be measure exactly simultaneously. Moreover, it is easy to reach an erroneous conclusion, namely that ``a system in this physical state is in a superposition of some real states, until it is measured''. Further theoretical calculations demonstrate that there are two new transitions and that $\mathrm{e}^{3}$ and $2\mathrm{e}^{3}$ are characteristic values of these two transitions, respectively, where $\mathrm{e}$ is Euler's number. Considerable experimental data shows that the characteristic value of glass transition appears to be concentrated near $2\mathrm{e}^{3}$ and the characteristic value of another transition (for example, the splitting of relaxation peak) appears to be concentrated near $\mathrm{e}^{3}$.

We give a new proof of a theorem due to Shumakovitch and Wang on base point independence of Khovanov--Rozansky homology in characteristic $p$. Some further symmetries of $\mathfrak{gl}(p)$-homology in characteristic $p$ are also discussed., Comment: 14 pages, comments welcome!

Let $F$ be a locally compact non-Archimedean field, and $\bf G$ a connected quasi-split reductive group over $F$. We are interested in complex irreducible smooth generic representations $\pi$ of ${\bf G}(F)$. When $F$ has positive characteristic, we prove important properties which previously were only available for $F$ of characteristic 0. The first one is the tempered $L$-function conjecture of Shahidi, stating that when $\pi$ as above is tempered, then the $L$-functions attached to $\pi$ by the Langlands-Shahidi method have no pole for ${\rm Re}(s)>0$. We also establish the standard module conjecture of Casselman and Shahidi, saying that if $\pi$ is written as the Langlands quotient of a standard module, then it is in fact the full standard module. Finally, for a split classical group $\bf G$ we prove a useful result on the unramified unitary spectrum of ${\bf G}(F)$.

In this article, the authors determine the optimal regularity of characteristic functions in Besov-type and Triebel--Lizorkin-type spaces under restrictions on the measure of the $\delta$-neighborhoods of the boundary. In particular, the necessary and sufficient conditions for the membership in these spaces of characteristic functions of the snowflake domain and also some spiral type domains are obtained., Comment: 59 pages, Submitted

In this paper, we consider a generalization of McKay correspondence in positive characteristic regarding Euler characteristic of crepant resolutions given by finite subgroups of the special linear group. As the main result, we prove that the generalization holds for groups with a specific semidirect product structure. Apart from such groups, two certain examples for groups with more difficult structure are also given., Comment: 17 pages

In this paper, we study the regularity of solutions to uniformly degenerate elliptic equations in bounded domains under the condition that the characteristic polynomials have varying characteristic exponents., Comment: 60 pages

Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint eigenvalue probability density function of these two ensembles are not known. Nonetheless, as carried out in the recent works of Liu and Zhang, Akemann et al.~and Kulkarni et al., by considering averages of products of characteristic polynomials, analytic progress can be made. Here we show that an approach based on the theory of zonal polynomials provides an alternative to the diffusion operator or supersymmetric Grassmann integrations methods of these works. It has the advantage of not being restricted to a Gaussian unitary invariant measure on the matrix spaces. To illustrate this, as an extension, we consider averages of products and powers of characteristic polynomials for complex symmetric, and complex self dual random matrices subject to a spherical measure. In the case of powers, when comparing against the corresponding real Ginibre, respectively quaternion Ginibre averages with a spherical measure, one finds the qualitative feature of a decreasing (increasing) profile as the magnitude of the argument of the characteristic polynomial increases. This is analogous to the findings of the second two of the cited works in the Gaussian case., Comment: 16 pages

In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-character varieties, and connections between the algebraic geometry of the character variety and the topology of the 3-manifold are established via group actions on trees. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over an arbitrary algebraically closed field. Examples include a change in the $A$-polynomial in characteristic 2, a closed Haken hyperbolic 3-manifold with no detected essential surface, and a closed Haken hyperbolic 3-manifold with an essential surface only detected in characteristic 2., Comment: v2, 48 pages

Let $p$ be a prime number. We consider diagonal $p$-permutation functors over a (commutative, unital) ring $\mathsf{R}$ in which all prime numbers different from $p$ are invertible. We first determine the finite groups $G$ for which the associated essential algebra $\mathcal{E}_\mathsf{R}(G)$ is non zero: These are groups of the form $G=L\rtimes \langle u\rangle$, where $(L,u)$ is a $D^\Delta$-pair. When $\mathsf{R}$ is an algebraically closed field $\mathbb{F}$ of characteristic 0 or $p$, this yields a parametrization of the simple diagonal $p$-permutation functors over $\mathbb{F}$ by triples $(L,u,W)$, where $(L,u)$ is a $D^\Delta$-pair, and $W$ is a simple $\mathbb{F}\mathrm{Out}(L,u)$-module. Finally, we describe the evaluations of the simple functor $\mathsf{S}_{L,u,W}$ parametrized by the triple $(L,u,W)$. We show in particular that if $G$ is a finite group and $\mathbb{F}$ has characteristic $p$, the dimension of $\mathsf{S}_{L,1,\mathbb{F}}(G)$ is equal to the number of conjugacy classes of $p$-regular elements of $G$ with defect isomorphic to $L$.

We investigate deformations of skew group algebras that arise from a finite cyclic group acting on a polynomial ring in positive characteristic, where characteristic divides the order of the group. We allow deformations which deform both the group action and the vector space multiplication. We fully characterize the Poincare-Birkhoff-Witt deformations which arise in this setting from multiple perspectives: a necessary and sufficient condition list, a practical road map from which one can generate examples corresponding to any choice of group algebra element, an explicit formula, and a combinatorial analysis of the class of algebras.

We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of several specific types of hyperplane arrangements via much simpler arguments. Our theorem also generalizes to type B., Comment: 11 pages