The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic $\neq 2$). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete $p$-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of the unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.

We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.

In this article we give two combinatorial properties of elements satisfying the stuffle relations; one showing that double shuffle elements are determined by less than the full set of stuffle relations, and the other a cyclic property of their coefficients. Although simple, the properties have some useful applications, of which we give two. The first is a generalization of a theorem of Ihara on the abelianizations of elements of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$ to elements of the double shuffle Lie algebra in a much larger quotient of the polynomial algebra than the abelianization, namely the trace quotient introduced by Alekseev and Torossian. The second application is a proof that the Grothendieck–Teichmüller Lie algebra $\mathfrak{grt}$ injects into the double shuffle Lie algebra $\mathfrak{ds}$, based on the recent proof by H. Furusho of this theorem in the pro-unipotent situation, but in which the combinatorial properties provide a significant simplification.

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched $G$-covers of a fixed base $X$ in terms of homomorphisms from a suitable fundamental group of $X$ together with $G$-inertia structures on $X$. As applications, we show that finite embedding problems for graphs have proper solutions and prove a Grunwald–Wang type result stating that an arbitrary collection of fibers may be realized by a global cover.

The central topic is this question: is a given $k$-étale algebra $\prod_l E_l/k$ the specialization of a given $k$-cover $f : X \to B$ at some unramified point $t_0 \in B(k)$? Our main tool is a twisting lemma that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields.

This paper gives a geometric interpretation of the generalized (including the regularization relation) double shuffle relation for multiple $L$-values. Precisely it is proved that Enriquez' mixed pentagon equation implies the relations. As a corollary, an embedding from his cyclotomic analogue of the Grothendieck–Teichmüller group into Racinet's cyclotomic double shuffle group is obtained. It cyclotomically extends the result of our previous paper [F3] and the project of Deligne and Terasoma which are the special case $N=1$ of our result.

Suppose that a finite group $G$ is realized in the Cremona group $\mathrm{Cr}_m (k)$, the group of $k$-automorphisms of the rational function field $K$ of $m$ variables over a constant field $k$. The most general version of Noether's problem is then to ask, whether the subfield $K^G$ consisting of $G$-invariant elements is again rational or not. This paper treats Noether's problem for various subgroups $G$ of $\mathfrak{S}_6$, the symmetric group of degree 6, acting on the function field $\boldsymbol{Q}(s, t, z)$ over $k = \boldsymbol{Q}$ of the moduli space $\mathcal{M}_{0,6}$ of $\mathbb{P}^1$ with ordered six marked points. We shall show that this version of Noether's problem has an affirmative answer for all but two conjugacy classes of transitive subgroups $G$ of $\mathfrak{S}_6$, by exhibiting explicitly a system of generators of the fixed field $\boldsymbol{Q}(s,t,z)^G$. In the exceptional cases $G \cong \mathfrak{A}_6, \mathfrak{A}_5$, the problem remains open.

In this survey paper, we explain a moduli space of finite flat models and a dimensional bound of the moduli spaces. We also explain some variant of the moduli space and a conjecture.

We discuss a number of statements in profinite Teichmüller theory, usually in parallel with their discrete counterparts. Some of these can be shown to hold true, others can be brought back to a few conjectures which we state explicitly and are mostly related to the homotopy type of certain simplicial profinite complexes.

We establish the faithfulness of a geometric action of the absolute Galois group of the rationals that can be defined on the discriminantal variety associated to a finite complex reflection group, and review some possible connections with the profinite Grothendieck–Teichmüller group.

The lifting problem that we consider asks: given a smooth curve in characteristic $p$ and a group of automorphisms, can we lift the curve, along with the automorphisms, to characteristic zero? One can reduce this to a local question (the so-called local lifting problem) involving continuous group actions on formal power series rings. In this expository article, we overview much of the progress that has been made toward determining when the local lifting problem has a solution, and we give a taste of the work currently being undertaken. Of particular interest is the case when the group of automorphisms is cyclic. In this case the lifting problem is expected to be solvable—this is the Oort conjecture.

We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of Artin–Mazur and Sullivan which extends also to non-connected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmüller theory where equivariant completions are applied. Along the way we collect and survey the most important known results of Artin–Mazur, Sullivan and Rector about profinite completion of spaces from a modern point of view. So this article is in part of expository nature.

In my talk at the Galois Theoretic Arithmetic Geometry meeting, I described recent joint work with Akio Tamagawa on a finiteness conjecture regarding abelian varieties whose $\ell$-power torsion is constrained in a particular fashion. In the current article, we introduce the conjecture and provide some geometric motivation for the problem. We give some examples of the exceptional abelian varieties considered in the conjecture. Finally, we prove a new result—that the set $\mathscr{A}(\mathbb{Q},2,3)$ of $\mathbb{Q}$-isomorphism classes of dimension 2 abelian varieties with constrained 3-power torsion is non-empty, by demonstrating an explicit element of the set.

In this paper we formulate a refined version of the Oort conjecture on liftings of cyclic Galois covers between curves. We introduce the notion of fake liftings of cyclic Galois covers between curves; their existence would contradict the Oort conjecture, and we study the geometry of their semi-stable models. Finally, we introduce and investigate some examples of the smoothening process, which ultimately aims to show that fake liftings do not exist. This in turn would imply the Oort conjecture.

We discuss the role which might be played in anabelian geometry by the étale homotopy type of Artin and Mazur. We quote some results around the question how local the construction is. Furthermore, we show that, after completion away from the residue characteristics, the functor étale homotopy type factors through the motivic $\mathrm{A}^1$-homotopy category of Morel–Voevodsky.

Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of its fundamental group extension. We refine Nakamura's approach to these sections and obtain an anabelian theorem for hyperbolic genus 0 curves over quite general fields, for example $\mathbb{Q}^{\mathrm{ab}}$.

Let $\Sigma$ be a smooth projective surface, let $f':S'\to\Sigma$ be a double cover of $\Sigma$ and let $\mu :S\to S'$ be the canonical resolution of $S'$. Put $f = f'\circ\mu$. An irreducible curve $D$ on $\Sigma$ is said to be a splitting curve with respect to $f$ if $f^*D$ is of the form $D^+ + D^- + E$, where $D^+ \neq D^-$, $D^- = \sigma_f^* D^+$, $\sigma_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $\mu$. In this article, we consider "reciprocity" concerning splitting curves when $\Sigma$ is a rational ruled surface.

Let $\pi$ be a pro-$\ell$ completion of a free group, and let $\pi = [\pi]_1 \supset [\pi]_2 \supset [\pi]_3 \supset \ldots$ denote the lower central series of $\pi$. Let $G$ be a profinite group acting continuously on $\pi$. First suppose that the action is given by a character. Then the boundary maps $\delta_n: H^1 (G,\pi/[\pi]_n)\to H^2 (G, [\pi]_n/[\pi]_{n+1})$ are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that $\pi_1$ sections of $\mathbb{P}^1_k - \{0,1,\infty\}$ satisfy the condition that associated $n^{th}$ order Massey products in Galois cohomology vanish. For the $\pi_1$ sections coming from rational points, these conditions imply that $$\langle x^{-1},\ldots,x^{-1},(1-x)^{-1},x^{-1},\ldots,x^{-1}\rangle = 0$$ where $x$ in $H^1 (\mathrm{Gal}(\overline{k}/k), \mathbb{Z}_\ell(\chi))$ is the image of an element of $k^*$ under the Kummer map. For the $\pi_1$ sections coming from rational tangent vectors at infinity, these conditions imply that $$\langle x^{-1},\ldots,x^{-1},(-x)^{-1},x^{-1},\ldots,x^{-1}\rangle = 0.$$

In this paper we are studying Lie algebras associated with Galois representations on the fundamental groups of $\mathbb{P}^1_{\bar{\mathbb{Q}}} \setminus ( \{0,\infty\} \cup \mu_{2^n})$ and $\mathbb{P}^1_{\bar{\mathbb{Q}}} \setminus ( \{0,\infty\} \cup \mu_{3^n})$.

We will give a survey on the theory of $p$-adic multiple zeta values and $p$-adic multiple $L$-values.

Let $\Sigma$ be a nonempty set of prime numbers. In the present paper, we continue our study of the pro-$\Sigma$ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields of characteristic zero. Our first main result asserts, roughly speaking, that if an F-admissible automorphism [i.e., an automorphism that preserves the fiber subgroups that arise as kernels associated to the various natural projections of the configuration space under consideration to configuration spaces of lower dimension] of a configuration space group arises from an F-admissible automorphism of a configuration space group [arising from a configuration space] of strictly higher dimension, then it is necessarily FC-admissible, i.e., preserves the cuspidal inertia subgroups of the various subquotients corresponding to surface groups. After discussing various abstract profinite combinatorial technical tools involving semi-graphs of anabelioids of PSC-type that are motivated by the well-known classical theory of topological surfaces, we proceed to develop a theory of profinite Dehn twists, i.e., an abstract profinite combinatorial analogue of classical Dehn twists associated to cycles on topological surfaces. This theory of profinite Dehn twists leads naturally to comparison results between the abstract combinatorial machinery developed in the present paper and more classical scheme-theoretic constructions. In particular, we obtain a purely combinatorial description of the Galois action associated to a [scheme-theoretic!] degenerating family of hyperbolic curves over a complete equicharacteristic discrete valuation ring of characteristic zero. Finally, we apply the theory of profinite Dehn twists to prove a "geometric version of the Grothendieck Conjecture" for—i.e., put another way, we compute the centralizer of the geometric monodromy associated to—the tautological curve over the moduli stack of pointed smooth curves.