This paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

The problem of resolution of singularities in positive characteristic can be reformulated as follows: fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps. The first consists of the transformation of $X$ to a hypersurface with $n$-fold points in the so-called monomial case. The second step consists of the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to present a notion of strong monomial case which parallels that of monomial case in characteristic zero: if a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

For an arbitrary connected reductive group $G$, we consider the motivic integral over the arc space of an arbitrary $ \mathbb{Q} $-Gorenstein horospherical $G$-variety ${X}_{\Sigma } $ associated with a colored fan $\Sigma $ and prove a formula for the stringy $E$-function of ${X}_{\Sigma } $ which generalizes the one for toric varieties. We remark that, in contrast to toric varieties, the stringy $E$-function of a Gorenstein horospherical variety ${X}_{\Sigma } $ may be not a polynomial if some cones in $\Sigma $ have nonempty sets of colors. Using the stringy $E$-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.

The main result of this paper states that if $k$ is a field of characteristic $p\gt 0$ and $A/ k$ is a central simple algebra of index $d= {p}^{n} $ and exponent ${p}^{e} $, then $A$ is split by a purely inseparable extension of $k$ of the form $k( \sqrt[{p}^{e} ]{{a}_{i} }, i= 1, \ldots , d- 1)$. Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most $d- 1$ cyclic algebras of degree ${p}^{e} $. This gives a drastic improvement upon previously known upper bounds.

For a $ \mathbb{Z} $-cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.

Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on ${T}^{2} \times {S}^{3} $ and certain related 5-manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.