Let $G_0$ be a real semisimple Lie group. It acts naturally on every complex flag manifold $Z=G/Q$ of its complexification. Given an Iwasawa decomposition $G_0 = K_0 A_0 N_0$, a $G_0$-orbit $γ⊂Z$, and the dual $K$-orbit $κ⊂Z$, Schubert varieties are studied and a theory of Schubert slices for arbitrary $G_0$-orbits is developed. For this, certain geometric properties of dual pairs $(γ,κ)$ are underlined. Canonical complex analytic slices contained in a given $G_0$-orbit γ which are transversal to the dual $K_0$-orbit $\gamma \cap \kappa$ are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space $Ω_W$($D$) is a Stein domain that contains the universally defined Iwasawa domain $Ω_I$. This is one of the main ingredients in the proof that $Ω_W(D)=Ω_{AG}$ for all but a few Hermitian exceptions. In the Hermitian case, $Ω_W(D)$ is concretely described in terms of the associated bounded symmetric domain.

Let $X$ be a geometrically rational (or, more generally, separably rationally connected) variety over a finite field $K$. We prove that if $K$ is large enough, then $X$ contains many rational curves defined over $K$. As a consequence we prove that $R$-equivalence is trivial on $X$ if $K$ is large enough. These results imply the following conjecture of J.-L. Colliot-Thélène: Let $Y$ be a rationally connected variety over a number field $F$. For a prime $P$, let $Y_P$ denote the corresponding variety over the local field $F_P$. Then, for almost all primes $P$, the Chow group of 0-cycles on $Y_P$ is trivial and $R$-equivalence is trivial on $Y_P$.

Let $K$ be a number field containing the group $μ_n$ of $n$th roots of unity, and let $S$ be a set of primes of $K$ including all those dividing $n$ and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal $S$-ramified extension of $K$ with coefficients in $μ_n$, which yields a pairing on a subgroup of $K^x$ containing the $S$-units. In this general situation, we determine a formula for the cup product of two elements that pair trivially at all local places.

Our primary focus is the case in which $K=\mathbb{Q}({\mu }_p)$ for $n=p$, an odd prime, and $S$ consists of the unique prime above $p$ in $K$. We describe a formula for this cup product in the case that one element is a pth root of unity. We explain a conjectural calculation of the restriction of the cup product to $p$-units for all $p≤10,000$ and conjecture its surjectivity for all $p$ satisfying Vandiver's conjecture. We prove this for the smallest irregular prime $p=37$ via a computation related to the Galois module structure of $p$-units in the unramified extension of $K$ of degree $p$.

We describe a number of applications: to a product map in $K$-theory, to the structure of $S$-class groups in Kummer extensions of $K$, to relations in the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}({\mu }_p)$ unramified outside $p$, to relations in the graded $ℤ_p$-Lie algebra associated to the representation of the absolute Galois group of $\mathbb{Q}$ in the outer automorphism group of the pro-$p$ fundamental group of $P^1(\overline{\mathbb{Q}})-\{0,1,\infty \}$, and to Greenberg's pseudonullity conjecture.

We prove Gaussian estimates for heat kernels on semisimple Lie groups by using the method of Bloch wave representation. We also give a large time asymptotic expansion for heat kernels on compact extensions of abelian Lie groups.

It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ parameterized by a Pisot number $θ$ is countable. Combined with results of R. Salem and P. Sarnak, this proves that for every fixed $θ > 1$ the spectrum of the convolution operator $f\mapsto {\mu }_{\theta }\ast f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in (1,\infty )$, namely, $\overline{\{\widehat{{\mu }_{\theta }}(n):n\in \mathbb{Z}\}}$. Our result answers the question raised by Sarnak in [8]. We also consider the sets $\overline{\{\widehat{{\mu }_{\theta }}(rn):n\in \mathbb{Z}\}}$ for $r > 0$ which correspond to a linear change of variable for the measure. We show that such a set is still countable for all $r\in \mathbb{Q}(\theta )$ but uncountable (a nonempty interval) for Lebesgue-a.e. $r>0$.

The second moment

$${\displaystyle \sum _{q\le Q}{\displaystyle \sum _{a=1}^q{(\psi (x;q,a)-\rho (x;q,a))}^2}}$$

is investigated with the novel approximation

$$\rho (x;q,a)={\displaystyle \sum _{\begin{array}{c}n\le x\\ n\equiv a(\text{mod\hspace{0.17em}}q)\end{array}}{F}_R(n),}$$

where

$$F_R(n)={\displaystyle \sum _{r\le R}\frac{\mu (r)}{\varphi (r)}{\displaystyle \sum _{\begin{array}{c}b=1\\ (b,r)=1\end{array}}^{r}e(bn/r),}}$$

and it is shown that when $R\le {\text{log}}^{A}x$, this leads to more precise conclusions than in the classical Montgomery-Hooley case.

The third moment

$${\displaystyle \sum _{q\le \mathrm{Q}}{\displaystyle \sum _{a=1}^q{(\psi (x;q,a)-\rho (x;q,a))}^3}}$$

is investigated with the novel approximation

$$\rho (x;q,a)={\displaystyle \sum _{\begin{array}{c}n\le x\\ n\equiv a(\text{mod}q)\end{array}}{F}_R(n),}$$

where

$$F_R(n)={\displaystyle \sum _{r\le R}\frac{\mu (r)}{\varphi (r)}{\displaystyle \sum _{\begin{array}{c}b=1\\ (b,r)=1\end{array}}^{r}e(bn/r),}}$$

and it is shown that when $R\le {\text{log}}^{A}x$, this leads to more precise conclusions than those obtained by Hooley in the classical case.

In this paper we study biholomorphic maps between Teichmüller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If $M$ and $N$ are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmüller spaces Teich($M$) and Teich($N$), then $M$ and $N$ are quasiconformally related. Also, every such biholomorphic map is geometric. In particular, we have that every automorphism of the Teichmüller space Teich($M$) must be geometric. This result generalizes the previously known results (see [2], [5], [7]) and enables us to prove the well-known conjecture that states that the group of automorphisms of Teich($M$) is isomorphic to the mapping class group of $M$ whenever the surface $M$ is not of exceptional type. In order to prove the above results, we develop a method for studying linear isometries between $L^1$-type spaces. Our focus is on studying linear isometries between Banach spaces of integrable holomorphic quadratic differentials, which are supported on Riemann surfaces. Our main result in this direction (Theorem 1.1) states that if $M$ and $N$ are Riemann surfaces of nonexceptional type, then every linear isometry between $A^1(M)$ and $A^1(N)$ is geometric. That is, every such isometry is induced by a conformal map between $M$ and $N$.

By applying the methods of V. Markovic [7] to the special case of Riemann surfaces of finite type, we obtain a transparent new proof of a classical result about isometries between the spaces of $L^1$ holomorphic quadratic differentials on such surfaces.

We obtain an asymptotic expansion and some regularity results for the Bergman kernel on the intersection of two balls in $ℂ^2$.