We study the boundary properties of the Green function of bounded simply connected domains in the plane. Essentially, this amounts to studying the conformal mapping taking the unit disk onto the domain in question. Our technique is inspired by a 1995 article of Jones and Makarov [11]. The main tools are an integral identity as well as a uniform Sobolev embedding theorem. The latter is in a sense dual to the exponential integrability of Marcinkiewicz-Zygmund integrals. We also develop a Grunsky identity, which contains the information of the classical Grunsky inequality. This Grunsky identity is the case where $p=2$ of a more general Grunsky identity for $L^p$-spaces

We show that if $M$ is an arithmetic hyperbolic $3$-manifold, the set $\mathbb{Q}L(M)$ of all rational multiples of lengths of closed geodesics of $M$ both determines and is determined by the commensurability class of $M$. This implies that the spectrum of the Laplace operator of $M$ determines the commensurability class of $M$. We also show that the zeta function of a number field with exactly one complex place determines the isomorphism class of the number field

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin-Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [8] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [8] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results

We construct nonsmooth points on unitary eigenvarieties. More precisely, we construct points so that the local ring of the irreducible components of the eigenvariety through this point is nonsmooth and not even a unique factorization domain (UFD). We use those points to construct geometrically several independent extensions in the relevant Selmer group

Fix a bounded domain $\Omega \subset {\mathbb{R}}^d$, a continuous function $F:\partial \Omega \to \mathbb{R}$, and constants $\epsilon >0$ and $1<p,q<\infty$ with $p^{-1}+q^{-1}=1$. For each $x\in \Omega$, let $u^{\epsilon }(x)$ be the value for player I of the following two-player, zero-sum game. The initial game position is $x$. At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector $v\in \stackrel{̲}{B}(0,\epsilon )$ to add to the game position, after which a random noise vector with mean zero and variance $(q/p)|v{|}^2$ in each orthogonal direction is also added. The game ends when the game position reaches some $y\in \partial \Omega$, and player I's payoff is $F(y)$.

We show that (for sufficiently regular $\Omega$) as $\epsilon$ tends to zero, the functions $u^{\epsilon }$ converge uniformly to the unique $p$-harmonic extension of $F$. Using a modified game (in which $\epsilon$ gets smaller as the game position approaches $\partial \Omega$), we prove similar statements for general bounded domains $\Omega$ and resolutive functions $F$.

These games and their variants interpolate between the tug-of-war games studied by Peres, Schramm, Sheffield, and Wilson [15], [16] ($p=\infty$) and the motion-by-curvature games introduced by Spencer [17] and studied by Kohn and Serfaty [9] ($p=1$). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about $p$-capacity and $p$-harmonic measure

Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism $U_l$ from the pointed derivator $\mathsf{HO}(\mathsf{dgcat})$ associated with the Morita homotopy theory of dg categories to a triangulated strong derivator $M_{\mathrm{dg}}^{\mathrm{loc}}$ such that $U_l$ commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle, and is universal for these properties.

Similarly, we construct the universal additive invariant of dg categories, that is, the universal morphism of derivators $U_a$ from $\mathsf{HO}(\mathsf{dgcat})$ to a strong triangulated derivator $M_{\mathrm{dg}}^{\mathrm{add}}$ that satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen's $K$-theory becomes corepresentable in the target of the universal additive invariant. This is the first conceptual characterization of Quillen and Waldhausen's $K$-theory (see [34], [43]) since its definition in the early 1970s. As an application, we obtain for free the higher Chern characters from $K$-theory to cyclic homology