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In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G. For parabolic subgroups associated to even nilpotents, we parameterize (G,P)-opers by an object generalizing the base of the Hitchin fibration.…

We present a conjecture about the asymptotic representation of certain series. The conjecture implies the Riemann hypothesis and it would also indicate the simplicity of the non-trivial zeros of the zeta-function.

Given two finite-dimensional representations $\rho$ and $\sigma$ of $\mathsf{SU}(n)$, when is there $n \in \mathbb{N}$ such that $\rho^{\otimes n}$ is isomorphic to a subrepresentation of $\sigma^{\otimes n}$? When is there a third…

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let…

We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which allows us, in…

We show that for every $\epsilon>0$, there exists a compact lamination by $\epsilon$-holomorphic surfaces in the complex projective plane, minimal, and that carries hyperbolic holonomy. We call $\epsilon$-holomorphic a real 2-dimensional…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…

We initiate and develop the theory of complex harmonic maps to holomorphic Riemannian symmetric spaces, which we make use of to study complex analytic aspects of higher Teichm\"uller theory, with a focus on rank $2$ Hitchin components.…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite…

The purpose of this note is to construct a sequence of spin hyperbolic surfaces $\Sigma_n$ with genus going to infinity and with a uniform spectral gap for the Dirac operator. Our construction is completely explicit. In particular, the…

We disprove Hitchin's conjecture to the effect that for a generic complex structure on a simply connected spin complex surface the square root of the canonical bundle has no more cohomology then is predicted by the Riemann--Roch theorem.…

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B, $$ with…

Let $\Sigma$ be a surface with $\chi (\Sigma) < 0$, and a representation $\rho $ from the fundamental group $\pi_1 (\Sigma)$ into $ \rm{SL} (2 , \mathbb{C})$. We define the \emph{trace systole} of $\rho$, denoted $\mathrm{tys} (\rho)$ as…

We prove that given a Hitchin representation in a real split rank 2 group $\mathsf G_0$, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin…

Every almost Hermitian structure $(g,J)$ on a four-manifold $M$ determines a hypersurface $\Sigma_J$ in the (positive) twistor space of $(M,g)$ consisting of the complex structures anti-commuting with $J$. In this note we find the…

We study the asymptotic behaviour of 1-parameter subgroups with respect to Hofer's metric when the underlying symplectic manifold is an open surface of infinite area. We prove that, depending on the topology of the level sets of the…

In recent years, the Tian-Zelditch asymptotic expansion for the equivariant components of the Szeg\"{o} kernel of a polarized complex projective manifold, and its subsequent generalizations in terms of scaling limits, have played an…