Related papers: Lagrange Spectra in Teichm\"uller Dynamics via ren…
We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichm\"uller moduli space.
Given a number field $K$, we show that certain $K$-integral representations of closed surface groups can be deformed to being Zariski dense while preserving many useful properties of the original representation. This generalizes a method…
According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to the…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…
We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. This covers, in particular, the well-known situation for spaces of Besov and Triebel-Lizorkin…
We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra…
In this paper, let $L=L_{0}+V$ be a Schr\"{o}dinger type operator where $L_{0}$ is higher order elliptic operator with complex coefficients in divergence form and $V$ is signed measurable function, under the strongly subcritical assumption…
The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it…
We study the asymptotic behavior as L \to \infty of the finite-volume spectral shift function for a positive, compactly-supported perturbation of a Schr\"odinger operator in d-dimensional Euclidean space, restricted to a cube of side length…
We prove a semisimplicity result for the boundary, in the corresponding Deligne-Mumford compactification, of a totally geodesic subvariety of a moduli space of Riemann surfaces. At the level of Teichm\"uller space, this semisimplicity…
This paper explores the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and gives unconditional proofs. Namely, we prove the existence of a universal bound for the…
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed…
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1…
The purpose of this paper is to study spectral properties of non-self-adjoint Schr\"odinger operators $-\Delta-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type…
We provide a short proof of a sharp rearrangement estimate for a generalized version of a potential of Wolff--Havin--Maz'ya type. As a consequence, we prove a reduction principle for that integral operators, that is, a characterization of…
This paper studies the infinitesimal variation of the Lefschetz decomposition associated with a compatible sl_2-representation on a graded algebra. This allows to prove that the Jordan-Lefschetz property holds infinitesimally for the…
Reconstructions of potential in Schrodinger equation with data in the diffusion frequency domain have been successfully obtained within Lippmann-Schwinger-Lanczos (LSL) approach, however limited resolution away from the sensor positions…
While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably…
We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of…
In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In…