Related papers: Why bootstrapping for $J$-holomorphic curves fails…

We investigate the quadratic Schr\"odinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schr\"odinger potentials under mild assumptions on the marginals that are…

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and…

For a small polarised deformation of a constant scalar curvature K\"ahler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature…

We prove $L^p$-parabolic a-priori estimates for $\partial_t u + \sum_{i,j=1}^d c_{ij}(t)\partial_{x_i x_j}^2 u = f $ on $R^{d+1}$ when the coefficients $c_{ij}$ are locally bounded functions on $R$. We slightly generalize the usual…

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational…

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical…

Let $C\subset \mathbb{P}^{g-1}$ be a general curve of genus $g$ and let $k$ be a positive integer such that the Brill-Noether number $\rho(g,k,1)\geq 0$ and $g > k+1$. The aim of this short note is to study the relative canonical resolution…

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichm\"uller theory. In the case of elliptic curves, we compare…

By introducing a class of meromorphic functions with certain ramification structures on $\Bbb{CP}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These…

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as…

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli…

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this…

In the present work, we consider a fractional p-Kirchhoff equation in the entire space R^N featuring doubly nonlinearities, involving a generalized nonlocal Choquard subcritical term together with a local critical Sobolev term; the problem…

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away…

We use convex integration techniques to provide examples of failure of weighted Calder\'{o}n-Zygmund estimates for degenerate linear elliptic PDEs when the weights are in $A_p$, $p > 2$.

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding…

Concerned with elliptic operators with stationary random coefficients governed by linear or nonlinear mixing conditions and bounded (or unbounded) $C^1$ domains, this paper mainly studies (weighted) annealed Calder\'on-Zygmund estimates,…

We consider Galerkin approximations of holomorphic Fredholm operator eigenvalue problems for which the operator values don't have the structure "coercive+compact". In this case the regularity (in sense of [O. Karma, Numer. Funct. Anal.…

Let X be a curve over a number field K with genus g>=2, $\pp$ a prime of O_K over an unramified rational prime p>2r, J the Jacobian of X, r=rank J(K), and $\scrX$ a regular proper model of X at $\pp$. Suppose r<g. We prove that…

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…