相关论文: Weierstrass filtration on Teichm\"uller curves and…

We compute the Euler characteristics of the recently discovered series of Gothic Teichm\"{u}ller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichm\"{u}ller curves. Contrary…

We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two…

We use the Burgess bound and Selberg sieve to obtain an upper bound on the second moment of sums over an interval $[u+1,u+h]$ of Legendre symbols modulo primes $p$ in a dyadic interval $[Q,2Q]$. The bound is nontrivial and gives a power…

Costantini and Kappes gave an algebraic equation of the universal family over the Kenyon-Smillie (2,3,4)-Teichm\"uller curve. This equation gives rise to a family of projective plane quartic curves with three singular members. These…

We construct a Harder-Narasimhan filtration for rank $2$ tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving…

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar…

Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{H})$ is the set of asymptotic rays to the embedding of $T(\mathbb{H})$ in the space of geodesic currents; the boundary is identified with the projective bounded measured…

We develop a reduction theory for the representation of $\mathrm{SL}_n$ on pairs of symmetric $n\times n$ matrices. We apply this theory to the pencils of quadrics arising from divisors on hyperelliptic curves. We use these results to show…

We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichm\"uller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct.…

We study the problem of $d$-gonality of the modular curve $X_0(N)$. As a result, we can give an upperbound of the level $N$ by means of $d$. This generalizes Ogg's result on hyperelliptic modular curves ($d = 2$). As a corollary of this…

In this article, we review our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or…

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions…

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational…

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…

We compute the Harder-Narasimhan filtration of vector bundles $f_*\mathcal O_Y$ for certain finite morphisms $f\,:\,Y\,\longrightarrow\, X$ and in some other cases.

We give upper bounds for the Bergman kernels associated to tensor powers of a smooth positive line bundle in terms of the rate of growth of the Taylor coefficients of the K\"ahler potential. As applications, we obtain improved off-diagonal…

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring…

The slopes of maximal subbundles of rank $s$ divided by the degree of the map under various pull backs form a bounded collection of numbers called the $s$-spectrum of the bundle. We study the supremum of the $s$-spectrum and determine it in…

Here we prove that for a smooth projective variety $X$ of arbitrary dimension and for a vector bundle $E$ over $X$, the Harder-Narasimhan filtration of a Frobenius pull back of $E$ is a refinement of the Frobenius pull-back of the…