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We prove the Shafarevich conjecture for very irregular varieties of dimension less than half the dimension of their Albanese variety, subject to some mild numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in the…

The additive square problem is a relatively famous open problem in the area of combinatorics on words: Does there exist an infinite word over a finite alphabet, such that no two consecutive blocks of the same length have the same sum? In…

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 3) contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

We provide elementary proofs for the terms that are left in the work of Kelly Bickel, Sandra Pott, Maria C. Reguera, Eric T. Sawyer, Brett D. Wick who proved the sharp weighted $A_2$ bound for Haar shifts and Haar multiplier. Our proofs use…

We prove an explicit version of Burgess' bound on character sums for composite moduli.

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.

Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A…

The main result of this paper is the proof for elliptic modular threefolds of conjectures on the existence and structure of a filtration on the Chow groups of smooth projective varieties. In the form we prove them these conjectures were…

Let H be a semisimple algebaric group and let X be a smooth projective curve defined over an algebraically closed field k. In the first part of this paper we show that the moduli of semistable principal H-bundles exists once given a…

In this note, we will give a short proof of an identity for cubic partitions.

We describe an explicit version of Hilbert's irreducibility theorem using a generalization of Gallagher's larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation…

We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the G\"artner-Ellis theorem and sharp large deviations tools.

In this paper, we prove that the moduli space $\overline{M}_{X}(\nu)$ of $H$-Gieseker semistable sheaves on a smooth cubic threefold $X$ with Chern character $\nu=(4,-H,-\frac{5}{6}H^{2},\frac{1}{6}H^{3})$ is non-empty, smooth and…

We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix…

We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.

We give an alternative proof of Suslin's equi-dimensionalization moving lemma using a different geometric construction. The new construction provides better control of the degrees of the polynomials describing the geometric procedure. The…

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

In this note we compute length, support and dimension of syzygy modules of certain modules. This partially answers questions asked by Huneke et al.