Related papers: Counting $3$-dimensional algebraic tori over $\mat…

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the…

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the…

We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.

Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on $ S ^ 1 $, called the Sym point. For a given spectral curve the…

We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. Both of these are optimal, aside…

Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$…

In this paper, we give the explicit structure of $ \otimes^{3} H $ and $ \wedge^{3} H $ where $ H $ is a generalized Heisenberg Lie algebra of rank at most $ 2. $ Moreover, for a non-abelian nilpotent Lie algebra $ L, $ we obtain an upper…

Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the…

We classify the possible torsion structures of rational elliptic curves over quintic number fields. In addition, let E be an elliptic curve defined over Q and let G = E(Q)_tors be the associated torsion subgroup. We study, for a given G,…

We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional…

Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n…

The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture…

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of…

In a recent paper, we proved that for any large enough odd modulus $q\in \mathbb{N}$ and fixed $\alpha_2\in \mathbb{N}$ coprime to $q$, the congruence \[ x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q} \] has a solution of…

Let $A_{\pi}(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\rm SL_3(\mathbb{Z})$ and $ \omega(x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum…

We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T…

We show that for any $n$-dimensional lattice $\mathcal{L} \subseteq \mathbb{R}^n$, the torus $\mathbb{R}^n/\mathcal{L}$ can be embedded into Hilbert space with $O(\sqrt{n\log n})$ distortion. This improves the previously best known upper…

Uniqueness and reconstruction in the three-dimensional Calder\'on inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schr\"odinger operators $-\Delta +q $. We study the Born approximation of $q$ in…

This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise…

Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity…