Related papers: The Minimal Model Program for threefolds in charac…
We introduce a $2$-approximation algorithm for the minimum total covering number problem.
We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicity are almost optimal: they are essentially the same as…
We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.
We show how to reduce a general, strictly-feasible LP problem, into a min-max problem, which can be solved by the algorithm from the third section of my thesis.
We prove identifiability of the tree parameters of the 3-class Jukes-Cantor mixture model. The proof uses ideas from algebraic statistics, in particular: finding phylogenetic invariants that separate the varieties associated to different…
We extend the Cone Theorem of the Log Minimal Model Program to log varieties with arbitrary singularities.
Higher-dimensional analogs of the predictable degree property and column reducedness are defined, and it is proved that the two properties are equivalent. It is shown that every multidimensional convolutional code has, what is called, a…
We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism…
The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was…
We give characterizations of sharp minimizers that emphasize their geometric properties. These include tilt invariance and weak upper gradient conditions. We relate sharp minimality to cusps in nonsmooth manifolds when interpreted locally…
We prove boundedness of rationally-connected threefolds in $\mathbb P^6$ under some extra-assumptions.
We provide a simple proof of the radial symmetry of any nonnegative minimizer for a general class of quasi-linear minimization problems.
We prove Iitaka's $C_{n,m}$ conjecture for $3$-folds over the algebraic closure of finite fields. Along the way we prove some results on the birational geometry of log surfaces over nonclosed fields and apply these to existence of relative…
We study rationality properties of real singular cubic threefolds.
This paper proves that the 5-canonical map of a smooth minimal 3-fold is birational when the geometric genus is bigger than 2. A combination of the results in this paper and that of Ein-Lazarsfeld-Lee, the possible exceptional cases are…
Let $(X,\Delta)$ be a normal pair with a projective morphism $X \to Z$ and let $A$ be a relatively ample $\mathbb{R}$-divisor on $X$. We prove the termination of some minimal model program on $(X,\Delta+A)/Z$ and the abundance conjecture…
We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.
We prove a semistable reduction theorem for principal bundles on curves in almost arbitrary characteristics. For exceptional groups we need some small explicit restrictions on the characteristic.
We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and…
We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete…