Rational points on algebric varieties

数论的基本目标之一是找到某些类型的方程的整数解，这些方程被称为丢芬图方程。这些方程通常可以用几何的方式来解释，因此几何技术可以用来解决这个问题。保罗·沃伊塔根据丢芬图方程的几何性质，对丢芬图方程的解进行了广泛的推测。在我未来的研究中，我建议对这些猜想进行研究，尝试改进我之前对它们的各种特殊情况的证明，并尝试利用现有的结果进一步深入了解丢番图方程的解。

One of the fundamental goals of number theory is to find solutions in whole numbers to certain kinds of equations called Diophantine equations. These equations can often be interpreted in a geometric manner, and therefore geometric techniques can be brought to bear on the solution of the problem. Paul Vojta has made some wide-ranging conjectures on what kinds of solutions Diophantine equations should have, based on what kind of geometrical properties the equations have. In my future research, I propose to study these conjectures, try to improve on my previous proofs of various special cases of them, and try to use existing results to gain further insight into the solutions of Diophantine equations.