Mathematical Sciences: Problems in Arithmetics Geometry and Complexity Theory

该奖项支持伊利诺伊大学芝加哥分校杰里米·泰特尔鲍姆教授在算术代数几何方面的研究。泰特尔鲍姆博士将在例外零猜想的指导下，继续他对p-进上半平面的几何和算术的研究，该猜想将某些函数的值，即p-进L函数，与某些曲线的解析几何联系起来。此外，他还将继续从代数几何的角度研究算法的计算复杂性。算术代数几何是一门结合了代数几何和数论技术的学科。在最初的公式中，代数几何处理的是可以在平面上由最简单的方程定义的图形，即多项式。数论始于整数和一个整数被另一个整数整除等问题。这两个学科看似相距甚远，实际上从最早的时候起就相互影响，但在过去的四分之一个世纪里，相互影响大大增加。算术代数几何领域现在使用了所有现代数学的技术，并在其自身边界之外产生了相应的影响。

This award supports the research in arithmetic algebraic geometry of Professor Jeremy Teitelbaum of the University of Illinois at Chicago. Dr. Teitelbaum will pursue his studies of the geometry and arithmetic of the p-adic upper half plane, guided by the Exceptional Zero Conjecture, which relates the values of certain functions, namely p-adic L-functions, to the analytic geometry of certain curves. In addition, he will continue his study of the computational complexity of algorithms from algebraic geometry. Arithmetic algebraic geometry is a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.