Conferences on quadratic and higher degree forms

数论有着悠久而丰富的历史，一直是，并将继续是一个充满活力的研究领域。数论中有许多重要的结果和问题，多年来对这些结果和问题的研究导致了数学内外几个领域的重大发展。一个很好的例子是美丽的定理，即每个正整数至多是四个平方的和。同样，在将整数表示为立方、四次和等方面也有了重要的结果。这推动了对二次型和高次型的更广泛的研究，这对代数、数论和组合学的许多部分都产生了深远的影响。在2008-09学年，将在佛罗里达大学举办两个国际会议：(I)2008年秋季关于高等学位形式的会议；(Ii)2009年春季关于二次型、平方和、theta函数和积分格的会议(和学生工作坊)。这两个会议将是第一次将重点放在二次和更高学位形式上，这些领域近年来取得了戏剧性的进展。最值得注意的是，在过去的几年里，曼朱尔·巴尔加瓦(普林斯顿)将著名的二次型高斯合成定律推广到了高次形式。自从19世纪高斯关于二次型的开创性工作以来，在更高阶型上就没有这种性质的进展了！此外，在过去的两年里，Bhargava和Jonathan Hanke解决了John Conway关于确定所有泛二次形式的猜想，即像代表所有正整数的四个平方之和的二次形式；关于泛二次形式问题的起源可以追溯到印度天才斯里尼瓦萨·拉马努扬的著作，他在20世纪初以其非凡的发现震惊了数学界。在关于二次型和相关主题的会议之前，将有一个学生工作坊，为学生准备在会议上提出的高级主题。这两次会议是为期一年的代数、数论和组合学(ANTC)综合课程的亮点，与佛罗里达大学在ANTC的优势和传统以及与Ramanujan工作相关的领域相一致。这些会议将为学生和年轻的研究人员提供一个机会，与在与二次和更高学位形式相关的领域取得重大进展的著名数学家进行交流。整个项目包括适当组合的时长讲座和较短的演讲，是为学生和年轻研究人员以及物理和计算机科学等相关学科的研究人员设计的。因此，所取得的更广泛影响将是巨大的。

Number theory, with its long and rich history, has always been, and continues to be, a vibrant area of research. There is a wealth of important results and problems in number theory, and the study of these over the years have led to major developments in several areas both within and outside of mathematics. A fine example is the beautiful theorem that every positive integer is a sum of at most four squares. Similarly, there are important results on the representation of integers as sums of cubes, fourth powers, etc. This motivated the more general study of quadratic and higher degree forms which has had deep impact in many parts of algebra, number theory and combinatorics. During the academic year 2008-09, two international conferences will be conducted at the University of Florida: (i) Conference on higher degree forms in Fall 2008, and (ii) Conference on quadratic forms, sums of squares, theta functions, and integral lattices (and Student Workshop), in Spring 2009. The two conferences would be the first to have focus on quadratic and higher degree forms, areas that have seen dramatic progress in recent years. Most notably, in the last few years, Manjul Bhargava (Princeton) has extended the celebrated Gauss composition law for quadratic forms, to forms of higher degree. There was no progress of this nature on higher degree forms since Gauss' pioneering work on quadratic forms in the nineteenth century! Also, within the last two years, Bhargava and Jonathan Hanke have solved a conjecture of John Conway concerning the determination of all universal quadratic forms, namely quadratic forms like the sum of four squares that would represent all positive integers; the origins of the problem on universal quadratic forms can be traced to the writings of the Indian genius Srinivasa Ramanujan, who astounded the mathematical world in the early twentieth century with his remarkable discoveries. The conference on quadratic forms and related topics will be preceded by a Student Workshop which will prepare the students for the advanced topics presented at the conference. The two conferences are the highlights of a year long comprehensive program on Algebra, Number Theory and Combinatorics (ANTC) and tie in with the strength and tradition at the University of Florida in ANTC and areas related to the work of Ramanujan. The conferences will provide an opportunity for students and young researchers to interact with eminent mathematicians who are making major advances on areas related to quadratic and higher degree forms. The entire program consisting of an appropriate mix of hour lectures and shorter presentations, has been planned for the benefit of students and young researchers, as well as researchers in allied disciplines like physics and computer science. Thus the broader impact achieved will be significant.