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.
