DescriptionEuler showed that the sum of the inverse squares over all natural numbers equals π2/6 and more generally, for any even integer n>1, that the corresponding sum of the inverses of n-th powers is a rational multiple of πn. Such sums, called zeta values, play an important role in number theory. In recent years, iterated sums of a similar kind, the so-called multiple zeta values (MZV's), have appeared in many different contexts, ranging from number theory, algebraic geometry or knot theory to Feynman integrals in quantum field theory.There is rather little known about individual evaluations of these MZV's; by contrast, they satisfy a wealth of interesting relations like two types of multiplication as well as a so-called co-multiplication, and moreover one knows many combinatorial identities among them. One of the basic problems of the theory is to exhibit a complete set of such relations, at least three candidates of which have been proposed (and to some extent have been related to each other). An important feature which guarantees a lot of structure is that each such multiple zeta value can be written both as an iterated sum and as an integral of a very specific kind (one iteratedly integrates rather simple rational functions). Some of the rich structure behind these numbers can be encoded in terms of, e.g., shuffle algebras, Lie algebras as well as Hopf algebras, the latter of which are delicate objects in which one can compose (i.e. multiply) and decompose (i.e. "co-multiply"). The algebras attached to MZVs have a strong combinatorial flavour. Moreover, there is a surprising connection to other--a priori completely unrelated--number theoretic objects, so-called modular forms, which give rise, via their "period polynomials", to special relations among multiple zeta values. These special relations in turn relate modular forms to conjectures on the dimension of rational vector spaces spanned by the MZVs. Intriguingly, multiple zeta values very often appear as coefficients in expansions of integrals attached to Feynman graphs which arise, e.g., in quantum field theory calculations. This hints at an interplay between number theory and physics, and in recent years such parallels and connections have indeed been discovered and developed. Moreover, the last decade witnessed several breakthrough results in the field, some of which have already made it into a textbook
Possible directionsAs there are many different directions and levels on which multiple zeta values can be studied, from the very concrete to rather abstract, this project can easily be tailored to the student's background. Depending on the topic chosen the project could just focus on theoretical investigations but it also allows for computer experiments to possibly find new structures or patterns.
PrerequisitesFamiliarity with analysis of sums and series, as well as partial fraction expansions, is recommended; for a study of the underlying algebraic structures, Algebra would be needed, and additional accompanying Number Theory would certainly be very helpful.
ResourcesA nice survey of certain aspects of the theory, was given by Zudilin in and there was a lecture series on the subject given by Waldschmidt
Original papers: Euler's seminal paper can be found here: Zagier `resurrected' MZV's and studied them in context with a larger class of zeta functions while Hoffman independently developped an algebraic theory of nested sums
Recently this textbook was written by Fresan and Burgos, and there is a rather comprehensive (and up-to-date!) web page of references for all kinds of results on multiple zeta values. |
email: Herbert Gangl