The dilogarithm function in geometry and number theory pdf

The research field number theory and geometry brings together people in the department with interests in arithmetic and various aspects of geometry, especially arithmetic and diophantine geometry. Introduction to modern number theory and arithmetic geometry. In number theory and algebraic geometry, london math. Relations between crystal bases, branching functions b. Frontiers in number theory, physics, and geometry ii on. Now enters geometry in 1980s, william thurstons seminal work established a strong connection between hyperbolic geometry and knot theory, namely that most knot complements are hyperbolic. Conformal field theory prime discrete groups number theory renormalization. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Donaldson branes, calibrations and supergravity 79 jerome p.

Pdf dilogarithm identities, partitions and spectra in. Douglas m theory, g 2manifolds and four dimensional physics 31 bobby s. Zagier iii discuss the conjecture that the values at s m of the dedekind zeta function fs for an arbitrary number field f can be expressed in terms of values of d. In this note, we give a survey of the in nitesimal version of the above theory. Frontiers in number theory, physics, and geometry ii pdf. Dilogarithm identities progress of theoretical physics. Geometry and string theory the dilogarithm function skip the navigation links home page all pages recently revised. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. The prime number theorem for arithmetic progressions ii 2 38 16. The dilogarithm function for complex argument 2809 the transformation 3.

Algebra, geometry and number theory msc leiden university. Because these numbers are not symbolic objects, dilog returns floatingpoint results. The dilogarithm function for complex argument leonard c. It will be observed that the notion of an open set is rather important here. Rogersramanujans type identities, asymptotic behavior of. Geometry and string theory the dilogarithm function.

The dilogarithm function also appears in hyperbolic geometry, conformal eld theory and the theory of cluster algebras. The algebra, geometry and number theory masters specialisation focuses on number theory in the broadest sense, ranging from discrete dynamical systems to arithmetic geometry and motives. Schilling 5 the kernel function and conformal mapping, s. After a century of neglect the dilogarithm appeared twenty years ago in works of gabrielovgelfandlosik ggl on a combinatorial formula for the rst pontryagin class, bloch on k theory and regulators bl1 and wigner on lie groups. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Conformal field theory prime discrete groups number theory. Zagier, the dilogarithm function in geometry and number theory,number theory and related topics, tata inst.

Essentially, a set xis open if every point xinside xhas points round it that are also in x. The blochwignerramakrishnan polylogarithm function don zagier maxplanckinsfitut fiir mathematik, gottfriedclarenstrasse 26, d5300 bonn 3, federal republic of germany to hans grauert the polylogarithm function 9 co n appears in many parts of mathematics and has an extensive literature 2. Traditionally the dilogarithm function li2zis defined by the taylor series espansion. Dilogarithm function for numeric and symbolic arguments. K2 and lfunctions of elliptic curves euler in 1735 discovered that and dirichlet in 1839 proved that we begin by reinterpreting these sums as special values of functions of number fields. Polylogarithm is connected to the infinite geometric progression sum. The geometry of knot complements abhijit champanerkar. Number theory immediately available upon purchase as print book. The number of rational approximations to algebraic numbers and the number of solutions of norm form equations 232240. The group organizes the number theory seminar and the annual number theory days, jointly with epf lausanne and university of basel. The notion of functions has been vastly generalized and their special values are the subject of the celebrated conjectures of birchswinnertondyer.

Zagier, the dilogarithm function in geometry and number theory, number theory and related topics, tata inst. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. Journal of number theory 19, 345373 1984 the inner structure of the dilogarithm in algebraic fields l. Frontiers in number theory, physics, and geometry ii. Compute the dilogarithm function for these numbers. The geometry of knot complements city university of new york. Congruence fix n 2n, a is called congruent to b modulo n if nja b.

In the padic analytic number theory of number elds, elliptic curves, and modular forms, \coleman maps provide the corresponding padic bridge. Contributions to automorphic forms, geometry, and number theory, 521560, johns hopkins univ. Liu tong introduction to modern number theory and arithmetic geometry classical number theory. There is a strong emphasis on algebraic methods and a direct interest in the computational aspects of the theory. Geometry number theory calculus probability basic mathematics logic classical mechanics electricity and magnetism computer science quantitative.

In recent years it has become much better known, due to its appearance in hyperbolic geometry and in algebraic k theory on the one hand and in. Number theory immediately available upon purchase as print book shipments may be delayed due to the covid19 crisis. How important is differential geometry for number theory. Number theoryalgebraic geometry representation theory. The survey 33 is an excellent introduction to some aspects of this function. The last relationship, which seems to be the most interesting property of the. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Dilogarithm identities, partitions and spectra in conformal eld theory 3 theorem b. The polylogarithm function is an important function for integration, and finding seemingly complicated sum. We show that all of these equations follows from the case of functions of degree three and antisymmetry relation. Msri model theory, arithmetic geometry and number theory.

Lewin department of electrical and computer engineering, university of colorado, boulder, colorado 80309 communicated by w. Nonetheless, it has many intriguing properties and has appeared in various branches of mathematics and physics such as number theory the study of asymptotic behaviour of partitions, e. The appearance of the dilogarithm and logarithms at oneloop order is predictable from unitarity considerations since this reproduces the behaviour of the oneloop scattering amplitude under single, or double twoparticle cuts in four dimensions. Frontiers in number theory, physics, and geometry ii on conformal field theories, discrete groups and renormalization. Number theory and geometry department of mathematics eth.

In geometry the common value of the two sides is called the donaldsonthomas dt invariant associated with the a 2 quiver, or it is interpreted as the simplest wallcrossing for. The dilogarithm function in geometry and number theory. Tata institute of fundamental research and oxford university press. The dilogarithm function, defined in the first sentence of chapter i, is a function which has been known for more than 250 years, but which for a long time was familiar only to a few enthusiasts. Number theory and geometry department of mathematics. The prospect of a gon proof for ternary hasseminkowski. In analysis and number theory it is a quantum version of the veterm identity for the dilogarithm function, see 5, 21, 22 and references therein. Naturally some of the convex bodies involved can though in practice arent always too bad get somewhat complicated or defy simple computations of volume, which of course is a.

Don zagier, the dilogarithm function, in frontiers in number theory. Conformal field theory and torsion elements of the bloch group. The dilogarithm function in geometry and number theory 274. Minkowski theory is a rich area of number theory that uses geometry extensively to study solutions to diophantine problems.

The result is an algebraic integral with parameters, and the physical problem of e cient evalua. The union of any number of open subsets is an open set. The dilogarithm function in geometry and number theory1 the dilogarithm function is. Bergman 6 introduction to the theory of algebraic functions of one variable, c. The inner structure of the dilogarithm in algebraic fields. Number theory in physics department of mathematics. The dilogarithm function in geometry and string theory. Bloch that elliptic dilogarithm is subject of big bunch of socalled steinberg functional equation parametrized by rational functions on an elliptic curve. Depending on its arguments, dilog returns floatingpoint or exact symbolic results. Pdf doc del clipboard journal article mr20586182005g. In mathematics, spences function, or dilogarithm, denoted as li 2 z, is a particular case of the polylogarithm. Gauntlett mtheory on manifolds with exceptional holonomy 127 sergei gukov special holonomy and beyond 159 nigel hitchin. Levin, a kronecker double series and the dilogarithm.

The dilogarithm function in geometry and number theory 5. Some exponential diophantine equations ii 258273 15. Rogers ramanujans type identities, asymptotic behavior of. Dilogarithm identities, partitions and spectra in conformal field theory, i article pdf available january 1993 with 35 reads how we measure reads. The dilogarithm function also appears in hyperbolic geometry. Structural properties of polylogarithms american mathematical. Thurston introduced tools from hyperbolic geometry to study knots that led to new geometric invariants, especially hyperbolic volume. Then we have fxconst lx we continue the function lx on all real axisr r1 f1g by the following rules. In fact, the lobachevsky function can be considered as an imaginary part of the dilogarithm function.