The dilogarithm function in geometry and number theory pdf

Traditionally the dilogarithm function li2zis defined by the taylor series espansion. 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. Conformal field theory and torsion elements of the bloch group. Bergman 6 introduction to the theory of algebraic functions of one variable, c. The geometry of knot complements abhijit champanerkar.

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. Zagier, the dilogarithm function in geometry and number theory, number theory and related topics, tata inst. The dilogarithm function in geometry and number theory. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Zagier, the dilogarithm function in geometry and number theory,number theory and related topics, tata inst. The last relationship, which seems to be the most interesting property of the.

Dilogarithm identities progress of theoretical physics. Pdf dilogarithm identities, partitions and spectra in. The union of any number of open subsets is an open set. The prime number theorem for arithmetic progressions ii 2 38 16. Geometry and string theory the dilogarithm function skip the navigation links home page all pages recently revised. Tata institute of fundamental research and oxford university press. 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. 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. Geometry number theory calculus probability basic mathematics logic classical mechanics electricity and magnetism computer science quantitative. There is a strong emphasis on algebraic methods and a direct interest in the computational aspects of the theory. The dilogarithm function in geometry and number theory1 the dilogarithm function is.

Schilling 5 the kernel function and conformal mapping, s. The dilogarithm function also appears in hyperbolic geometry, conformal eld theory and the theory of cluster algebras. 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. Congruence fix n 2n, a is called congruent to b modulo n if nja b. Minkowski theory is a rich area of number theory that uses geometry extensively to study solutions to diophantine problems. The dilogarithm function in geometry and number theory 274. The number of rational approximations to algebraic numbers and the number of solutions of norm form equations 232240. Lewin department of electrical and computer engineering, university of colorado, boulder, colorado 80309 communicated by w. Douglas m theory, g 2manifolds and four dimensional physics 31 bobby s. Some exponential diophantine equations ii 258273 15.

The dilogarithm function for complex argument leonard c. The dilogarithm function for complex argument 2809 the transformation 3. Linear operators and automorphic forms 241257 14 t. Conformal field theory prime discrete groups number theory renormalization. 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. 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. The polylogarithm function is an important function for integration, and finding seemingly complicated sum. Algebra, geometry and number theory msc leiden university. Introduction to modern number theory and arithmetic geometry. Bloch that elliptic dilogarithm is subject of big bunch of socalled steinberg functional equation parametrized by rational functions on an elliptic curve. Number theory in physics department of mathematics. Depending on its arguments, dilog returns floatingpoint or exact symbolic results.

Journal of number theory 19, 345373 1984 the inner structure of the dilogarithm in algebraic fields l. 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. Two related special functions are referred to as spences function, the dilogarithm itself. 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. In this note, we give a survey of the in nitesimal version of the above theory. The geometry of knot complements city university of new york. Donaldson branes, calibrations and supergravity 79 jerome p. Essentially, a set xis open if every point xinside xhas points round it that are also in x. Because these numbers are not symbolic objects, dilog returns floatingpoint results. Number theory immediately available upon purchase as print book.

In number theory and algebraic geometry, london math. 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. Thurston introduced tools from hyperbolic geometry to study knots that led to new geometric invariants, especially hyperbolic volume. Relations between crystal bases, branching functions b. It will be observed that the notion of an open set is rather important here. The prospect of a gon proof for ternary hasseminkowski. The dilogarithm function in geometry and number theory 5. In the padic analytic number theory of number elds, elliptic curves, and modular forms, \coleman maps provide the corresponding padic bridge. 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. We use cookies to distinguish you from other users and to provide you with a better experience on our websites.

Geometry and string theory the dilogarithm function. We show that all of these equations follows from the case of functions of degree three and antisymmetry relation. Number theory immediately available upon purchase as print book shipments may be delayed due to the covid19 crisis. Dilogarithm function for numeric and symbolic arguments. Frontiers in number theory, physics, and geometry ii. The intersection of a nite number of open subsets is an open set. Number theoryalgebraic geometry representation theory. The group organizes the number theory seminar and the annual number theory days, jointly with epf lausanne and university of basel.

Conformal field theory prime discrete groups number theory. The result is an algebraic integral with parameters, and the physical problem of e cient evalua. 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.

Polylogarithm is connected to the infinite geometric progression sum. Dilogarithm identities, partitions and spectra in conformal field theory, i article pdf available january 1993 with 35 reads how we measure reads. Contributions to automorphic forms, geometry, and number theory, 521560, johns hopkins univ. Dilogarithm identities, partitions and spectra in conformal eld theory 3 theorem b. 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. Structural properties of polylogarithms american mathematical. Frontiers in number theory, physics, and geometry ii on conformal field theories, discrete groups and renormalization. Don zagier, the dilogarithm function, in frontiers in number theory. Compute the dilogarithm function for these numbers. The survey 33 is an excellent introduction to some aspects of this function. Larsens alternative, moments, and the monodromy of lefschetz pencils. Connection between dilogarithm identities and algebraic ktheory torsion in k 3 r is discussed. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. 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.

The dilogarithm function also appears in hyperbolic geometry. The notion of functions has been vastly generalized and their special values are the subject of the celebrated conjectures of birchswinnertondyer. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. 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. The dilogarithm function in geometry and string theory.

Pdf doc del clipboard journal article mr20586182005g. Msri model theory, arithmetic geometry and number theory. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. The inner structure of the dilogarithm in algebraic fields. Number theory and geometry department of mathematics. How important is differential geometry for number theory. Rogers ramanujans type identities, asymptotic behavior of.