Results 1  10
of
14
From polygons and symbols to polylogarithmic functions
 JHEP 1210 (2012) 075, arXiv:1110.0458 [mathph
"... Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an ..."
Abstract

Cited by 46 (7 self)
 Add to MetaCart
(Show Context)
Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of nontrivial elements in the kernel of the symbol map for arbitrary weight.
Multiple logarithms, algebraic cycles and trees. (to appear
 in “Frontiers in Number Theory, Physics and Geometry”, Les Houches Proceedings
"... This is a short exposition—mostly by way of the toy models “double logarithm” and “triple logarithm”—which should serve as an introduction to the article [3] in which we establish a connection between multiple polylogarithms, rooted trees and algebraic cycles. 1 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
This is a short exposition—mostly by way of the toy models “double logarithm” and “triple logarithm”—which should serve as an introduction to the article [3] in which we establish a connection between multiple polylogarithms, rooted trees and algebraic cycles. 1
THE ALGEBRA OF CELLZETA VALUES
"... Abstract. In this paper, we introduce cellforms on M0,n, which are topdimensional differential forms which diverge along the boundary of exactly one cell (connected component) of M0,n(R). We show that the cellforms generate the topdimensional cohomology group of M0,n, so that there is a natural d ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we introduce cellforms on M0,n, which are topdimensional differential forms which diverge along the boundary of exactly one cell (connected component) of M0,n(R). We show that the cellforms generate the topdimensional cohomology group of M0,n, so that there is a natural duality between cells and cellforms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cellzeta values, which generate a Qalgebra called the cellzeta algebra. By the main result of [5], the cellzeta algebra is equal to the algebra of multizeta values. The cellzeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cellzeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the muchstudied double shuffle relations. 1.
Contents
"... Abstract. We define “weighted multiplicative presheaves ” and observe that there are several weighted multiplicative presheaves that give rise to motivic cohomology. By neglect of structure, weighted multiplicative presheaves give symmetric monoids of presheaves. We conjecture that a suitable stabil ..."
Abstract
 Add to MetaCart
Abstract. We define “weighted multiplicative presheaves ” and observe that there are several weighted multiplicative presheaves that give rise to motivic cohomology. By neglect of structure, weighted multiplicative presheaves give symmetric monoids of presheaves. We conjecture that a suitable stabilization of one of the symmetric monoids of motivic cochain presheaves has an action of a caterad of presheaves of acyclic cochain complexes, and we give some fragmentary evidence. This is a snapshot of work in progress.
POLYGONAL COMBINATORICS FOR ALGEBRAIC CYCLES
"... This text is a somewhat vulgarized version of a portion of our joint work [4] with Goncharov and Levin. We want to give evidence for the claim that polygons and their internal structure are very (mixed Tate) motivic, at least if we work over a field. Definition 1.1. Let R be a set. An Rdeco polygon ..."
Abstract
 Add to MetaCart
This text is a somewhat vulgarized version of a portion of our joint work [4] with Goncharov and Levin. We want to give evidence for the claim that polygons and their internal structure are very (mixed Tate) motivic, at least if we work over a field. Definition 1.1. Let R be a set. An Rdeco polygon pi is an oriented polygon with a distinguished root side and a decoration {sides of pi} → R. We indicate the root side of a polygon by a double line. The orientation is expressed by a bullet at one of the two vertices of the root side, with the convention that the first vertex of the polygon is that bulleted one and the last side is the root side. The set R that we typically have in mind is the multiplicative group of a field. Example 1.2: Here is an Rdeco 6gon pi = [ a1,..., a6]. The root side is drawn by a double line, the first vertex is marked by a bullet, and the orientation is counterclockwise. a3 a2 111111 a4 a5
Research Statement for Susama Agarwala
"... I am interested in geometric and algebraic questions motivated by high energy physics, particularly renormalization. I study renormalization using combinatorial Hopf algebra in the program established by Connes and Kreimer in [11]. The study of combinatorial Hopf algebras leads me to work on problem ..."
Abstract
 Add to MetaCart
(Show Context)
I am interested in geometric and algebraic questions motivated by high energy physics, particularly renormalization. I study renormalization using combinatorial Hopf algebra in the program established by Connes and Kreimer in [11]. The study of combinatorial Hopf algebras leads me to work on problems in many different fields, such as noncommutative geometry, number theory, and even control theory. I am also interested motives, particularly as they apply to multiple polylogarithms and field theories in configuration space. Combinatorial Hopf algebras have led me to study the process of renormalization certain types of field theories [2, 1, 5]. These algebras have applications in the study of multiple polylogarithms [18, 3]. They have an important role to play in noncommutative geometry [15]. In ongoing work, I am studying their relationship with gauge structures found in control theory. In more speculative work, I am investigating whether these Hopf algebras and their related structures may shed light on the Fundamental Lemma, at least in the case of GL2(k). Combinatorial Hopf algebras can be expressed as the Hopf algebra of rooted trees, Hrt. These are nonplanar oriented trees with one marked vertex that all edges are oriented away from. The coproduct structure is defined by making a certain type of cuts on these trees. These structures first appeared in the problem