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Search Results: 1 - 10 of 9047 matches for " Francis Brown "
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Feynman Amplitudes and Cosmic Galois group
Francis Brown
Physics , 2015,
Abstract: The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods.
Dedekind Zeta motives for totally real fields
Francis Brown
Mathematics , 2008, DOI: 10.1007/s00222-012-0444-x
Abstract: Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.
The massless higher-loop two-point function
Francis Brown
Mathematics , 2008, DOI: 10.1007/s00220-009-0740-5
Abstract: We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge.
Depth-graded motivic multiple zeta values
Francis Brown
Mathematics , 2013,
Abstract: We study the depth filtration on motivic multiple zeta values, and its relation to modular forms. Using period polynomials for cusp forms for PSL_2(Z), we construct an explicit Lie algebra of solutions to the linearized double shuffle equations over the integers, which conjecturally describes all relations between depth-graded motivic multiple zeta values (modulo zeta(2)). The Broadhurst-Kreimer conjecture is recast as a statement about the homology of this Lie algebra.
Multiple Modular Values for SL_2(Z)
Francis Brown
Mathematics , 2014,
Abstract: Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the upper half plane generalising the iterated Shimura integrals of Manin. In this paper, some first properties of the underlying theory are established in the case of the full modular group: in particular, the relationship with special values of L-functions of modular forms at all positive integers; and the action of the Hodge-motivic Galois group via a certain group of automorphisms.
Motivic periods and the projective line minus three points
Francis Brown
Mathematics , 2014,
Abstract: This is a review of the theory of the motivic fundamental group of the projective line minus three points, and its relation to multiple zeta values.
Single-valued periods and multiple zeta values
Francis Brown
Mathematics , 2013,
Abstract: The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. In this paper, the properties of this algebra are studied from the point of view of motivic periods.
Notes on Motivic Periods
Francis Brown
Mathematics , 2015,
Abstract: The second part of a set of notes based on lectures given at the IHES in 2015 on Feynman amplitudes and motivic periods.
Irrationality proofs for zeta values, moduli spaces and dinner parties
Francis Brown
Mathematics , 2014,
Abstract: A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction yields Ap\'ery's approximations to $\zeta(2)$ and $\zeta(3)$, and for larger $n$, an infinite family of small linear forms in multiple zeta values with an interesting algebraic structure. It also contains a generalisation of the linear forms used by Ball and Rivoal to prove that infinitely many odd zeta values are irrational.
Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve
Francis Brown
Mathematics , 2015,
Abstract: This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements $\sigma_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and completely solve the double shuffle equations for multiple zeta values in depths two and three.
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