## Programme

### VenueMarylin and James Simons Conference Center

 Monday December 3 Tuesday December 4 Wednesday December 5 Thursday December 6 Friday December 7 9h00 -- 10h00 Dixon Heslop Henn Bern Kosower 10h00 -- 10h30 Coffee break 10h30 -- 11h30 Duhr Kreimer Schnetz Trnka Smirnov 11h45 -- 12h45 Bjerrum-Bohr Schlotterer Green Arkani-Hamed Paulos 12h45 -- 15h00 Lunch 15h00 -- 16h00 Deligne Bloch Brown Mason moderated discussions 16h15 -- 17h15 Gangl Weinzierl Zagier Caron-Huot 17h30 -- 19h00 moderated discussion Moderator: Einan Gardi moderated discussions Moderator: Mikhail Kapranov moderated discussions Moderator: James Drummond moderated discussions

The videos of the talks are online at the Dailymotion playlist Amplitudes and Periods playlist.
The links to individual videos are provided below.

## Titles of Talks

 Speaker Title and Abstract Video / Files N. Arkani-Hamed Scattering Amplitudes are the Positive Grassmannian video Z. Bern Surprises in UV behavior of gravity theoriesThis talk will describe surprising UV cancellations in half-maximal supergravity. We explain these cancellations as a consequence of the duality between color and kinematics and the associated double copy structure of gravity. We also will comment on ongoing calculations that should shed significant light on the UV properties of supergravity theories. video | pdf N.E.J. Bjerrum-Bohr Amplitude relations in Yang-Mills theory and Gravity We will discuss the existence of color-kinematics relations between amplitudes in Yang-Mills theory and the implications for gravity amplitudes. We will discuss how such relations are connected to monodromy relations in string theory and to the algebraic structure of vertices in Yang-Mills theories and gravity. video | pptx S. Bloch Motives associated to graphs A number of modern mathematical ideas (configuration spaces, motivic fundamental groups, Potts models in statistical mechanics) admit a common generalization, associating to an algebraic variety $X$ and a graph $G$ with vertices labeled by points of $X$ a "motive" $H(X,G)$. In fact there are two constructions which are dual in the case $X$ smooth and projective. I will discuss various questions related to generators and relations for cohomology and periods for such motives. video F. Brown Anatomy of an associator The problem of constructing explicit rational associators was raised by Drinfel'd in 1990 and would have many applications in knot theory, Grothendieck-Teichmuller theory, the Kashiwara-Vergne problem, and deformation quantization. A conjecturally equivalent problem is to give an explicit construction of all rational solutions to the equations satisfied by multiple zeta values. A solution to the latter problem yields an exact algorithm for reducing multiple zeta values into any given basis. In this talk I will outline some elementary but conjectural recipes for solving both these problems. There are some analogies with Quantum Field Theory. video S. Caron-Huot Feynman integrals and their geometry in space-time The value of a Feynman integral can be related to the geometry of the space-time region where its propagators vanish. I'll discuss how such relations may be turned into an algorithm for computing Feynman integrals, emphasizing the unified recursive form of one-loop integrals in all (integral) dimensions, and also some two- loop examples. video P. Deligne Motifs et périodes J'essayerai de montrer comment la philosophie des motifs permet de conjecturer, et parfois de déterminer, la "nature" d'intégrales de quantités algébriques. Exemples de "natures" : multiples rationnels d'une puissance de $\pi$, logarithmes de rationnels, valeurs multizêtas,.... video L. Dixon Single-valued harmonic polylogarithms and the multi-Regge limit (Part I) The natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. Using these functions, and factorization formulas in Fourier-Mellin moment space due to Fadin, Lipatov, Prygarin and Schnitzer, we determine the six-gluon MHV remainder function in multi-Regge kinematics in the leading-logarithmic approximation (LLA) through ten loops, and the next-to-LLA (NLLA) terms through nine loops. The LLA approximation to the six-gluon NMHV amplitude is evaluated analogously through ten loops. All loop orders formulae for the LLA results have been found more recently by Pennington. video | pdf C. Duhr Single-valued harmonic polylogarithms and the multi-Regge limit (Part II) We discuss how we can constrain the symbol of the four-loop MHV remainder function in N=4 Super Yang-Mills and we construct the most general symbol consistent with all known constraints on the remainder function. We then use our result to extract the NNLL and NNNLL corrections to the impact factor and the NNLL correction to the BFKL eigenvalue by constructing a set of functions whose Fourier-Mellin transforms span the space of all single-valued harmonic polylogarithms up to weight 6. video H. Gangl Functional equations for weight 4 multiple polylogarithms Using Goncharov's symbols, we relate weight 4 multiple polylogarithms of different depth to each other, and in particular give a candidate for the corresponding higher Bloch group. video E. Gardi Multi-parton Webs: non-abelian exponentiation theorem for the multi-parton case pdf M.B. Green Supersymmetry and duality constraints on string scattering amplitudesThis talk will discuss the constraints imposed by supersymmetry and duality on the moduli-dependent coefficients of the low energy expansion of string scattering amplitudes. These coefficients transform as automorphic forms under the relevant duality group, as will be illustrated, in particular, for the case of SL(2). Connections will be made between these automorphic forms and properties of perturbative and nonperturbative string theory. video J. Henn Quantum field theory, harmonic polylogarithms, and multiple zeta values In this talk I will discuss the analytic computation of certain classes of quantum field theory integrals that occur in scattering amplitudes and Wilson loops. We show that an infinite class of ladder integrals can be evaluated in terms of harmonic polylogarithms in one variable. Intriguingly, up to the order explicitly evaluated, no multiple zeta values of depth higher than one appear, and the results have a curious positivity property. I will also outline an algorithm for computing more general classes of integrals, and give several examples of physical interest. P. Heslop Perturbative results in $N=4$ SYM I will discuss some of the recent advances in computing amplitudes, Wilson loops and correlation functions in perturbation theory in $N=4$ SYM. After introducing some developments at the level of the integrand, I will then largely focus on analytic results for amplitudes. These are given in terms of polylogarithms with arguments involving twistors for which the symbol has been invaluable. I will focus on a simpler sector of the theory obtained by restricting external particles to lie in a 2d plane. Using multi-collinear limits, at 2 loops all $n$-point amplitudes can be obtained and at higher loops strongly constrained. video D. Kosower Maximal Unitarity at Two Loops video | pptx D. Kreimer From scalar to gauge field theory: two graph and one corolla polynomialScalar field theory amplitudes can be studied through the two Kirchhoff polynomials of the corresponding graphs. The integrand for renormalized amplitudes is particularly appealing when expressed in terms of these polynomials. The transition to gauge theory can be obtained using the recently discovered corolla polynomial. It allows to obtain the full gauge amplitudes in a covariant gauge from scalar field theory with only cubic interactions. video L. Mason Einstein and conformal gravity amplitudes in twistor-string theoryAccording to a conjecture of Berkovits and Witten, twistor-string theory contains in its gravitational sector a non-minimal version of conformal supergravity. This contains Einstein gravity and using an argument of Maldacena one can attempt to construct Einstein amplitudes. We show that one can obtain the Hodges MHV formula for Einstein tree amplitudes by taking only maximally connected trees from the contractions amongst worldsheet correlators. This also leads to a motivation for the new Cachazo-Skinner formula for the full tree-level supergravity $S$-matrix which will be briefly reviewed along with its proof. video M. Paulos Loops, Polytopes and Splines We uncover an unexpected connection between the physics of loop integrals and the mathematics of spline functions. One loop integrands are Laplace transforms of splines. This clarifies the geometry of the associated loop integrals, since a $n$-node spline has support on an $n$-vertex polyhedral cone. One-loop integrals are integrals of splines on a hyperbolic slice of the cone, yielding polytopes in $AdS$ space. Splines thus give a geometrical counterpart to the rational function identities at the level of the integrand. Spline technology also allows for a clear, simple, algebraic decomposition of higher point loop integrals in lower dimensional kinematics in terms of lower point integrals - e.g. an hexagon integral in 2d kinematics can be written as a sum of scalar boxes. Higher loops can also be understood directly in terms of splines - they map onto spline convolutions, leading to an intriguing representation in terms of hyperbolic simplices integrated over other hyperbolic simplices. We finish with speculations on the interpretation of one-loop integrals as partition functions, inspired by the use of splines in counting points in polytopes. video O. Schlotterer Motivic Multiple Zeta Values and Superstring Amplitudes I will discuss the mathematical structure of tree level amplitudes among massless open superstring states. String corrections to these amplitudes take a striking and elegant form once the contributions from different classes of multiple zeta values are disentangled. This novel organization of the alpha prime expansion makes use of a Hopf algebra structure underlying the motivic version of multiple zeta values: It induces an isomorphism which casts the amplitudes into a very symmetric form and represents the generalization of the symbol of a transcendental function. Equipped with these open string results, we can better understand the decoupling of certain multiple zeta values from closed superstring tree amplitudes. video O. Schnetz Proof of the zig-zag conjecture In quantum field theory primitive Feynman graphs give---via the period map---rise to renormalization scheme independent contribution to the beta function. While the periods of many Feynman graphs are multiple zeta values there exists the distinguished family of zig-zag graphs whose periods were conjectured in 1995 by D. Broadhurst and D. Kreimer to be certain rational multiples of odd single zetas. In joint work with F. Brown it was possible in 2012 to prove the zig-zag conjecture using the theory of graphical functions, single valued multiple polylogarithms and a theorem by D. Zagier on multiple zeta values of the form $\zeta(2,\ldots,2,3,2,\ldots,2)$. video V. Smirnov How to prove expansion by regions? If a given Feynman integral depends on kinematic invariants and masses which essentially differ in scale, a natural idea is to expand it in ratios of small and large parameters. As a result, the integral is written as a series of simpler quantities than the original integral itself and it can be substituted by a sufficiently large number of terms of such an expansion. For limits typical of Euclidean space (for example, the off-shell large-momentum limit or the large-mass limit), one can write down the corresponding asymptotic expansion in terms of a sum over certain subgraphs of a given graph. This prescription of expansion by subgraphs has been mathematically proven. For limits typical of Minkowski space, the universal strategy of expansion by regions is available. A geometrical algorithm to reveal regions relevant to a given limit was recently developed. According to it, regions correspond to bottom convex envelopes in the space of weights associated with polynomials characteristic for a given Feynman integral. Still expansion by regions has an experimental status. To prove it is a natural mathematical problem. Expansion by regions is also formulated for parametrical integrals not necessarily related to Feynman integrals. Two other physically inspired mathematical problems are briefly reviewed, with a summary of results obtained by mathematicians and physicists. video J. Trnka Scattering Amplitudes and the Positive Grassmannian. video S. Weinzierl Loop integrals, Hodge structures and differential equations In this talk I will discuss how ideas from the theory of mixed Hodge structures can be used to obtain differential equations for loop integrals. In particular I will discuss the two-loop sunrise graph in two dimensions and show how these methods lead to a differential equation which is simpler than the ones obtained from integration-by-parts. video D. Zagier Modular forms, periods, and differential equations video