Wed, 06 Jul 2022
12:00
C2

Pushing Forward Rational Differential Forms

Robert Moermann
(University of Hertfordshire)

Note: we would recommend to join the meeting using the Zoom client for best user experience.

Abstract

The scattering equations connect two modern descriptions of scattering amplitudes: the CHY formalism and the framework of positive geometries. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry in the CHY moduli space. In this talk, I consider the general problem of pushing forward rational differential forms via the scattering equations. I will present some recent results (2206.14196) for achieving this without ever needing to explicitly solve any scattering equations. These results use techniques from computational algebraic geometry, and they extend the application of similar results for rational functions to rational differential forms.

Mon, 26 Nov 2018

14:15 - 15:15
L4

Amplituhedron meets Jeffrey-Kirwan residue

Tomasz Lukowski
(University of Hertfordshire)
Abstract

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional
conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations

Thu, 20 Jan 2000

14:00 - 15:00
Comlab

Cheap Newton steps for discrete time optimal control problems: automatic differentiation and Pantoja's algorithm

Prof Bruce Christianson
(University of Hertfordshire)
Abstract

In 1983 Pantoja described a stagewise construction of the exact Newton

direction for a discrete time optimal control problem. His algorithm

requires the solution of linear equations with coefficients given by

recurrences involving second derivatives, for which accurate values are

therefore required.

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Automatic differentiation is a set of techniques for obtaining derivatives

of functions which are calculated by a program, including loops and

subroutine calls, by transforming the text of the program.

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In this talk we show how automatic differentiation can be used to

evaluate exactly the quantities required by Pantoja's algorithm,

thus avoiding the labour of forming and differentiating adjoint

equations by hand.

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The cost of calculating the newton direction amounts to the cost of

solving one set of linear equations, of the order of the number of

control variables, for each time step. The working storage cost can be made

smaller than that required to hold the solution.

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