Triangle Seminars
Tuesday, 10 Feb 2009
Chaotic Dynamics and Modelling
Celso Grebogi
(University of Aberdeen)
Abstract:
The talk is based on a 1990 Phys. Rev. milestone paper
Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one basic goal is to develop models whose solutions accurately reflect the nature of the physical process. But a dynamical model uses simplifying assumptions and approximations. The question of whether a model accurately reflects nature is one constantly faced by scientists. I will argue that there exist levels of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent chaotic processes in nature using deterministic models. Specifically, in such cases, no model of such a system produces solutions of reasonable length that are realized by nature. Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.
The talk is based on a 1990 Phys. Rev. milestone paper
Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one basic goal is to develop models whose solutions accurately reflect the nature of the physical process. But a dynamical model uses simplifying assumptions and approximations. The question of whether a model accurately reflects nature is one constantly faced by scientists. I will argue that there exist levels of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent chaotic processes in nature using deterministic models. Specifically, in such cases, no model of such a system produces solutions of reasonable length that are realized by nature. Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.
Posted by: KCL
Electrostatic equilibrium problems and zeros of orthogonal polynomials
Mourad Ismail
(University of Central Florida)
Wednesday, 11 Feb 2009
Vacua of Gauged Extended Supergravities
📍 London
Thomas Fischbacher
(Southampton)
Abstract:
This talk briefly explains why finding the vacuum solutions of gauged maximal supergravity models is a mathematically challenging problem, and which techniques exist that are powerful enough to actually solve it. As these techniques require joint effort between physicists and computer scientists, the problem is explained in a way that should be
accessible to both groups.
In models of extended supergravity with more than one
gravitino, the symmetry group transforming the gravitini into one another can be promoted to a local symmetry. In order to maintain supersymmetry, such a deformation mandates the introduction of a potential for the scalar fields whose stationary points correspond to vacua with spontaneously broken symmetry. The most famous such model
is the SO(8)-gauged N=8 supergravity in four dimensions by de Wit and Nicolai, which also can be obtained by dimensional reduction of M-theory on the 7-sphere. The mathematical problem in the determination of the vacua lies in the complexity of the potential, which is a function on a high-dimensional Riemannian symmetric space based on an E-series exceptional Lie group. While previous group
theoretic arguments allowed a partial investigation of the problem only, there are powerful semi-numeric computational techniques that indeed seem to be strong enough to solve the problem in full. These methods are explained using as an example the most challenging supergravity potential, that of N=16 in D=3 with SO(8)xSO(8) gauge group, and some first data on new vacua of N=8 supergravity in D=4 are
presented.
This talk briefly explains why finding the vacuum solutions of gauged maximal supergravity models is a mathematically challenging problem, and which techniques exist that are powerful enough to actually solve it. As these techniques require joint effort between physicists and computer scientists, the problem is explained in a way that should be
accessible to both groups.
In models of extended supergravity with more than one
gravitino, the symmetry group transforming the gravitini into one another can be promoted to a local symmetry. In order to maintain supersymmetry, such a deformation mandates the introduction of a potential for the scalar fields whose stationary points correspond to vacua with spontaneously broken symmetry. The most famous such model
is the SO(8)-gauged N=8 supergravity in four dimensions by de Wit and Nicolai, which also can be obtained by dimensional reduction of M-theory on the 7-sphere. The mathematical problem in the determination of the vacua lies in the complexity of the potential, which is a function on a high-dimensional Riemannian symmetric space based on an E-series exceptional Lie group. While previous group
theoretic arguments allowed a partial investigation of the problem only, there are powerful semi-numeric computational techniques that indeed seem to be strong enough to solve the problem in full. These methods are explained using as an example the most challenging supergravity potential, that of N=16 in D=3 with SO(8)xSO(8) gauge group, and some first data on new vacua of N=8 supergravity in D=4 are
presented.
Posted by: KCL
A deformed detour in AdS/CFT space
Teresia Mansson
(KTH, Stockholm)
Abstract:
N=4 super Yang-Mills theory has a remarkable feature of being integrable.
This discovery has made it possible to achieve strong evidence for the AdS/CFT
duality.
But what about its siblings: marginal deformations of N=4 SYM
that preserve conformal symmetry. In this case much less is known about the
conjectured supergravity dual. In particular, the Leigh-Strassler deformations
are of this kind, which are only integrable for some special values of the
deformation parameters, or in some restricted subsectors.
In order to gain more insight into the theory, one would like to understand
the symmetries of the theory better. We know that the symmetries of
integrable systems can be described by Hopf algebras. Here we would like
to show that a Hopf algebra structure actually emerges for the generic
Leigh-Strassler deformation.
N=4 super Yang-Mills theory has a remarkable feature of being integrable.
This discovery has made it possible to achieve strong evidence for the AdS/CFT
duality.
But what about its siblings: marginal deformations of N=4 SYM
that preserve conformal symmetry. In this case much less is known about the
conjectured supergravity dual. In particular, the Leigh-Strassler deformations
are of this kind, which are only integrable for some special values of the
deformation parameters, or in some restricted subsectors.
In order to gain more insight into the theory, one would like to understand
the symmetries of the theory better. We know that the symmetries of
integrable systems can be described by Hopf algebras. Here we would like
to show that a Hopf algebra structure actually emerges for the generic
Leigh-Strassler deformation.
Posted by: IC