Triangle Seminars
Tuesday, 16 Feb 2010
Models of evolution on structured populations
Mark Broom
(City University)
Abstract:
We investigate two examples of models of populations with structure. These are different in character, with the common theme that the structure has an important influence on population outcomes. In the first part we consider a model of kleptoparasitism, the stealing of food from one animal by another. We investigate a model where individuals are allowed to fight in groups of more than two, as often occurs in real populations, but which has not featured in previous theoretical models. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable strategies (ESSs) for the challenging behaviour of the participants. We show that ESSs can only come from a restricted subset of the possible strategies and that there is always at least one ESS. We show that there can be multiple ESSs, and indeed that the number of ESSs is unbounded. Finally we discuss the biological circumstances when particular ESSs occur in terms of key parameters such as the availability of food and the cost of fighting. The second part of the talk concerns the study of evolutionary dynamics on populations with some non-homogeneous structure, a topic in which there is a rapidly growing interest. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. This process is a Markov chain and we prove several mathematical results. For example we prove that for all but a restricted set of graphs, (almost) all states are accessible from the possible initial states. We then consider graphs within this restricted set or with considerable symmetry. To find the fixation probability of a line graph we relate this to a two-dimensional random walk which is not spatially homogeneous. We investigate our solutions numerically and find that for mutants with fitness greater than the resident, the existence of population structure helps the spread of the mutants. Thus it may be that models assuming well-mixed populations consistently underestimate the rate of evolutionary change.
We investigate two examples of models of populations with structure. These are different in character, with the common theme that the structure has an important influence on population outcomes. In the first part we consider a model of kleptoparasitism, the stealing of food from one animal by another. We investigate a model where individuals are allowed to fight in groups of more than two, as often occurs in real populations, but which has not featured in previous theoretical models. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable strategies (ESSs) for the challenging behaviour of the participants. We show that ESSs can only come from a restricted subset of the possible strategies and that there is always at least one ESS. We show that there can be multiple ESSs, and indeed that the number of ESSs is unbounded. Finally we discuss the biological circumstances when particular ESSs occur in terms of key parameters such as the availability of food and the cost of fighting. The second part of the talk concerns the study of evolutionary dynamics on populations with some non-homogeneous structure, a topic in which there is a rapidly growing interest. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. This process is a Markov chain and we prove several mathematical results. For example we prove that for all but a restricted set of graphs, (almost) all states are accessible from the possible initial states. We then consider graphs within this restricted set or with considerable symmetry. To find the fixation probability of a line graph we relate this to a two-dimensional random walk which is not spatially homogeneous. We investigate our solutions numerically and find that for mutants with fitness greater than the resident, the existence of population structure helps the spread of the mutants. Thus it may be that models assuming well-mixed populations consistently underestimate the rate of evolutionary change.
Posted by: KCL
Wednesday, 17 Feb 2010
Causality and Photon Propagation in Curved Spacetime
📍 London
Graham Shore
(Swansea )
Abstract:
We discuss the effect of vacuum polarization on the propagation of photons in curved spacetime in QED. A compact formula is presented for the full frequency dependence of the refractive index for any background in terms of the Van Vleck-Morette matrix for its Penrose limit. This shows explicitly how the superluminal propagation found in the low-energy effective action is reconciled with causality.
The geometry of null geodesic congruences is found to imply
a novel analytic structure for the refractive index and
Green functions of QED in curved spacetime, which preserves
their causal nature but violates familiar axioms of S-matrix theory and dispersion relations. The Kramers-Kronig dispersion relation and the optical theorem for QFT in curved spacetime are discussed critically. The significance of the Penrose limit for black hole spacetimes and their relation to homogeneous plane waves is explained and unexpected features of light propagation in a number of spacetimes are described.
We discuss the effect of vacuum polarization on the propagation of photons in curved spacetime in QED. A compact formula is presented for the full frequency dependence of the refractive index for any background in terms of the Van Vleck-Morette matrix for its Penrose limit. This shows explicitly how the superluminal propagation found in the low-energy effective action is reconciled with causality.
The geometry of null geodesic congruences is found to imply
a novel analytic structure for the refractive index and
Green functions of QED in curved spacetime, which preserves
their causal nature but violates familiar axioms of S-matrix theory and dispersion relations. The Kramers-Kronig dispersion relation and the optical theorem for QFT in curved spacetime are discussed critically. The significance of the Penrose limit for black hole spacetimes and their relation to homogeneous plane waves is explained and unexpected features of light propagation in a number of spacetimes are described.
Posted by: KCL
Gradient formula for the beta function of 2d quantum field theory
Anatoly Konechny
(Heriot-Watt)
Abstract:
I will explain a gradient formula for beta functions of two-dimensional
quantum field theories. The gradient formula has the form derivative c = - (gij+Delta gij+bij) bj where bj are the beta functions, c and gij are the Zamolodchikov c-function and metric, bij is an antisymmetric tensor introduced by H. Osborn and Delta gij is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.
I will explain a gradient formula for beta functions of two-dimensional
quantum field theories. The gradient formula has the form derivative c = - (gij+Delta gij+bij) bj where bj are the beta functions, c and gij are the Zamolodchikov c-function and metric, bij is an antisymmetric tensor introduced by H. Osborn and Delta gij is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.
Posted by: IC
Thursday, 18 Feb 2010
Solitons in Holographic Superfluids
Sean Nowling
(Helsinki)
Abstract:
The gauge/gravity duality provides a new set of tools for exploring
the physics near 2+1 dimensional quantum critical points. In many
highly quantum states of matter, soliton configurations yield clues
about the physics of both large and small length scales. I will
review a holographic model for a relativistic superfluid. I will then
discuss dark soliton and vortex solutions supported by this
superfluid.
The gauge/gravity duality provides a new set of tools for exploring
the physics near 2+1 dimensional quantum critical points. In many
highly quantum states of matter, soliton configurations yield clues
about the physics of both large and small length scales. I will
review a holographic model for a relativistic superfluid. I will then
discuss dark soliton and vortex solutions supported by this
superfluid.
Posted by: QMW