Dynamics and renormalization in models of coagulation and branching

The Department is delighted to welcome Bob R Pego (Carnegie Mellon University) as a Nelder fellow in June 2015.

Background:
Bob's research interests are broad and include the dynamics in infinite-dimensional physical systems, universal scaling behavior in models of clustering and coarsening, stability of nonlinear waves, numerical methods for incompressible viscous flow. If you are interested to meet Bob, please, do so by sending an email to rpego@andrew.cmu.edu.

Abstract:
Bob will deliver a minicourse on Dynamics and renormalization in models of coagulation and branching. the lectures will take place on Wednesday 17th Jue and Thursday 18th June. These lectures will survey recent progress regarding the emergence of coherent scaling behavior in important kinetic models of aggregation, random shock clustering, and branching. Dynamical concepts motivated by the theory of stable laws and infinite divisibility in probability play a significant role. Also surprisingly useful is a set of tools, related to the Levy-Khintchine formula, that concern Bernstein functions (primitives of Laplace transforms).

Lecture 1: Wednesday 17 June, 10:00 -11:10, Huxley 340

Solvable models of coagulation

• The problem of universal behavior in complex systems.
• Some fundamental models of clustering and coagulation and their interrelations: Smoluchowski's coagulation equations, ballistic aggregation, Burgers' turbulence model (shock-wave clustering).
• Dynamic self-similarity in the simplest cases.
• Role of regular variation and Laplace transforms:Tauberian theorems and scaling rigidity.

Lecture 2: Wednesday 17 June, 11:20-12:30, Huxley 340

Random shocks and complete integrability

• Invariance of solutions with Levy increments, Markov processes.
• Lax equation for generators.
• Validity for Levy increments and for white- noise data.
• Menon's theory of integrability and inverse scattering for a finite-state approximation on the Markov group.

Lecture 3: Thursday 18 June, 14:00-15:10, Huxley 139

 A general framework for dynamic scaling analysis.

• Characterization theorems for domains of attraction.
• Analogy to stable laws of probability.
• Self-similar behavior and its lack in coagulation equations.
• Scaling dynamics in general for additive coagulation equations.
• The scaling attractor and its measure representation.
• Bernstein functions and topology for Levy triples.
• Conjugacy with dilational dynamics.
• Signatures of chaos.
• Analogy to infinite divisibility.

Lecture 4: Thursday 18 June, 15:20-16:30, Huxley 139

Dynamics of branching processes.

• Discrete-to-continuum limits, continuous-state branching processes.
• Ex- istence of universal Galton-Watson processes.
• Characterization of do- mains of attraction for critical CBSP through renormalized limits.

Further information can be found here.

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