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 Prairie Analysis Seminar 2015



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Department of Mathematics
Kansas State University
Manhattan, Kansas

September 25-26, 2015                                                        

Principal Lecturer:

David Kinderlehrer

Carnegie Mellon University

Lecture 1: Introducing mass transport

Abstract: Elements of mass transport. Description of solving equations by the implicit scheme - the Jordan, K, Otto problem of convergence, summarizing the flow interchange work of Matthes-McCann-Savare’ to the special case of Fokker-Planck.

Lecture 2: Evolution of material microstructure and the discovery of the grain boundary character distribution

  Background of the issue. The theory of the grain boundary character distribution. Validation of the theory and perhaps a better dissipation relation.

Invited Speakers:
Robert McCann
University of Toronto
Title: The intrinsic dynamics of optimal transport

Abstract: The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere.  We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique.  This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probability densities is unique.

Yekaterina Epschtein
The University of Utah
Title: Grain boundary character distribution and mass transport paradigm

Abstract: Cellular networks are ubiquitous in nature. Most technologically useful
materials arise as polycrystalline microstructures, composed of a myriad of small crystallites, or grains, separated by interfaces, or grain boundaries. The energetics and connectivity of the grain boundaries network plays a crucial role in determining the properties of a material across a wide range of scales. Coarsening, or growth process is influenced ainly by the effort of the system to decrease the interfacial energy subject to spatial constraints. The recently discovered Grain Boundary Character Distribution (GBCD) indicates that the boundary network of a cellular structure, and, more generally, material texture has a natural order.

Grain Boundary Character Distribution (GBCD) is a new characterization of the texture which is found to be strongly correlated to the interfacial energy. In this talk GBCD is introduced and investigated by the use of a large scale simulations and mathematical analysis. We present the simplified critical event model and discuss an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, the GBCD behaves like the solution to a Fokker-Planck Equation.
This is joint work with K. Barmak, P. Bardsley, E. Eggeling, M. Emelianenko, D.Kinderlehrer, R. Sharp, and S. Ta'asan.

Contributed Talks:
There will be time allocated for short contributed talks by participants. Priority will be given to graduate students and those in early stages of their careers. Participants interested in given a contributed talk and/or receiving financial support should visit the registration page, or e-mail Marianne Korten.

Nathan Albin, KSU
Estela A. Gavosto, KU
Marianne Korten, KSU
Charles Moore, WSU
Rodolfo H. Torres, KU

Contact Information: marianne@math.ksu.edu, lecronjs@math.ksu.edu
This conference is supported by the National Science Foundation (NSF) and by the KSU Mathematics Department through the Isidore and Hilda Dressler Endowment for the Enrichment of Mathematics. 

The Prairie Analysis Seminar is a joint project of the Department of Mathematics of  Kansas State University  and the Department of Mathematics of the University of Kansas.

The picture of the Kansas Prairie is a courtesy of the Kansas Geological Survey.