Here is a list of research papers that have been suggested for reading and presenting:
- Liero, Mielke, Savare preprints on geometry of optimal transport with mass loss and creation. Global distance is the Hellinger-Kantorovich distance. http://arxiv.org/abs/1508.07941, http://arxiv.org/pdf/1509.00068.pdf.
- Numerics of Optimal Transport: Cuturi, Benamou, Carlier, Cuturi, Nenna, and Peyre, Oberman, Ruan, Merigot and Oudet (presenter: Slav). Also with connections to volume preserving transport, Merigot and Mirebeau (presenter: Antoine).
- Optimal transportation in random setting: Garcia Trillos and Slepcev, and Fournier Guillin (and references therein).
- Gradient flows and the relationship with the Fokker-Planck equation: Jordan, Kinderlehrer and Otto (presenter: David).
- Discrete optimal transportation and discrete gradient flows wrt. Wasserstein metric. An interesting question is to consider these in random setting. (Maas, Gigli and Maas, Erbar and Maas , Gozlan, Samson, Roberto and Tetali).
- Microscopic origins of gradient flows and connections with large deviations: Adams, Dirr, Peletier and Zimmer, Fathi, Duong, Laschos and Renger, Erbar, Maas, and Renger Liero, Mielke, Peletier, Renger On microscopic origins of generalized gradient structures (see also Peletier for background on large deviations and gradient flows) (presenter: Matt and Riccardo).
- Systems of equations as gradient flows: Reaction diffusion systems Liero and Mielke.
- Monge-Kantorovich can be recast in a fluid mechanics setting: Benamou and Brenier (presenter: Marco).
- Transport distances that penalize paths: Qinglan Xia (presenter: Kevin).
Good reference texts for background reading are:
- Otto, 2001, The geometry of dissipative evolution equations: the porous medium equation
- Villani, 2003, Topics in Optimal Transportation (link to publisher)
- Villani,2008, Optimal transport, old and new
- Santambrogio, 2015, Optimal Transport for Applied Mathematicians.
- Ambrosio, Gigli and Savare, 2008 (2nd edition), Gradient Flows in Metric Spaces and in the Space of Probability Measures. (link to publisher) In particular we will go over chapter 8 on absolutely continuous curves and the continuity equation.
- Ambrosio and Gigli, 2011, A User’s Guide to Optimal Transport. (contains a good part of AGS, but in less generality)
- Evans, 2001, Partial Differential Equations and Monge-Kantorovich Mass Transfer.
- Ambrosio, 1995, Lecture Notes on Optimal Transport Problems.
- Ambrosio and Pratelli, 2004, Existence and Stability Results in the L1 Theory of Optimal Transportation
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