## Content

### Curvature of spaces

• can only see thing through a mirror
• Riemann Hypothesis
• relation to prime numbers
• and also eigenvector of matrix describing large atoms (similar with bus drivers)
• in practice it’s true (not proof, cf physics, economics (joke))
• how to become famous mathematician
• prove great theorem
• find great problem
• find great concepts
• Riemannian geometry
• curvature
• hyperbolic stitching
• constant negative curvature
• Poincaré: the art of giving same name to different things
• Ricci curvature (cf general relativity)

## Slow mastering of chance

• Boltzmann
• sect of people visiting Boltzmann’s grave
• connection to statistics
• Bernoulli
• law of large numbers
• Abraham de Moivre
• exp(-x* *2)
• Buffon’s needle
• Laplace
• the Law of Errors
• Adolphe Quetelet – popularized law of random causes
• Francis Galton (if Greeks had known that law, they’d have personified it)
• Claude Channon
• draft: information or uncertainty
• rework by von Neumann: entropy
• entropy
• number of combinations is out of this world, so logarithmize it
• vacuum doesn’t suck, it’s power of statistics
• “Disorder and information is better than order and disinformation” (about a broadcasting company)
• partial differential equations
• Navier-Stokes equations are popular in Holywood industry
• maximum entropy – law of errors
• a Boltzmann’s equation for homogenous gas (is not homogenous)

### Optimization problems

• Cercignani Conjecture
• Toscani, Villani
• “Simple to state, difficult to prove. As you like it.”
• this idea was completely naïve and could never work but he found an interesting identity
• Leonid Kantorovich
• very real-life problems (A-bomb, taxi fares, Petrograd siege)
• book: The best uses of economic resources
• operational research, plywood
• linear programming used to be in French high school curriculum
• Monge-Kantorovich
• min-max flow theorem?
• the theorem looks like maximization of capitalistic profit

### Synthesis

• conclusion of topics
• prices (optimization)
• shapes of triangles (geometry)
• entropy (statistics)
• coworkers
• Felix Otto, John Lott
• visiting in USA, no duties, just going to lunch and socialize – it cost him fortune
• the lazy gas experiment
• initial and final state of gas
• optimize the way it transitions
• similar to movement in curved space
• Optimal transport, old and new (1000 pages book by him)
• new theorem every month
• importance of random encounters for research