## Canonical Supermartingle Couplings

ABSTRACT: Two probability distributions in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Frechet coupling of classical transport and its symmetric counterpart, the Antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward...

Read More## Yaglom limits can depend on the initial state

Abstract: To quote the economist John Maynard Keynes: “The long run is a misleading guide to current affairs. In the long run we are all dead.” It makes more sense to study the state of an evanescent system given it has not yet expired. For a substochastic Markov chain with kernel K on a state space S with killing this amounts to the study of of the Yaglom limit; that is the limiting probability the state at time n is y given the chain has not been absorbed; i.e. lim_{n\to\infty}K^n(x,y)/K^n(x,S). We given an example where the Yaglom limit depends on the starting...

Read More## Shearer’s inequality and the Infimum Rule

Abstract: We review subbadditivity properties of Shannon entropy, in particular, from the Shearer’s inequality we derive the infimum ruleâ for actions of amenable groups. We briefly discuss applicability of the infimum formula to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer’s inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the infimum fomula gives correct value of topological entropy. Joint work with Tomasz...

Read More## Teaching conics sections at the school.

ABSTRACT: We discuss a possible way of teaching conics sections at the level of the grade 6 and 7 of the elementary school through experimental mathematics in optics and equilibria.

Read More## History of Conics.

ABSTRACT: We shall discuss the history of the conics sections from the very beginning (350 BC) to Newton (1685) and their applications to equilibria, optics and celestial mechanics.

Read More## Twin peaks

Abstract: I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks. Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.

Read More