In the first part of the mini-course I shall discuss how the entropy production observable can be defined for any classical or quantum dynamical system as a derivative of the Radon-Nikodym cocycle. For the so-called open systems, which describe the interaction of several thermal reservoirs, this definition coincides with the standard thermodynamical definition in terms of the fluxes (heat, charge, mass...) across the system. After reviewing some basic properties of the entropy production observable and non-equilibrium steady states, I shall describe the large deviation theory of the entropy production observable. The main topic will be certain symmetries (Evans-Searls and Gallavotti-Cohen) of the moment generating functionals which can be interpreted as an extension of the Green-Kubo linear response formula to far from equilibrium steady states. The emphasis of the course will be on the mathematical structure of the theory. One novelty of the exposition is that the classical nd quantum case will be treated in parallel. The mini-course is based on some very recent joint work with Claude-Alain Pillet, Yan Pautrat, Yoshiko Ogata and Luc Rey-Bellet.

Note for Attendees

If there is sufficient interest, this talk will be expanded to a small mini-course.

Employing an affine version of the plactic algebra (which arises in theRobinson-Schensted-Knuth correspondence) one can define non-commutative Schur polynomials. The latter can be employed to construct a combinatorial ring with integer structure constants. This combinatorial ring turns out to be isomorphic to what is called the su(n) WZNW fusion ring in the physics and the su(n) Verlinde algebra (extension over C) in the mathematics literature. There is a simple physical description of this ring in terms of quantum particles hoping on the affine su(n) Dynkin diagram. Many of the known complicated results concerning the fusion ring can be derived in a novel and elementary way. Using the particle picture one also arrives at new recursion formulae for the structure constants which are dimensions of moduli spaces of generalized theta functions. I explain the close connection with the small quantum cohomology ring of the Grassmannian and present a simple reduction formula which allows to relate the structure constants of the su(n) Verlinde algebra with Gromov-Witten invariants.

Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle.

Weakly electric fish are fascinating animals that have evolved an electric sense that blends aspects of our senses of touch, vision and audition. Much is known about the relatively simple (compared to higher mammals) circuitry of their brains, the kinds of stimuli they respond to and their social communications/interactions. They are particularly well-suited to study principles of neural encoding and decoding because of the availability of electrophysiological recordings at many successive processing stations, enabling mathematical modeling of information transfer between stations. This talk will review past and current research on this topic from the experimental-theoretical collaboration of Len Maler, John Lewis and Andre Longtin at the University of Ottawa. We will focus especially on the role of feedback and how it interacts with stochastic spatio-temporal stimuli to induce oscillatory neural activity.

Abstract: Given a reasonable filtration of spaces F_0, F_1,..., F_n, and a homology theory H, we define what it means to split the filtration with respect to H and then give a criterion for when this is possible. We strongly refine this criterion
to decide when the filtration splits stably (and hence splits with respect to any homology theory). Many examples will be
discussed (like Steenrod splitting, snaith splitting, configuration spaces, commuting tuples in lie groups, Miller
splitting of unitary groups, etc). This work is by and with Stylian Zanos (Lille).

Starting from the standard contact structure of the supercircle, S^{1|1}, one considers the subgroups E(1|1), Aff(1|1), and SpO(2|1) of the group, K(1), of its contactomorphisms that respectively define its Euclidean, affine, and projective geometries. The notion of p|q-transitivity allows one to systematically construct the characteristic invariants of each geometry, in particular the super cross-ratio. One deduces the nontrivial associated 1-cocycles of K(1), e.g., the superschwarzian. The case of the supercircle S^{1|2} is also studied. The aim of this talk is to present in a synthetic fashion these geometric objects which are somewhat scattered in the literature. This is joint work with J.-P. Michel.

## Note for Attendees

If there is sufficient interest, this talk will be expanded to a small mini-course.