**Abishek Dhar:** The quantum first passage and first return problems.

*The lectures will provide an introduction to the problem of first passage and first return in quantum systems. This question leads one to the study of the dynamics where the unitary quantum evolution is interrupted by repeated projective measurements. **We will discuss the *resulting effective dynamics and apply them to obtain results for the first passage problem.

**Tristan Benoist:** Introduction to quantum trajectories.

*In this course, after the definition of the Markov chain associated to a Quantum trajectories and the probability measures defining the laws of the sequences of measurement outcomes, we will present their main properties. We will particularly focus on their long time asymptotic properties (ergodicity, law of large numbers, central limit theorem, large deviations,...). One that is specific to quantum trajectories is the phenomenon of purification. We will particularly see the role it plays in the classification of invariant measures of quantum trajectories. If time remains we will discuss some results in statistics and introduce some current open problems.*

**Uwe Franz:** Semigroups of positive maps.

*We will recall the definitions and characterisations of k-positive and k-superpositive maps, and show that their cones are mutually dual. Then we will study semigroups of such maps acting on the matrix algebras Mn(C). We will show how to characterize the generators of semigroups of k-positive and k-superpositive maps, using a Schoenberg-type correspondence due Schürmann.*

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**Benoit Collins:** Equivariance and positivity.

*We will review a few recent results about techniques from group symmetries to describe new positive functions and methods to detect various types of entanglement. We will also discuss questions related to positivity and k-positivity from the point of view of free probability and random matrix theory. This is based on joint works with Bardet, Sapra, Osaka on the one hand, and Hayden, Nechita on the other hand.*

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**B V Rajarama Bhat:** C*-convexity and completely positive maps.

*C*-convexity is a quantized version of classical convexity **where scalars of convex combinations get replaced by elements of **C*-algebras. **This concept has a natural role in the study of completely positive **maps and positive operator valued measures. **We characterize C*-extreme points of various C*-convex sets and **explore connections with **the theory of nest algebras.*

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**Ion Nechita:** A mathematical introduction to quantum entanglement.

*Entanglement is a type of correlation arising in quantum theory which does not have a classical counterpart. I will introduce entanglement by discussing the different ways one can construct state spaces for a bipartite system starting from the individual sets of states. I will then discuss entanglement detection (and entanglement criteria), followed by entanglement measures. Finally, I will briefly mention multipartite entanglement.*

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Lecture videos & slides