June 19th // Monday SkolTech |
10:30 – 11:30 | Sergei Nechaev |
From RMT to CFT |

coffee |
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11:50 – 12:50 | Ion Nechita |
On some applications of Random Matrices in Quantum Information Theory. Lecture 1 | |

lunch |
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14:00 – 15:40 | Discussion |
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coffee |
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16:10 – 16:40 | Stephane Dartois |
Blobbed topological recursion for tensor models | |

16:50 – 17:20 | Boris Bychkov |
Degrees of strata of Hurwitz spaces | |

17:20 – 17:50 | Helder Larraguivel |
Why matrix models and topological recursion? | |

18:00 – 20:00 | Welcome party |
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June 20th // Tuesday SkolTech |
10:30 – 11:30 | Ion Nechita |
On some applications of Random Matrices in Quantum Information Theory. Lecture 2 |

coffee |
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11:50 – 12:50 | Petr Dunin-Barkowski |
Proving topological recursion combinatorially for Hurwitz-type problems | |

lunch |
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14:00 – 15:30 | Discussion |
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15:30 – 16:30 | Leonid Chekhov |
ABCD of topological recursion and unitary transformations | |

coffee |
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16:50 – 17:20 | Gleb Koshevoy |
Canonical basis combinatorics and quantum Toda lattice | |

June 21st // Wednesday SkolTech |
10:30 – 11:30 | Ion Nechita |
On some applications of Random Matrices in Quantum Information Theory. Lecture 3 |

coffee |
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11:50 – 12:50 | Maxim Kazarian |
Symplectic geometry of topological recursion | |

afternoon free |
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June 22nd // Thursday Independent Univ.,11 Bolshoy Vlasyevskiy Pereulok, Moscow |
10:30 – 11:30 | Leonid Chekhov, Maxim Kazarian |
Double-colored maps: complex matrix models, topological recursion and canonical transformations |

coffee |
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11:50 – 12:50 | Ezra Getzler |
Moduli spaces of Riemann surfaces and topological field theory | |

lunch |
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14:00 – 15:30 | Discussion |
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15:30 – 16:30 | Piotr Sulkowski |
Topological recursion and quantum curves | |

coffee |
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16:50 – 17:20 | Javier Cuesta |
On the stability of the quantum Darmois-Skitovich theorem | |

17:20 – 17:50 | Pavlo Gavrylenko |
Fredholm determinant representations of the general solutions of Painlev’e equations | |

18:00 | Conference Dinner at the Independent Univ. |
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June 23rd // Friday Independent Univ.,11 Bolshoy Vlasyevskiy Pereulok, Moscow |
10:30 – 11:30 | Piotr Sulkowski |
Quantum curves as singular vectors of CFT |

coffee |
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11:50 – 12:50 | Vladimir Bazhanov |
Non-linear sigma-model and quantum-classical duality | |

lunch |
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14:00 | closing |

**Vladimir Bazhanov** // Non-linear sigma-model and quantum-classical duality

**Boris Bychkov** // Degrees of strata of Hurwitz spaces

*I will talk about computation of degrees of strata of codimension 2 in Hurwitz space of rational functions. On of the key steps in the proof is the binomial identity for Abel polynomials. As a consequences I’ll present new formulae for series of double Hurwitz numbers*

**Leonid Chekhov** // ABCD of topological recursion and unitary transformations

*I will describe an abstract topological recursion based on spectral curves. I show how topological recursion implies the existence of a set of differential constraints on the partition function. I next discuss the interplay between local and global models and how they are related via unitary transformations of Bogolyubov’s (Givental) form. Time permitting I’ll give two basics examples: GUE and Legendre ensemble*

**Leonid Chekhov, Maxim Kazarian** // Double-colored maps: complex matrix models, topological recursion and canonical transformations

**Javier Cuesta** // On the stability of the quantum Darmois-Skitovich theorem

*The Darmois-Skitovich theorem is a simple characterization of the normal distribution in terms of the independence of linear forms. We present a non-commutative version of this theorem in the context of Gaussian bosonic states (GBS), which has a clear physical realization. In this work, we show the robustness of a characterization of these GBS and give explicit estimates on the constants appearing in this stability problem*

**Petr Dunin-Barkowski** // Proving topological recursion combinatorially for Hurwitz-type problems

*We discuss the approach to proving topological recursion for Hurwitz-type problems combinatorially via semi-infinite wedge formalism. The examples of successfull application of this technique include the cases of simple Hurwitz numbers, orbifold Hurwitz numbers, monotone Hurwitz numbers, r-spin Hurwitz numbers and, surprisingly, colored HOMFLY polynomials of torus knots*

**Pavlo Gavrylenko** // Fredholm determinant representations of the general solutions of Painleve equations

*In this talk I will consider the toy example of Fredholm determinant of the special kind which can be computed explicitly. Then I will demonstrate how can it be generalized to the block Fredholm determinant in order to produce the general solution of the Painleve VI equation, and then how this solution transforms under the degenerations of Painleve VI to Painleve V and III*

**Ezra Getzler** // Moduli spaces of Riemann surfaces and topological field theory

*This talk will give a survey of the role of moduli spaces of Riemann surfaces in the study of topological field theories, emphasizing the point of view based on the theory of modular operads. I will focus on my work with Losev and Shadrin on the relationship between the operations “(homotopy) quotient by a circle action” and “coupling to gravity”. In particular, I will outline a proof of the theorem “Topological field theories up to homotopy are topological field theories”, which turns out to be a result in Teichm”uller theory: the proof uses ideas of Ivanov*

**Maxim Kazarian** // Symplectic geometry of topological recursion

*The topological recursion is govern by the geometry of Lagrangian Grassmannian in the infinitedimensional symplectic space and the linear symplectic group acting on it. To be more concrete, we illustrate this general assertion in the particular problem of enumeration of maps and hypermaps*

**Gleb Koshevoy** // Canonical basis combinatorics and quantum Toda lattice

*Fock and Goncharov conjectured that the algebra of functions on a cluster variety has a canonical vector space basis parameterized by the tropical points of the mirror cluster variety. Unfortunately, this conjecture is usually false. Gross, Hacking, Keel, and Kontsevich (2014) using methods developed in their study of mirror symmetry proved the conjecture in corrected form. For the cluster algebra of the based affine space G/N (G=SL_n), the basis is parametrized by tropical points defined through their potential W. Using combinatorics of the canonical basis constructed by Lusztig, we show that the same potential (up to a unimodular monomial transformation) is presented in the Givental mirror symmtery for G/B*

**Helder Larraguivel** // Why matrix models and topological recursion?

*In this brief talk I will argue the importance of matrix models from the point of view of path integrals and partition functions. As well as how does the topological recursion appears naturally to compute combinatorial coefficients in an asymptotic series expansion of the matrix integral by means of working out the Airy integral as an example. Finish by introducing the concept of an Airy structure and showing how it generalizes the previews version of the topological recursion*

**Sergei Nechaev** // From RMT to CFT

*It is known that some correlation functions in RMT can be formulated as the reunion probability in the ensemble of directed “vicious” random walks. We consider a model in which by changing the stretching of walks, one can pass from the CFT with the central charge c = -2 (no stretching) to the RMT (full stretching)*

**Ion Nechita** // On some applications of Random Matrices in Quantum Information Theory

*Abstract: My goal is to present some recent results in QIT which make use of random matrices. After a brief introduction to random matrix theory, I will present the method of moments, one of the most successful methods used to study the spectra of large random matrices. This will be the occasion to discuss integration over Gaussian spaces and over unitary groups; a diagrammatic method will be discussed. On the QIT side, I will focus on two main topics, random quantum states and random quantum channels. I will then prove two recent results, one on the asymptotic eigenvalue distribution of the partial transposition of random quantum states, and another on the output set of random quantum channels. Both will require some terminology and results from free probability, which will also addressed.*

Useful reference: B. Collins and I. Nechita – Random matrix techniques in quantum information theory, J. Math. Phys. 57, 015215 (2016); dx.doi.org/10.1063/1.4936880; arxiv.org/abs/1509.04689*Plan of lectures:*

Lecture 1.

– Introduction to Random Matrices

– – Gaussian random variables and integration. The Wick formula

– – – The Haar measure on the unitary group. The Weingarten formula

Lecture 2.

– Random density matrices. The induced measure

– – The asymptotic distribution of eigenvalues

– – – The partial transposition of random quantum states. Free probability theory

Lecture 3.

– Random quantum channels obtained from random isometries

– – The maximal output entropy of quantum channels. The additivity question

– – – Product of conjugate random quantum channels

– – – – The asymptotic output set of a random quantum channel

**Piotr Sulkowski** // Topological recursion and quantum curves

**Piotr Sulkowski** // Quantum curves as singular vectors of CFT

(Leonid Chekhov)