These special lecture series in probability and mathematical finance are made possible by the generous support of the Minerva Research Foundation. Time and location vary.

In Fall 2016, the speakers are René Carmona (October 20 to 28), Robin Pemantle (November 2 to 16), and Nicolai Reshetikhin (December 5 to 9).

Date and Time | Speaker | Title |
---|---|---|

Thursday, October 20 4:10-5:25 pm SSW 903 |
René Carmona (Princeton) | Minerva Lecture: Introduction to Mean Field Games and the Two Pronged Probabilistic Approach SlidesSince its inception about a decade ago, the theory of Mean Field Games has rapidly developed into one of the most significant and exciting sources of progress in the study of the dynamical and equilibrium behavior of large systems. The introduction of ideas from statistical physics to identify approximate equilibria for sizeable dynamic games created a new wave of interest in the study of large populations of competitive individuals with "mean field" interactions. The lectures will rely on examples from economic growth theory, flocking, herding and congestion models for crowd behavior, systemic risk, cyber security, bank runs and liquidity crises, information percolation on social networks, ... to introduce the mathematical challenges raised by the intractability of most of these large scale equilibrium problems. We shall quickly review the original partial differential equations approach to the solution of these stochastic games, and introduce a probabilistic approach based on analysis on spaces of probability measures, the theory of forward/backward stochastic differential equations, and the optimal control of McKean-Vlasov stochastic differential equations. We shall show how these tools can be brought to bear in an effort to solve some of these challenging equilibrium problems. |

Tuesday, October 25 4:10-5:25 pm Math 507 |
René Carmona (Princeton) | Minerva Lecture: Calculus over Wasserstein Space and Control of McKean-Vlasov Equations Slides |

Thursday, October 27 4:10-5:25 pm SSW 903 |
René Carmona (Princeton) | Minerva Lecture: Master Equations, Games with Common Noise, and with Major and Minor PlayersSlides |

Friday, October 28 12-1 pm Math 520 |
René Carmona (Princeton) | Minerva Lecture: Finite State Space Games and Games of TimingSlides |

Wednesday, November 2 2:30-4:30 pm Math 507 |
Robin Pemantle (UPenn) | Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros:
applications to probability, combinatorics and computer scienceUnivariate Theory I: Zeros of random polynomials and power series Slides Univariate Theory II: Zeros and coefficients of polynomials in one variable Slides |

Wednesday, November 9 2:30-4:30 pm Math 507 |
Robin Pemantle (UPenn) | Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros:
applications to probability, combinatorics and computer scienceMultivariate Theory I: Boolean variables and the strong Rayleigh property Slides Multivariate Theory II: Applications: random trees, determinantal measures and sampling Slides |

Wednesday, November 16 2:30-4:30 pm Math 507 |
Robin Pemantle (UPenn) | Minerva Lecture: Polynomials, their coefficients, and the locations of their zeros: applications to probability, combinatorics and computer scienceGeometry I: Hyperbolic polynomials Slides Handout Geometry II: Multivariate generating functions Slides Handout |

Monday, December 5 5:30-7:30 pm Math 520 |
Nicolai Reshetikhin (Berkeley) | Minerva Lecture: Limit shapes in integrable models in statistical mechanicsThe lectures will start with a discussion of local models in statistical mechanics. Main examples are dimer models and the 6-vertex model. Other models will be mentioned as well. Then we will proceed to the discussion of details of dimer models. They are one of the oldest integrable models in statistical mechanics, the Ising model being a particular case. The integrability in dimer models and the 6-vertex model has a slightly different nature which we will see in Lectures 4, 5, and 6. In Lectures 1 and 2, details of the Kasteleyn solution will be given and the discussion of the thermodynamic limit will begin. The limit shape phenomenon, also known as an "arctic circle", will be introduced at the end of Lecture 2. Lecture 3 will be focused on the structure of limit shapes and correlation functions in dimer models in the thermodynamic limit. The 6-vertex model on a cylinder will be defined and studied in Lectures 4 and 5. Here the spectrum of the corresponding transfer-matrix and the Bethe ansatz will be presented, followed by the exposition of what is known about this spectrum in the thermodynamic limit. Lecture 5 will conclude with conjectures about the limit shape emergence in the 6-vertex model and its description by a variational principle. In Lecture 6 we will see that there is a strong indication about the integrability of the Euler-Lagrange PDE describing the limit shape of the 6-vertex model. We will also see that in the special case of the 6-vertex model, known as the stochastic point, these PDE become the Burgers equation. The stochastic point of the 6-vertex model is closely related to the ASEP model in stochastic processes. This part of lectures is based on joint results with Ananth Sridhar. At the end some speculations about correlation functions in the 6-vertex model and about other models will be given. Lecture 1 Slides Lecture 2 Slides |

Wednesday, December 7 5:30-7:30 pm Math 520 |
Nicolai Reshetikhin (Berkeley) | Minerva Lecture: Limit shapes in integrable models in statistical mechanicsLecture 3 Slides Lecture 4 Slides |

Friday, December 9 11:45 am - 1:45 pm Math 520 |
Nicolai Reshetikhin (Berkeley) | Minerva Lecture: Limit shapes in integrable models in statistical mechanicsLecture 5 Slides Lecture 6 Slides |