IMPRS Combo III

The IMPRS Combo Cosmology and Quantum Information Theory is an interdisciplinary and introductory workshop for PhD students and PostDocs in their early career. It aims to bring together all IMPRS students and encourage them to learn about a wider spectrum of mathematics. We will have three lecture series. Two of which are focused on the mathematical side, namely on information theory by Joan Simon Soler and on the Kähler geometry of random curves by Yilin Wang. The third lecture series, by Matthew Cordes, takes a more general standpoint and introduces the ethics behind math research. The lectures will start on an introductory level so that almost no additional pre-knowledge is required. We will provide basic introductions/repetitions to the most crucial topics to our Combo.

Matthew Cordes (Heriot Watt University): Ethics for Mathematicians

In these talks we will discuss the ethics related to the practice of mathematics. We will explore several applications of mathematics and their impacts on society and questions inherent to the academic mathematical community.

Joan Simon Soler (University of Edinburgh): Information, Entropy and Gravity: From Coins to Black Holes

Information theory cuts across different scientific research disciplines. The purpose of this course is to develop the mathematics quantifying classical information, such as for a coin, to describe its more relevant differences when extending this framework to quantum mechanics and, time permitting, to point out how these conceptual ideas are helping us to understand some potential deep aspects in gravitational physics, such as black holes. The course will not assume any prior knowledge on information theory, quantum mechanics, differential geometry and general relativity.

Yilin Wang (IHES, Paris): From the universality to the Kähler geometry of random curves

In probability theory, universality is the phenomenon where random processes converge to a common limit despite microscopic differences. For instance, the random walk, under mild conditions, converges to the same Brownian motion seen from afar, regardless of the law of each independent step. This phenomenon underlies the appearance of the random simple curve, called SLE, as the universal scaling limit of interfaces in conformally invariant 2D systems. SLE plays a central role in 2D random conformal geometry and a probabilistic approach to 2D quantum gravity and conformal field theory. On the other hand, a subfamily of relatively regular simple curves forms the Weil-Petersson Teichmüller space and has an essentially unique Kähler geometry. To describe these geometric structures we invoke the group structure and Kähler structure which is described via infinitesimal variations of the curves. Although these two worlds look very different we will explain how they are tied together via the Loewner energy.

Course Material

Riemannian geometry (focus: hyperbolic), Brownian motion (focus: on manifolds)

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