Weekly Bulletin

Hyperbolic operators and Lorentzian signature lie at the heart of relativistic physics, encoding propagation and the causal structure. Yet, many approaches to quantum field theory and quantum gravity are more naturally formulated in terms of elliptic operators and Euclidean path integrals. The conventional bridge between the two is Wick rotation, in which the time coordinate is analytically continued to imaginary values. This prescription, however, hides an important structural distinction: passing from Euclidean to Lorentzian physics is not merely a change of variable, but a change in the character of the underlying differential equations and therefore in the notion of admissible data, propagation, and causality. We investigate whether this transition can instead occur dynamically. Starting from a Euclidean quantum-gravity path integral in which spacetime itself emerges from fluctuations, we ask how the inclusion of interactions can alter the principal structure of the effective theory. We identify a path for elliptic behavior to give way to hyperbolic behavior and explore the conditions under which an effective Lorentzian signature and causal structure emerge. From this perspective, Wick rotation need not be imposed as an external analytic continuation: it may arise as an effective dynamical transition from ellipticity to hyperbolicity.

More information: https://eth-its.ethz.ch/activities/seminars-and-talks-2026-to-2027.htmlcall_made

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

Discrete Exterior Calculus has become a standard tool for computations on meshes, yet it remains mostly limited to scalar-valued forms. Many geometric and physical systems, however, involve bundle-valued forms together with a connection, for which no widely adopted discrete framework exists.This talk will introduce a discrete analogue of the exterior covariant derivative that extends DEC to forms with values in a vector bundle. We will explore the geometric ideas behind the construction, the role of local frame fields in numerical evaluation, and how the resulting operator preserves key structural properties such as Bianchi identities while converging —in contrast to previous approaches— to its smooth counterpart under refinement.

We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the viability domain of this target problem, we show that the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the viability domain also allows us to obtain the value of screening contracts. This is a joint work with Ibrahim Ekren and Liwei Huang.