May 5 - August 15, 2025
Organizers: Roland Bauerschmidt, Eveliina Peltola, Ellen Powell, Rémi Rhodes, Eero Saksman
Description: Quantum field theories (QFTs) have been successfully applied, throughout the last 70 years, to model and analyze diverse physical phenomena; in particular, critical behavior in statistical mechanics, and interactions of fundamental particles. However, a rigorous mathematical framework to construct and understand these theories is still limited. This program will further pursue this direction, building on recent developments towards QFT coming from random geometry and probability theory. In particular, the goal is to bring together researchers with different viewpoints and expertise on this multifaceted topic.
With this combined expertise, the program aims at addressing some of the main challenges and key open questions in the field, including in the following areas:
- advances in quantum gauge theories in 3D and 4D
- rigorous constructions of exactly solvable quantum field theories, especially in 2D
- more generally, analysis of probabilistic aspects of quantum field theories
- the theory of phase transitions, in particular for systems with continuous symmetry, and related phenomena
The program will include an introductory summer school and three workshops. For further information concerning these events, please see the above-mentioned links to the corresponding websites.
The online application platform to participate in this Trimester Program has been closed.
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Aru Juhan Bauerschmidt Roland Berestycki, Nathanael Camia, Federico Chandra, Ajay Chevyrev, Ilya Duch, Pawel Ganguly, Shirshendu Giuliani, Alessandro Gordina, Maria Guillarmou, Colin Gwynne, Ewain Helmuth, Tyler Holden Nina Huang, Yi-Zhi Klevtsov, Semyon Kupiainen, Antti Lacoin Hubert Lévy, Thierry Qian, Wei Schippers, Eric Sepulveda, Avelio Shen, Hao Sun, Xin Viklund, Fredrik Wang, Yilin Wu, Hao Wu, Wei Zeitouni, Ofer Zhu, Rongchan Zhu, Xiangchan |
EPFL Courant Institute of Mathematical Sciences University of Vienna New York University Abu Dhabi Imperial College London University of Edinburgh EPFL SB MATH UC Berkeley Università degli Studi Roma Tre University of Connecticut Univ. Paris Saclay University of Chicago Durham University New York University Rutgers University University of Strasbourg Helsinki University IMPA Sorbonne Université City University of Hong Kong University of Manitoba Universidad de Chile University of Wisconsin-Madison Peking University KTH Royal institute of technology Institut des Hautes Etudes Scientifiques Tsinghua University NYU Shanghai Weizmann Institute Beijing Institute of Technology Chinese academy of Sciences |
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May 12 - 16, 2025
Venue: HIM lecture hall (Poppelsdorfer Allee 45, Bonn)
Organizers: Roland Bauerschmidt, Eveliina Peltola, Ellen Powell, Rémi Rhodes, Eero Saksman
Lecturers:
- Morris Ang (University of California San Diego)
- Malin Palö Forsström (Chalmers University of Technology)
- Christophe Garban (University of Lyon)
- Antti Kupiainen (University of Helsinki)
Description:
Quantum field theories (QFTs) have been successfully applied to model and analyze diverse physical phenomena; in particular, critical behavior in statistical mechanics, and interactions of fundamental particles.
However, a rigorous mathematical framework to construct and understand these theories is still limited.
In the past two decades, making sense of Conformal Field Theory (CFT) from a probabilistic perspective has become a question of considerable interest, and significant breakthroughs have been made; for example, in contructing the Liouville CFT, and CFT quantities from critical lattice models.
Other important models of QFTs, that have attracted considerable attention in recent years, are gauge theories, which form the basis for the standard model, for which an important approach goes through lattice approximations.
Also, the macroscopic features of many critical lattice models are believed to be described by CFTs.
Interfaces in these models have been shown to be described by the Schramm–Loewner Evolution (SLE) random curves.
Another central theme in this area, motivated by the aim of describing large-scale behavior
of random planar map models, is that of so-called Liouville quantum gravity (LQG).
The relationship of SLE and LQG has been recently discovered to be rather intimate.
This Summer School explores these exciting areas of research at the intersection of probability theory and mathematical physics.
The online application platform to participate in the school will be accessible from November 4, 2024 to January 26, 2025 (11:59 PM CEST): https://math-events.uni-bonn.de/e/him2025-tp2-school
Early career Trimester Program participants who visit HIM during the period of this school are eligible to attend this event.
May 19 - 23, 2025
Venue: HIM lecture hall (Poppelsdorfer Allee 45, Bonn)
Organizers: Roland Bauerschmidt, Eveliina Peltola, Ellen Powell, Rémi Rhodes, Eero Saksman
Description:
Random geometry refers to the study of random geometric structures, such as curves and surfaces.
One of the most notable examples is the Schramm–Loewner Evolution (SLE), which has been shown to describe the scaling limits of interfaces in 2D critical models.
Significant progress has been made through its interplay with the Gaussian Free Field (GFF), deepening the understanding of the geometric properties of both SLE and the GFF.
The macroscopic features of lattice models are believed to be described by quantum field theories (QFTs).
Critical lattice models are expected to enjoy an additional conformal symmetry, and such QFTs are termed conformal field theories (CFTs).
For instance, the Ising model, a prototypical example with an order-disorder phase transition, has seen mathematical breakthroughs that have fully established its continuum description.
However, extending these results to other models, such as the Potts or random cluster models, remains a major challenge, e.g. due to the absence of similar powerful integrability techniques.
Another major theme in this workshop is Liouville quantum gravity (LQG),
a concept that can be approached through Liouville CFT, random planar maps, or the "mating-of-trees" framework.
These perspectives have recently seen significant advances and are increasingly interconnected, providing new insights into large-scale random geometry behavior.
This workshop will focus on these central themes and the challenges they pose, including:
- the construction of CFTs from lattice models
- probabilistic approaches to CFTs, such as the LQG, Liouville CFT, and its implications for 2D quantum gravity
- the development of new techniques for understanding scaling limits of
general lattice models
In case you are interested in participating in this event, please apply via the Trimester Program application platform. Please observe the application deadline.
Trimester Program guests, who were invited and have confirmed to be at HIM during the period of this workshop, are eligible to attend this event.
June 23 -27, 2025
Venue: HIM lecture hall (Poppelsdorfer Allee 45, Bonn)
Organizers: Roland Bauerschmidt, Eveliina Peltola, Ellen Powell, Rémi Rhodes, Eero Saksman
Description:
Two-dimensional conformal field theories (CFTs) are iconic examples of integrable quantum field theories.
Since their introduction by Belavin, Polyakov, and Zamolodchikov in 1984, they have advanced through developments in theoretical physics and mathematics, particularly via the conformal bootstrap method.
However, the connection between the constructive approach to CFT, through statistical physics models, and the probabilistic foundations of the bootstrap remains unclear.
Probability theory also has a role to play beyond exact solvability; more generally, in integrable QFT but the present understanding is then even more sparse.
This workshop will explore recent progress in the field, addressing key topics such as:
- minimal models and their connection to lattice models and
Schramm-Loewner evolution (SLE)
- probabilistic constructions of CFTs, notably the Liouville model and its implications for 2D quantum gravity
- bosonization and its role in integrability, particularly in the Sine-Gordon
and Thirring models
- the study of nonlinear sigma models and their importance in understanding
quantum spin chains
In case you are interested in participating in this event, please apply via the Trimester Program application platform. Please observe the application deadline.
Trimester Program guests, who were invited and have confirmed to be at HIM during the period of this workshop, are eligible to attend this event.
July 28 - August 1, 2025
Venue: HIM lecture hall (Poppelsdorfer Allee 45, Bonn)
Organizers: Roland Bauerschmidt, Eveliina Peltola, Ellen Powell, Rémi Rhodes, Eero Saksman
Description:
Quantum gauge theories, such as Yang-Mills theories with gauge group SU(N), play a fundamental role in particle physics and are central to one of the Clay Millennium Prize problems.
Lattice approximations of these theories reveal deep connections with spin systems,
which are models from condensed matter physics that generalize the Ising model.
A striking parallel between these fields is observed in the behavior of 2D spin systems and 4D gauge theories, both of which display asymptotic freedom and mass generation.
However, the mathematical understanding of non-abelian gauge theories and spin systems remains a significant challenge.
This workshop aims to bring together experts to explore these developments and tackle open problems, including:
- long-time study of stochastic Yang-Mills flow
- the development of probabilistic approaches for non-abelian and continuous symmetry models
- connections between lattice gauge theory and random geometric methods
In case you are interested in participating in this event, please apply via the Trimester Program application platform. Please observe the application deadline.
Trimester Program guests, who were invited and have confirmed to be at HIM during the period of this workshop, are eligible to attend this event.