Algebraic, analytic, geometric structures emerging from quantum field theory
Organisers: Anmin Li, Buhui Chen, Li Guo, Sylvie Paycha, Lie Sheng and Bin Zhang
March 1st to March 15th, Sichuan University, Chengdu, China
Since the 80-90s of the last century, theoretical physics, especially quantum field theory, has entered a period of rapid fusion with mathematics, with the most recent mathematical research applied to quantum field theory, while observations and conclusions of physics pose further challenges to mathematics and motivations for its advances. Many structures which emerged from quantum field theory involve broad aspects of mathematics and make their study a very active areas. The aim was to provide a platform for the exchange among scholars from within and outside of China,of the most recent progresses in these topics at the frontier between mathematics and physics.
1-3 March 2024: Introductory courses (6hrs each)
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Algebra and combinatorics (Li Guo, Rutgers, Newark):
Hopf algebras, algebraic combinatorics, non associative algebras -
Analysis and geometry (Sylvie Paycha, Potsdam, Germany):
wave front sets, pseudodifferential and Fourier integral operators, gauge theory, index theory
Sylvie Paycha: Prerequisites in geometry and functional analysis -
Geometry and topology (Bin Zhang, Chengdu, China):
manifolds, vector bundles, principal bundles, connections, characteristic classes
4-8 March 2024: Mini-courses (5hrs each)
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Carlo Bellingeri (Nancy, France)
Title: Hopf algebras and randomness
Abstract: In this series of lectures, I will give an introduction to the theory of Hopf algebras and their applications to some new and old problems in stochastics. I'll start from the usual setting of graded connected Hopf algebras and explain how these objects represent the abstract framework to describe features of random objects such as random variables, stochastic processes and random fields. A fundamental example to understand these constructions is given by the well-known Connes-Kreimer Hopf algebra. I will review its main properties and show how the explicit knowledge of its primitive elements allows the construction of a consistent pathwise stochastic calculus on general stochastic processes. Some content of this mini-course will be based on an ongoing work with Emilio Rossi Ferrucci and Nikolas Tapia. -
Clément Cren (Göttingen, Germany)
Title: Pseudodifferential calculus and deformation groupoids
Abstract: In recent years the theory of deformation groupoids has given new insights on pseudodifferential calculus. We know since the early work of Connes the link between the tangent groupoid and the analytical index of such operators. More recent work have deepened this bond by showing how one could define pseudodifferential operators in a completely intrinsic fashion using the tangent groupoid (even without using distributions).
In this lecture we will define the tangent groupoid of a manifold and show how to construct pseudodifferential operators from it. No prior knowledge on pseudodifferential operators nor groupoids is necessary. We will compare the two different approaches to pseudodifferential calculus from the tangent groupoid (the one of Debord-Skandalis on one hand and van Erp-Yuncken on the other). Depending on time we might see how this approach generalizes to other pseudodifferential calculi, more suited for a sub-riemannian setting for instance, or explore the link between the deformations of the tangent groupoid and Getzler's rescaling technique. -
Alex Ferreira (Vitória, Brazil)
Title: Deriving quantum dynamics from maps between Manifolds
Abstract: In this minicourse I will introduce a generalized dynamics, given through maps between manifolds, the source space being the configuration space and the target representing the internal structure of the field. From which, under some simple assumptions, we can derive the Schroedinger equation, giving interesting geometrical interpretations to some of its uncanny properties, such as non-locality. The minicourse will start with a thorough review of manifold mapping theory and its applications in physics. Subsequently, I will introduce the Schroedinger map, establishing the foundation for the generalized dynamics, and show how classical and quantum mechanics emerge with some possible generalizations. -
Yannic Vargas (Graz, Austria)
Title: Introduction to combinatorial species through algebraic combinatorics, geometry and physics
Abstract: This short course is intended to provide an introduction to the theory of combinatorial species, with a focus on various (new) applications that have emerged from this theory over the last two decades. In 1981, André Joyal introduced this notion as "a combinatorial theory of formal power series," and since then, the field has garnered significant attention as a formal foundation for reinterpreting combinatorial aspects of various problems in Hopf algebras, hyperplane arrangements and quantum field theory.
Yannic Vargas Slides 1
Yannic Vargas Slides 2
Yannic Vargas Slides 3 -
Fabrizio Zanello (Potsdam, Germany)
Title: Introduction to perturbative Algebraic Quantum Field Theory on Minkowski spacetimes
Abstract: In these lectures we present the algebraic approach to Quantum Field Theory in its simplest framework, namely, on flat Lorentzian spacetimes. We first introduct observables as elements of an abstract algebra, prior to any representation on some Hilbert space. Then we outline the appropriate analytical framework, essentially based on microlocal analysis thechniques, which allows to interpret a quantum field theory as a deformation quantization of its classical counterpart. Finally, with these notions at hand, we approach the interaction picture from the point of view of the so-called causal perturbation theory and, if time permits, we formulate in a mathematically rigorous way the renormalization problem.
Fabrizio Zanello Slides
Slides:
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11-15 March (Workshop, 50 mins talks)
Carlos Amendola (Berlin, Germany)
Carlo Bellingeri (Nancy, France)
Pierre Clavier (Mulhouse, France)
Clément Cren (Göttingen, Allemagne)
Fay Dowker (London, England)
Alessandra Frabetti (Lyon, France)
Alexandre Fereira (Vitoria, Brazil)
Yanyong Hong (Hangzhou, China)
Naihong Hu (Shanghai, China)
Jean-David Jacques (Potsdam, Germany)
Cuipo Jiang (Shanghai, China)
Honglei Lang (Beijing, China)
Gihyun Lee (Gent, Belgium)
Fang Li (Hangzhou, China)
Jiasheng Li (Paris, France) Slide 1 Slide 2
Si Li (Beijing, China)
Yong Li (Beijing, China)
Sarah-Jean Meyers (Oxford, England)
Martin Peev (London, England)
Raphael Ponge (Chengdu, China)
Yunhe Sheng (Jilin, China)
Alberto Richtsfeld (Potsdam, Germany)
David Sauzin (Beijing and Paris)
Herprit Singh (Edinburgh, Scotland)
Sumati Surya (Bengaluru, India)
Reiko Toriumi (Okinawa, Japan)
Yannic Vargas (Graz, Austria)
Hang Wang (Shanghai, China)
Xin Wang (Shandong, China)
Xioameng Xu (Beijing, China)
Jorge Zanelli (Valdivia, Chile)
Fabrizio Zanello (Potsdam, Germany)
Honglian Zhang (Shanghai, China)
Yuanyuan Zhang (Kaifeng, Henan, China)
Guodong Zhou (Shanghai, China)
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