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Elliptics Summer School 2023

→ Europe/Zurich
UZH Irchel

UZH Irchel

Zurich
Robin Marzucca (University of Zurich), Kay Schönwald (University of Zuerich), Vasily Sotnikov (University of Zurich (UZH))
Description

The appearance of elliptic curves in calculations in high-energy physics has long been a major bottleneck which has brought together researchers from both mathematics and physics to investigate the properties of integrals over elliptic curves. These interdisciplinary collaborations have been very fruitful, and have led to various descriptions of iterated integrals over elliptic curves as well as a better understanding of the mathematical properties underlying these functions. With many challenges still to overcome and integrals over elliptic curves becoming relevant also for gravitational wave scattering, the study of elliptic curves is a highly interesting and active field of research.

The lectures will be held by experts from both Physics and Mathematics and will deliver an introduction to the mathematical properties of elliptic curves and iterated integrals over them, and will prepare the participants to take part in highly relevant research opportunities.

Introductory lectures will be given by Robin Marzucca, Kay Schönwald and Vasily Sotnikov. The expert lectures will be given by

  • Johannes Brödel
  • Claire Burrin
  • Lorenzo Tancredi
  • Stefan Weinzierl

 

The elliptic summer school is supported by the UZH Graduate Campus and the European Research Council (ERC) Grant 101019620 (ERC Advanced Grant TOPUP)

Poster.pdf
  • Monday 5 June
    • 09:00 → 09:30
      Welcome 30m
    • 09:30 → 10:30
      Introductory Lecture 1: Scattering Amplitudes and Feynman integrals 1h

      We briefly overview the context in which special functions related to elliptic curves appear in particle physics and discuss our motivation to study their mathematical properties.
      We review certain elementary properties of Feynman integrals and showcase how they can give rise to new classes of functions.
      We then introduce multiple polylogarithms, a well understood class of functions which serves as a role model for more complicated classes of functions that will be discussed in this school.

      Speaker: Vasily Sotnikov (University of Zurich (UZH))
      presentation.pdf
    • 10:30 → 12:00
      Coffee 1h 30m
    • 12:00 → 13:00
      Introductory Lecture 2: Differential Equations and Iterated Integrals 1h

      We introduce differential equations that Feynman integrals satisfy and discuss how and when they can be formally solved through iterated integrals. We discuss general properties of iterated integrals and their relation to the known classes of functions.
      Finally, we showcase how elliptic periods appear as solutions of the homogenous differential equations.

      Speaker: Kay Schönwald (University of Zuerich)
    • 13:00 → 14:30
      Lunch 1h 30m
    • 14:30 → 15:30
      Robin Marzucca - Introduction to Elliptic Curves 1h

      I will give an overview of different representations of elliptic curves which will be useful throughout the lectures and will discuss abelian differentials defined on them.

      Speaker: Robin Marzucca (University of Zurich)
      Intro_To_Elliptic_Curves.pdf
    • 15:30 → 17:00
      Lorenzo Tancredi 1h 30m
      Speaker: Prof. Lorenzo Tancredi
      1. empls.pdf
  • Tuesday 6 June
    • 09:00 → 10:30
      Claire Burrin - Introduction to Modular Forms I 1h 30m

      I will give an overview of classical modular forms, groups, and symbols. If time is left, I will discuss some recent conjectures of Mazur and Rubin that have attracted a lot of attention among number theorists.

      Speaker: Prof. Claire Burrin
    • 10:30 → 11:00
      Coffee 30m
    • 11:00 → 12:30
      Lorenzo Tancredi 1h 30m
      Speaker: Prof. Lorenzo Tancredi
      2. torus picture and modular forms.pdf
    • 12:30 → 14:00
      Lunch 1h 30m
    • 14:00 → 17:00
      Exercises & Coffee 3h
      Exercises_Elliptics23_Tancredi.pdf
      Exercises_introduction.pdf
  • Wednesday 7 June
    • 09:00 → 10:30
      Claire Burrin - Introduction to Modular Forms II 1h 30m

      I will give an overview of classical modular forms, groups, and symbols. If time is left, I will discuss some recent conjectures of Mazur and Rubin that have attracted a lot of attention among number theorists.

      Speaker: Prof. Claire Burrin
    • 10:30 → 11:00
      Coffee 30m
    • 11:00 → 12:30
      Stefan Weinzierl - Construction of epsilon-factorised differential equations for elliptic Feynman integrals and beyond 1h 30m

      An epsilon-factorised differential equation allows us to construct solutions for a given Feynman integral to all order in the dimensional regularisation parameter epsilon. The task of computing a Feynman integral is therefore reduced to finding a transformation, which puts the differential equation for a Feynman integral into an epsilon-factorised form. In this lecture I will show how such a transformation can be found for elliptic Feynman integrals and Calabi-Yau Feynman integrals.

      Speaker: Prof. Stefan Weinzierl
    • 12:30 → 14:00
      Lunch 1h 30m
    • 14:00 → 17:00
      Exercises & Coffee 3h
      elliptic_exercise_01_Weinzierl.pdf
      elliptic_exercise_02_Weinzierl_2.pdf
      Exercises_Elliptics23_Tancredi.pdf
      Exercises_introduction
    • 18:00 → 20:30
      Apero 2h 30m
  • Thursday 8 June
    • 09:00 → 10:30
      Johannes Brödel - Numbers on elliptic curves 1h 30m

      In analogy to multiple zeta values, which appear as special values of polylogarithms one can define so-called elliptic multiple zeta values. In this lecture, the basics notions about multiple zeta values will be discussed before moving on to elliptic multiple zeta values, which appear as special values of elliptic polylogarithms. When discussing algebraic properties of those objects, a link will be made to their single-valued analogues: single-valued multiple zeta values and modular graph functions.

      Speaker: Dr Johannes Brödel
    • 10:30 → 11:00
      Coffee 30m
    • 11:00 → 12:30
      Stefan Weinzierl - Construction of epsilon-factorised differential equations for elliptic Feynman integrals and beyond 1h 30m

      An epsilon-factorised differential equation allows us to construct solutions for a given Feynman integral to all order in the dimensional regularisation parameter epsilon. The task of computing a Feynman integral is therefore reduced to finding a transformation, which puts the differential equation for a Feynman integral into an epsilon-factorised form. In this lecture I will show how such a transformation can be found for elliptic Feynman integrals and Calabi-Yau Feynman integrals.

      Speaker: Prof. Stefan Weinzierl
    • 12:30 → 14:00
      Lunch 1h 30m
    • 14:00 → 17:00
      Exercises & Coffee 3h
      elliptic_exercise_02_Weinzierl_2.pdf
      Prob01_Broedel.pdf
  • Friday 9 June
    • 09:00 → 10:30
      Johannes Brödel - Beyond the elliptic curve 1h 30m

      Numerous structures, objects and algorithms have been made accessible and have been understood for elliptic curves: abelian differentials, the Kronecker function, various flavours of iterated integrals and elliptic multiple zeta values. In this lecture, I will discuss the framework and boundary conditions in which the obvious question for similar structures on surfaces of genus two might be answered. We will explore higher-genus versions of theta functions, generalization of the Fay identity and have a short peek on Siegel modular forms. If time permits, we might discuss what it needs for a genus-two version of the well-known Kronecker function at genus one.

      Speaker: Dr Johannes Brödel
    • 10:30 → 11:00
      Coffee 30m
    • 11:00 → 12:30
      Exercises 1h 30m
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