ICMS 2020 Session: Computational Algebraic Analysis

ICMS 2020 Session: Computational Algebraic Analysis

ICMS 2020: Home, Sessions

Organizers

• Christoph Koutschan (Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria)
• Anna-Laura Sattelberger (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany)

Aim and Scope

The theory of D-modules enables an investigation of homogeneous partial linear differential equations with polynomial coefficients by algebraic methods. Many (special) functions arising in a mathematician's or physicist's daily life can be understood by their annihilating ideal in the Weyl algebra. The class of holonomic functions is a prominent, widely-used example for this approach. Applications include maximum likelihood estimates in statistics, volume computations of compact semi-algebraic sets, high precision evaluation of holonomic functions, and local optimization methods. There is a variety of computer algebra systems that allow computations in non-commutative rings of linear partial differential operators. The aim of this session is to give an overview of the broad range of applications, to introduce the participants to existing software, and to discuss recent developments in this field.

Topics (including, but not limited to)

• Bernstein–Sato theory
• Creative telescoping algorithms
• High-precision numerical evaluation of holonomic functions
• Holonomic gradient method and holonomic gradient descent
• Holonomic sequences
• Symbolic integration and summation
• Volume computation of compact semi-algebraic sets

Schedule

The COVID-19 pandemic effects ICMS 2020 as every planned conference. Despite the extraordinary conditions, ICMS 2020 will happen – as a virtual conference. The speakers are kindly asked to upload recordings of their talks in advance, and the audience is supposed to watch these videos prior to the live discussion session. For Session D "Computational Algebraic Analysis", the discussions on the talks take place during three slots:

• Thursday, July 16, 14:00 – 14:30:
  Anna-Laura Sattelberger
  Saiei-Jaeyeong Matsubara-Heo
  Francisco Jesús Castro Jiménez (talk from Session A)
• Thursday, July 16, 15:50 – 16:20:
  Francisco Jesús Castro Jiménez (talk from Session D)
  Bernd Sturmfels
• Thursday, July 16, 16:30 – 17:10:
  Viktor Levandovskyy
  Paul Görlach
  András Cristian Lőrincz

Submission Guidelines

For details, see the official guidelines. The most important points are:

• If you wish to give a talk at ICMS, you need to submit a title and short abstract to the session organizers (until February 23 at latest – the earlier the better).

• Those speakers which have been accepted to the session may submit an extended abstract to the LNCS proceedings volume (4–8 pages, due March 16). Please use this EasyChair link for the submission.

• The PC decision on the extended abstracts will be communicated on April 27, and the camera-ready papers are due on May 9, 2020.

Talks/Abstracts

(in alphabetical order)

• Francisco Jesús Castro Jiménez (Universidad de Sevilla, Spain):
  

  On the b-function associated with some irregular GKZ-hypergeometric ideals

  A result of M. Saito, B. Sturmfels and N. Takayama gives a closed formula for the generic b-function associated with a real weight vector and a regular GKZ-hypergeometric ideal in the complex Weyl algebra of order n. In this talk we propose an analogous formula in the irregular case, also valid for non-generic b-functions, when the hypergeometric ideal is associated with a smooth monomial curve in the three dimensional affine space. This talk is based on a joint work with Helena Cobo.

• Paul Görlach (Technische Universität Chemnitz, Germany):
  

  The holonomic gradient method in statistics

  The evaluation and optimization of holonomic functions can be aided by a systematic usage of its annihilating differential operators, known as the holonomic gradient method. In statistics, its importance revolves around the computation of normalizing constants and maximum likelihood estimates. We survey recent developments in this area with a focus on existing implementations.

• András Cristian Lőrincz (Humboldt-Universität zu Berlin, Germany):
  

  Algebraic Analysis of Rotation Data

  In this talk we present tools from algebraic analysis for statistical inference in the Fisher model for the rotation group. Based on the holonomic gradient method, we develop algorithms for maximum likelihood estimation and apply them for data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO(3) to arbitrary Lie groups. This is joint work with Michael F. Adamer, Anna-Laura Sattelberger, and Bernd Sturmfels.

• Viktor Levandovskyy (RWTH Aachen University, Germany):
  

  D-modules and algorithmic algebraic analysis with Singular

  We review the needs of modern algebraic analysis in the realm of algorithms and their implementations. Some highlights from the rich experience of developing, implementing and applying such algorithms will be presented. A whole suite of functionalities, implemented in Singular:Plural, will be described.

• Saiei-Jaeyeong Matsubara-Heo (Kobe University, Japan):
  

  Computing cohomology intersection numbers of GKZ systems

  In the theory of special functions, a particular kind of multidimensional integral appears frequently. It is called the Euler integral. In order to understand the topological nature of the integral, twisted de Rham cohomology theory plays an important role. In this talk, we propose an algorithm of computing an invariant "cohomology intersection number". The algorithm is based on the fact that the Euler integral satisfies a GKZ system and utilizes algorithms to find rational solutions of differential equations. We also provide software to perform this algorithm. This talk is based on a joint work with Nobuki Takayama.

• Anna-Laura Sattelberger (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany):
  

  Computational Algebraic Analysis

  This talk is meant to be an introductory talk to the session on Computational Algebraic Analysis. We briefly describe the general mathematical setup, give a first overview of the broad range of applications and existing software, and discuss recent developments in this field.

• Bernd Sturmfels (University of California at Berkeley, U.S., and Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany):
  

  Primary Ideals and their Differential Equations

  An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. Recent work with Yairon Cid-Ruiz and Roser Homs furnishes a structure theory for primary ideals in this context. After a brief historical introduction, from Gröbner to Ehrenpreis and Palamodov, we present connections to D-modules and software in Macaulay2.