There will be several sessions taking place in the ICMS 2020. You can find the list of sessions below.
Gröbner bases offer a conceptual technology for dealing with nonlinear problems in various contexts. Appearing ubiquitously in the science and technology, they often constitute the only possibility to solve a concrete problem. Dealing with the high computational complexity in practical implementations need advances in theory. In this session, we will discuss the range of applicability of modern Gröbner bases, addressing generalizations to the coefficient rings, to the non-commutative and even non-associative algebras and rings, to name a few.
Real Algebraic Geometry is a field of substantial computational importance in the real world, even if much less taught than geometry over algebraically closed fields. Software tools for checking the satisfiability of real-algebraic formulas play an important role in a wide range of applications. Such tools are being developed in Mathematics in the form of computer algebra systems as well as in Computer Science as SAT-modulo-theories (SMT) solvers. However, there are numerous theoretical as well as practical obstacles to efficient algorithms that fit the applications. This session should be a place where theoretical and practical results in computational real algebraic geometry can be presented.
The range of applicability of numerical methods within algebraic geometry has been rapidly expanding over the past years. In this session we will focus on problems in the frontiers of algebraic geometry that have been addressed by numerical computations. Although developments in numerical methods themselves are frequently disseminated in mathematical forums, we find that algebraic geometry is rarely the point of emphasis. This sessions intends to advertise numerical computation to algebraic geometers while promoting the work of geometers who rely on numerical computation.
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.
Number Theory and Arithmetic Geometry share many common problems and techniques: Initially independent areas of mathematics, the similarity between number fields and algebraic curves over finite fields lead to the (common) theory of global fields. This combined viewpoint proved to be very fruitful - theory, algorithms and software were transported and extended in both directions. Algebraic curves over global fields provide further challenges in theory and computational practice, while relying on tools for global fields. The computational instrumentarium used and developed here spans a rather broad field: Exact symbolic computations from computer algebra need to be complemented with numerical computations using complex numbers, power series rings or p-adic rings, and with geometric methods for lattices. Algorithms are often designed and analysed based on randomisation and statistical traits. This session aims at showcasing some state of the art implementations as well as bringing together researchers with different applications and using different tools.
The goal of the session is to discuss recent advances and to facilitate new research in the development and implementation of group theoretic algorithms. This includes methods and algorithms for permutation group, matrix groups and finitely presented groups as well as various applications. Many recent advances in algorithmic group theory focus on the area of matrix groups. Here our aim is to review the state of the art of methods for finite and infinite matrix groups and to discuss the implementation of algorithms in the computer algebra systems GAP and Magma. The development of more general software systems in areas surrounding algorithmic group theory is also of significant interest. GAP and Magma have been around for a long time, more novel systems include Sage and Oscar. We plan to host talks on the state of the art of each of these systems with respect to methods in algorithmic group theory.
Classification problems in discrete geometry including geometry over finite fields have long been an active field for people interested in computations. Very often, the problems can be expressed in terms of orbits of (mostly) finite groups acting on discrete or finite sets. Posets play a big role because they allow to generate the objects of interest in small steps. Classification algorithms must take advantage of the symmetry, in order to be sufficiently fast to attack interesting problems. Large numbers of CPU cycles are spent on some problems. Parallel computing is often used to get more CPU-cycles in shorter time. Computer Algebra systems play an important role because they facilitate group theoretic algorithms. On the other hand, computational primitives (clique finding, exact cover etc.) provide the backbone in many of these hard problems. Problems in discrete geometry often can be translated into a combinatorial setting and standard software tools can be applied. Isomorph rejection then turns into graph isomorphism, a notoriously difficult problem in theoretical computer science. Canonical forms are used frequently, but they are often difficult to compute. Tools for graph isomorphism have been developed, and are widely used. There are also approaches where the isomorphism problem is attacked in the original setting, using the group action on the partially ordered set. These approaches avoid backtracking altogether, but they are more memory intense. Some of the software packages which are often used include Gap, Magma, Nauty, Dancing links, Partition backtrack etc. Recently, Orbiter is another system that can be added to the list. Many people have written special purpose software packages that are not well-known but deserve to be highlighted.
Convex polyhedra and lattice polyhedra occur in linear and discrete optimization as the feasible regions of linear programs. In algebraic geometry and its applications piecewise-linear shapes occur in the guise of polyhedral fans. Examples include secondary fans and Gröbner fans, which play major roles, e.g., in tropical geometry. This session wants to bring together people working on algorithms and software dealing with the above structures.
Homotopy Type Theory/Univalent Foundations (HoTT/UF) is a new type-theoretic foundation for mathematics based on novel connections between dependent type theory and homotopy theory. Recently there has been much interest in the constructive meaning of the univalence axiom, which has led to multiple new cubical proof assistants natively supporting univalence and higher inductive types. These proof assistants allow for the convenient formalization of abstract mathematics, especially synthetic homotopy theory, and also provide several features previously missing from many type-theoretic proof assistants, such as function extensionality and quotients. The goal of this session is to gather experts on HoTT/UF and its implementation to present recent results and discuss future directions.
The session will cover the intersection between Mathematical Software for exact or error-controllable computation, and Artificial Intelligence (AI). All talks in the session must be on the intersection, but they may focus on either direction: i.e. either (1) using Mathematical Software to improve Artificial Intelligence; or (2) using Artificial Intelligence to improve Mathematical Software. Topic (1) has been discussed in the literature for a long time but there is now a renewed focus as part of the drive for “explained AI”. Meanwhile, topic (2) follows recent parallel develops in the different ICMS communities: computer algebra systems, theorem provers, and SAT/SMT solvers for example have all been recently shown to give an improved performance when heuristic decisions are taken by ML rather than the user/developer. This sessions aims to bring together developers of different mathematical software with a common interest on the intersection with artificial intelligence.
Big Data is becoming increasingly important in fundamental mathematics research, with the production and manipulation of large datasets playing an essential role. The subject is in the throes of a data revolution, with new theoretical results being driven by millions of computer experiments which produce terabytes of data stored in huge problem-specific databases. Although the information contained in these databases is of enormous importance, thorough understanding of the datasets is often hindered by the poor performance of off-the-shelf database technologies in answering the types of broad unstructured queries of interest to mathematicians. The aim of this session is to bring together experts working with and developing large datasets in Pure Mathematics. We will explore the questions asked by these researchers, discuss the database systems they have developed, and their integration with existing Computer Algebra Systems.
Modern mathematics research includes more than just pen and paper. Mathematical research data such as numerical models, scripts, software, and other digital artifacts became the daily bread and butter of working mathematicians. However, professional organizations primarily focus on academic publications as a vehicle for knowledge transfer and reward distribution. In this ICMS session, we address the problem of sharing mathematical research data other than publications.
OpenMath is a language for exchanging mathematical formulae across applications (such as computer algebra systems). From 2010 its importance has increased in that OpenMath Content Dictionaries were adopted as a foundation of the MathML 3 W3C recommendation, the standard for mathematical formulae on the Web.
The last years have seen the emergence of the open source web-based interactive computing environment Jupyter (formerly IPython). Its flagship is the traditional notebook application that allows you to create and share documents that contain live code, equations, visualizations and narrative text; millions of such notebooks have been published online. The main novelty of Jupyter is that it can be used with dozens of programming languages, including Julia, Python, R, Caml, C++, or Coq. Thanks to this level of generality and to the use of open standards and modern web technologies, a wide ecosystem of related tools has appeared, e.g. for interactive book and slides authoring, hosting, collaborating, sharing, or publishing. Several mathematical systems (e.g. GAP, SageMath, Singular, OSCAR) have already adopted it as user interface of choice. The purpose of this session is to review and discuss the merits (and demerits!) of this ecosystem and its alternatives for mathematical research and education, notably with open science and reproducibility in mind.