The Clay Mathematics Institute (CMI) is a prestigious organization based in Cambridge, Massachusetts, established in 1998. Its primary goal is to increase and disseminate mathematical knowledge and to promote the study of mathematics. The institute is well-known for its formulation of the Millennium Prize Problems, a set of seven of the most important unsolved problems in mathematics.

The Arithmetical Hierarchy is a classification of decision problems (or sets of natural numbers) based on the complexity of their definitions in terms of logical formulas. It arises from the study of computability and formal logic, particularly in relation to first-order arithmetic. The hierarchy is built on the idea of quantifier alternation in logical statements.

The Heilbronn Institute for Mathematical Research is an organization based in the UK that focuses on theoretical research in mathematics, particularly in areas like number theory, combinatorics, and related fields. Founded in 2000, it was established with the aim of fostering collaboration among mathematicians and providing support for research activities. The institute is named after the German mathematician and philanthropist, Sir Klaus Heilbronn.

The Institut de Mathématiques de Toulouse (IMT) is a mathematics research institute located in Toulouse, France. It is affiliated with the University of Toulouse and is part of the larger educational and research consortium in the region. IMT focuses on a wide range of mathematical fields, including pure and applied mathematics. It serves as a hub for research, collaboration, and education in mathematics, hosting seminars, workshops, and conferences to promote mathematical research and community engagement.

The Institute for Computational and Experimental Research in Mathematics (ICERM) is a research institute associated with Brown University, focused on the intersection of mathematics, computation, and experimental research. Established in 2013, ICERM aims to foster collaboration among mathematicians, scientists, and engineers by providing a space for interdisciplinary research and computational experimentation.

The Institute for Experimental Mathematics (IEM) is a research institution that focuses on exploratory and experimental approaches to mathematics. Although specific details about particular institutions can vary, the general goals of such institutes typically include: 1. **Interdisciplinary Research**: Promoting collaboration between mathematicians and scientists from various fields to explore new mathematics that arise from experimental work.

The Institute for Mathematics and its Applications (IMA) is a research organization based in the United States that focuses on the application of mathematics to real-world problems. Established in 1982 and located in Minneapolis, Minnesota, the IMA aims to foster mathematical research and promote collaboration between mathematicians and other scientists and professionals. The IMA organizes conferences, workshops, and special events that bring together mathematicians and experts from various fields to address challenging problems.

The Institute of Statistical Mathematics (ISM) is a research institution located in Tokyo, Japan, dedicated to the field of statistical mathematics. It was established with the aim of promoting research in statistics and its applications, as well as advancing education and training in this area. The ISM conducts both theoretical and applied research in various domains of statistics, including but not limited to statistical theory, methodology, computational statistics, and statistical applications in fields such as social science, medicine, and environmental science.

The Interdisciplinary Center for Scientific Computing (IWR) is a research institution based at the University of Heidelberg in Germany. It focuses on the development and application of computational methods in scientific research across various disciplines, such as physics, biology, chemistry, and engineering. The center promotes interdisciplinary collaboration, enabling researchers from different fields to work together to solve complex scientific problems through computational techniques.

The Istanbul Center for Mathematical Sciences (ICMS) is a research institution located in Istanbul, Turkey, focusing on various areas of mathematics and its applications. It aims to promote mathematical research and education, facilitating collaboration among mathematicians both locally and internationally. The center often hosts seminars, workshops, and conferences, providing a platform for researchers to share their work and ideas.

The Institute for Pure and Applied Mathematics (IPAM) is a research institute located at the University of California, Los Angeles (UCLA). It focuses on interdisciplinary research in mathematics and its applications to various fields. Established to foster collaboration between mathematicians and scientists from diverse disciplines, IPAM serves as a venue for workshops, seminars, and research programs that bring together experts in pure mathematics and applied mathematics.

The Institute of Applied Physics and Computational Mathematics (IAPCM) is an academic institution often associated with research and education in applied physics and computational mathematics. While specific details may vary by country or region, institutions with similar names generally focus on: 1. **Research**: Conducting advanced research in areas such as applied physics, computational methods, numerical analysis, and related fields. This includes both theoretical studies and practical applications.

The Korteweg-de Vries Institute for Mathematics (KdVI) is a research institute located in Amsterdam, Netherlands, affiliated with the University of Amsterdam. It focuses on various areas of mathematics, including pure and applied mathematics, and emphasizes both research and education. The institute is named after the Korteweg-de Vries equation, a significant partial differential equation that arises in the study of shallow water waves and soliton theory.

The B. Verkin Institute for Low Temperature Physics and Engineering, located in Kharkiv, Ukraine, is a prominent research institution that specializes in low-temperature physics, condensed matter physics, and related fields. The Mathematical Division specifically is likely involved in theoretical and mathematical modeling related to the phenomena studied at the institute, including superconductivity, quantum mechanics, and other areas of condensed matter physics.

The Istituto Nazionale di Alta Matematica Francesco Severi, commonly known as INdAM, is an Italian research institute dedicated to advanced studies in mathematics. Established in 1939, it was named after the Italian mathematician Francesco Severi. The institute aims to promote research and education in various fields of mathematics, supporting both theoretical and applied mathematics. INdAM organizes conferences, workshops, and lecture series, providing a platform for mathematicians and researchers to collaborate and share their findings.

The János Bolyai Mathematical Institute is a prominent research institution located in Szeged, Hungary, and is part of the University of Szeged. It was established in honor of János Bolyai, a 19th-century Hungarian mathematician known for his contributions to geometry and the development of non-Euclidean geometry. The institute focuses on a wide range of mathematical disciplines, including but not limited to pure mathematics, applied mathematics, and mathematical education.

The Keldysh Institute of Applied Mathematics (KIAM) is a research institution in Russia that is part of the Russian Academy of Sciences. Established in 1991, the institute is named after the prominent mathematician Mstislav Keldysh, who made significant contributions to various fields of mathematics and applied mathematics. KIAM specializes in applying mathematical methods and computational techniques to solve problems in various domains, including physics, engineering, economics, and social sciences.

The Low Basis Theorem is a concept from algebraic geometry and commutative algebra, particularly within the context of syzygies, which are relations among generators of a module. The theorem deals with certain properties of a graded free resolution of a module over a polynomial ring.

The Max Planck Institute for Mathematics in the Sciences (MPI MiS) is a research institution located in Leipzig, Germany. It is part of the Max Planck Society, which is renowned for its advanced scientific research across various disciplines. The MPI MiS focuses on the application of mathematical methods to address problems in the natural and social sciences. Established in 1996, the institute aims to foster interdisciplinary collaboration and promote innovations in areas such as mathematical physics, computational science, and data analysis.

The NASU Institute of Mathematics is a research institution located in Ukraine, affiliated with the National Academy of Sciences of Ukraine (NASU). The institute focuses on various fields of mathematics, including pure and applied mathematics, mathematical modeling, and computational mathematics. It plays a significant role in advancing mathematical research in Ukraine and often collaborates with mathematicians and institutions around the world.