Cyclotomic Generating Functions

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a holistic introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

In this chapter we summarize the basic elements of algebraic geometry and commutative algebra that are useful in the study of (modular) invariant theory. Normally, these techniques are most useful in questions about the structure of rings of invariants, and, as well, they seem to be most useful in proving theorems that hold true for all groups, modular or not. It is worth noting that a large fraction (as much as one third — see the paper of Fisher (1966, Page 146)) of the papers in mathematics in the latter stages of the 19th century were studies of invariant theory. It is worth noting as well that commutative algebra was invented, discovered if you prefer, by Hilbert, in order to clarify and understand invariant theory more fully, see Fisher (1966). There are several excellent references, including Atiyah and Macdonald (1969), Dummit and Foote (2004), Eisenbud (1995), Lang (1984), Matsumura (1980), and a forthcoming book by Kemper, (2010).

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas. This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings: (1) The study of the structure and representation theory through a) Slodowy slices and the Joseph ideal, b) Equivalence of categories of representations and certain geometric counterparts, c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions, e) free Lie algebras and current algebras, f) Dirac cohomology, h) Gelfand-Zeitlin theory (2) Combinatorial Aspects of Lie Systems particularly through a) Saturation problems and Buildings, b) Affine and double Affine Hecke algebras, Cherednik algebras, c) Solvable lattice models (3) Geometric Structures including a) Representation theoretic methods in enumerative geometry, b) Multiplicity free symplectic reduction, c) Monodromy actions of braid groups, d) Associated varieties for Lie super algebras, e) affine Grassmannians, f) Schubert varieties and Demazure modules The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.

The general subject of the conference was geometric group theory, that is the study of groups via suitable actions on spaces endowed with additional structures, for instance a distance, a measure (class), or both of them with natural compatibilities. The interactions with other mathematical fields such as algebraic topology (e.g. via homological methods), functional analysis (e.g. via the theory of operator algebras), probability theory (e.g. via random walks) or even representation theory and number theory, play an increasing role in the development of this part of group theory.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory-one of the two main sources of the modern discipline of commutative algebra.' The other source was algebraic geometry. It was in the setting of these two subjects that many of the main concepts and results of commutative algebra evolved in the 19th century. Thus commutative algebra can be said to have been well developed before it was created. To set the scene for our story, a word about mathematics, and especially algebra, in the 19th century. The period witnessed! fundamental transformations in mathematics -in its concepts, its methods, and in mathematicians' attitude toward their subject. These included a growing insistence on rigor and abstraction; a predisposition for founding general theories rather than focusing on specific problems; an acceptance of nonconstructive existence proofs; the emergence of mathematical specialties and specialists; the rise of the view of mathematics as a free human activity, neither deriving from, nor dependent on, nor necessarily applicable to concrete setting; the rise of set-theoretic thinking; and the reemergence of (a new variant of) the axiomatic method. As for algebra, its subject-matter and methods changed beyond recognition. Algebra as the study of solvability of polynomial equations gave way to algebra as the study of abstract structures defined axiomatically. While in the past algebra was on the periphery of mathematics, it now (the 19th and early 20th centuries) became one of its central concerns. Moreover, algebra began to penetrate other mathematical fields (e.g. geometry, analysis, logic, topology, number theory) to such an extent that in the early decades of the 20th century one began to speak of the algebraization of mathematics ([3, p. 135] and [22]).

In this survey article, we discuss the theory of Koszul algebras and Koszul duality. Koszul property plays an important role in various topics such as commutative and noncommutative algebra, geometry, topology, Lie algebras, representation theory, combinatorics and semigroup rings. Most of the results in this survey article are already known or are easy corollaries of known results. We collect these results and we discuss some proofs. Several survey articles, notably [22, 13, 14, 15, 31], and a book on quadratic algebras [36] contain valuable material to the theory of Koszul algebras or Koszul duality. Keeping these survey articles in mind, the focus has been to collect new results on Koszulness of diagonal subalgebras, its interaction with Polya frequency sequence, and monomial projective curves.

Invariants, their relatives, and problems Algebraic finiteness Combinatorial finiteness Noetherian finiteness Homological finiteness Modular invariant theory Special classes of invariants The Steenrod algebra and invariant theory Invariant ideals Lannes's T-functor and applications Review of commutative algebra References Typography Notation Index.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work. The College Mathematics Journal: Vol. 30, No. 3, pp. 170-186.

Dear Reader: The Moscow Journal of Combinatorics and Number Theory was founded in 2010 by the Moscow Institute of Physics and Technology, and in 2018 it started being published by MSP (Mathematical Sciences Publishers), a nonprofit scientific publisher based in Berkeley, California. Our journal publishes original, high-quality research articles from a broad range of interests within combinatorics, number theory and allied areas. Since 2011 we have published over 100 papers. Among our authors are such mathematicians as Noga Alon, Antal Balog, Jean-Pierre Demailly, Dominic Foata, Peter Frankl, Aleksandar Ivic, Sergei Konyagin, Yuri Nesterenko, Janos Pach, Yakov Sinai, Andrzej Schinzel, Wolfgang Schmidt, Carlo Viola, Michel Waldschmidt, and many others. This issue 1 of volume 8 is the first issue to appear under MSP's aegis. It contains selected papers presented by the participants of the Vilnius Conference in Combinatorics and Number Theory, which was organized with support from our journal and took place at the Department of Mathematics and Informatics of the University of Vilnius, Lithuania, 16–22 July 2017. Previous conferences connected to our journal were held in Russia (Diophantine analysis, Astrakhan, 30 July to 3 August 2012), Lithuania (Palanga Conference in Combinatorics and Number Theory, 1–7 September 2013), again Russia (Moscow Workshop in Combinatorics and Number Theory, January 27 to 2 February 2014), and Denmark (Diophantine Approximation and Related Topics, Aarhus, 13–17 July 2015). A collection of papers from the Astrakhan conference appeared in issue 3–4 of volume 3 (2013), and papers related to the Aarhus conference in issue 2–3 of volume 6 (2016). We hope that you will enjoy this issue and support the journal both with your submissions and by recommending a subscription to your institutional library!

While computer algebra systems have dealt with polynomials and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent problem. In this paper we explain how one system solves this problem, what types and operators it is necessary to introduce and, in short, how one can construct a computational theory of commutative algebra. Of necessity, such a theory is rather different from the conventional, non-constructive, theory. It is also somewhat different from the theories of Seidenberg [1974] and his school, who are not particularly concerned with practical questions of efficiency.KeywordsIntegral DomainCommutative AlgebraComputer Algebra SystemUnique FactorisationGreat Common DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.