35 research outputs found
On Skew Braces (with an appendix by N. Byott and L. Vendramin)
Braces are generalizations of radical rings, introduced by Rump to study
involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation
(YBE). Skew braces were also recently introduced as a tool to study not
necessarily involutive solutions. Roughly speaking, skew braces provide
group-theoretical and ring-theoretical methods to understand solutions of the
YBE. It turns out that skew braces appear in many different contexts, such as
near-rings, matched pairs of groups, triply factorized groups, bijective
1-cocycles and Hopf-Galois extensions. These connections and some of their
consequences are explored in this paper. We produce several new families of
solutions related in many different ways with rings, near-rings and groups. We
also study the solutions of the YBE that skew braces naturally produce. We
prove, for example, that the order of the canonical solution associated with a
finite skew brace is even: it is two times the exponent of the additive group
modulo its center.Comment: 37 pages. Final versio
International Journal of Mathematical Combinatorics, Vol.2
The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences
On Multiplication Groups of Quasigroups
Quasigroups are algebraic structures in which divisibility is always defined. In this thesis we investigate quasigroups using a group-theoretic approach. We first construct a family of quasigroups which behave in a group-like fashion. We then focus on the multiplication groups of quasigroups, which have first appeared in the work of A. A. Albert. These permutation groups allow us to study quasigroups using group theory. We also explore how certain natural operations on quasigroups affect the associated multiplication groups. Along the way we take the time and special care to pose specific questions that may lead to further work in the near future
Geometric classification of 4d N= 2 SCFTs
The classification of 4d N= 2 SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected \u211a-factorial log-Fano variety with Hodge numbers h p,q = \u3b4 p,q . With some plausible restrictions, this means that the Coulomb branch chiral ring [InlineMediaObject not available: see fulltext.] is a graded polynomial ring generated by global holomorphic functions u i of dimension \u394 i . The coarse-grained classification of the CSG consists in listing the (finitely many) dimension k-tuples \u394 1 , \u394 2 , ef , \u394 k which are realized as Coulomb branch dimensions of some rank-k CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible \u394 1 , ef , \u394 k \u2019s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number N(k) of dimensions allowed in rank k is given by a certain sum of the Erd\uf6s-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large kN(k)=2\u3b6(2)\u3b6(3)\u3b6(6)k2+o(k2). In the special case k = 2 our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension k-tuples \u394 1 , ef , \u394 k are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. Since the arguments tend to be abstract, we illustrate the various aspects with several concrete examples and perform a number of explicit checks. We include detailed tables of dimensions for the first few k\u2019s
Unsolved Problems in Group Theory. The Kourovka Notebook
This is a collection of open problems in group theory proposed by hundreds of
mathematicians from all over the world. It has been published every 2-4 years
in Novosibirsk since 1965. This is the 19th edition, which contains 111 new
problems and a number of comments on about 1000 problems from the previous
editions.Comment: A few new solutions and references have been added or update
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Mathematical Methods for 4d N=2 QFTs
In this work we study different aspects of 4d N = 2 superconformal field theories. Not only we
accurately define what we mean by a 4d N = 2 superconformal field theory, but we also invent and
apply new mathematical methods to classify these theories and to study their physical content.
Therefore, although the origin of the subject is physical, our methods and approach are rigorous
mathematical theorems: the physical picture is useful to guide the intuition, but the full mathematical rigor is needed to get deep and precise results. No familiarity with the physical concept
of Supersymmetry (SUSY) is need to understand the content of this thesis: everything will be
explained in due time. The reader shall keep in mind that the driving force of this whole work
are the consequences of SUSY at a mathematical level. Indeed, as it will be detailed in part II, a
mathematician can understand a 4d N = 2 superconformal field theory as a complexified algebraic
integrable system. The geometric properties are very constrained: we deal with special K\ua8ahler
geometries with a few other additional structures (see part II for details). Thanks to the rigidity
of these structures, we can compute explicitly many interesing quantities: in the end, we are able
to give a coarse classification of the space of "action" variables of the integrable system, as well as
a fine classification -- only in the case of rank k = 1 -- of the spaces of "angle" variables.
We were able to classify conical special K\ua8ahler geometries via a number of deep facts of algebraic
number theory, diophantine geometry and class field theory: the perfect overlap between mathematical theorems and physical intuition was astonishing. And we believe we have only scratched
the surface of a much deeper theory: we can probably hope to get much more information than
what we already discovered; of course, a deeper study of the subject -- as well as its generalizations
-- is required.
A 4d N = 2 superconformal field theory can thus be defined by its geometric structure: its scaling
dimensions, its singular fibers, the monodromy around them and so on. But giving a proper and
detailed definition is only the beginning: one may be interested in exploring its physical content. In
particular, we are interested in supersymmetric quantities such as BPS states, framed BPS states
and UV line operators. These quantities, thanks to SUSY, can be computed independently of
many parameters of the theory: this peculiarity makes it possible to use the language of category
theory to analyze the aforementioned aspects. As it will be proven in part V, to each 4d N = 2
superconformal field theory we can associate a web of categories, all connected by functors, that
describe the BPS states, the framed BPS states (IR) and the UV line operators. Hence, following
the old ideas of \u2018t Hooft, it is possible to describe the phase space of gauge theories via categories,
since the vacuum expectation values of such line operators are the order parameters of the confinement/deconfinement phase transitions. Mathematically, the (quantum) cluster algebra of Fomin
and Zelevinski is the structure needed. Moreover, the analysis of BPS objects led us to a deep
understanding of generalized S-dualities. Not only were we able to precisely define -- abstractly and
generally -- what the S-duality group of a 4d N = 2 superconformal field theory should be, but we
were also able to write a computer algorithm to obtain these groups in many examples (with very
high accuracy)