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We introduce the abelian class group C_{ab}(G) of a reductive group scheme G over a ring A of arithmetical interest and study some of its properties. In particular, we show that if the fraction field of A is a global field without real…

We exploit a new theory of duality transformations to construct dual representations of models incompatible with traditional duality transformations. Hence we obtain a solution to the long-standing problem of non-Abelian dualities that…

These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those…

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and…

This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…

This extended preface [to the Book `Bayesian Nonparametrics', Cambridge University Press, 2010, by NL Hjort, CC Holmes, P Mueller, SG Walker] is meant to explain why you are right to be curious about Bayesian nonparametrics -- why you may…

Algebraic geometry for groups and Lie algebraic has been recently defined and studied by many authors on the purpose to study set defined by algebraic equations on abstract groups and Lie algebras. The purpose of this paper is to present a…

In the paper "An Abelian Loop for Non-Composites" (arXiv:110.14716), we introduced a group-like structure consisting of odd prime numbers and 1, with properties that allowed us to prove analogous results to well known theorems in Number…

I discuss particular solutions of the integrable systems, starting from well-known dispersionless KdV and Toda hierarchies, which define in most straightforward way the generating functions for the Gromov-Witten classes in terms of the…

There are several examples in the literature of finite non-abelian $p$-groups whose automorphism group is abelian. For some time only examples that were special $p$-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav…

We construct non-Abelian N=2 on-shell vector multiplets in five and in four dimensions. Closing of the supersymmetry algebra imposes dynamical constraints on the fields, and these constraints should be interpreted as equations of motion. If…

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

We study the non-abelian tensor square $G\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\otimes G$ is isomorphic to the direct product of…

A new non-Abelian gauge transformation for two-forms is introduced. Construction is based on a fixed map from the spacetime to the loop space which attachs a closed loop to each point of the spacetime. It is argued that this set-up is…

We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups.

We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…

Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved…

In this note, we investigate how different fundamental groups of presentations of a fixed algebra $A$ can be. For finitely many finitely presented groups $G_i$, we construct an algebra $A$ such that all $G_i$ appear as fundamental groups of…

We prove that the torsion subgroup of the abelian fundamental group is finite for a regular geometrically integral projective variety over a local field. We also study the structure of $SK_1(X)$ for a regular projective variety $X$ over a…

This is an introduction to advanced linear algebra, with emphasis on geometric aspects, and with some applications included too. We first review basic linear algebra, notably with the spectral theorem in its general form, and with the…