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The existence of indistinguishable quantum particles provides an explanation for various physical phenomena we observe in nature. We lay out a path for the study of indistinguishable particles in general probabilistic theories (GPTs) via…
I try to assess the weak and strong points of the standard model of electro-magnetic, weak and strong forces, how it can be derived from general relativity by generalizing Riemannian to noncommutative geometry and what post- and predictions…
Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
The electromagnetic theory is considered in the framework of the generally covariant approach, that is applied to the analysis of electromagnetism in noninertial coordinate and frame systems. The special-relat\-ivistic formulation of…
Geometric modeling by constraints, whose applications are of interest to communities from various fields such as mechanical engineering, computer aided design, symbolic computation or molecular chemistry, is now integrated into standard…
We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these…
A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that admit some non-trivial symmetry. When $d=2$ there is a large literature on this…
It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…
This is mainly a brief review of some key achievements in a `hot'' area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying {\em integrable} quantum field theory models with {\em…
There exist two versions of the Kadomtsev-Petviashvili equation, related to the Cartesian and cylindrical geometries of the waves. In this paper we derive and study a new version, related to the elliptic cylindrical geometry. The derivation…
Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum…
Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this…
Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians…
We propose the model of massive spinning particle traveling in four-dimensional Minkowski space. The equations of motion of the particle follow from the fact that all the classical paths of the particle lie on a cylinder whose position in…
A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices…
In this note we review some of the uses of framed quivers to study BPS invariants of Donaldson-Thomas type. We will mostly focus on non-compact Calabi-Yau threefolds. In certain cases the study of these invariants can be approached as a…
This paper gives elements in the (skew) center of the generalized quantum group corresponding to its irreducible finite dimensional modules. Finally we give a conjecture stating that those must form a basis of the center.
In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Kra jewski diagrams [5].…