Program 2019

Abstract: This talk will discuss how the Dirac equation arises from Clifford algebraic considerations and how examining the action of the Dirac operator on a plane wave gives rise to an algebraic reformulation of the Dirac equation (equivalent to the original) that has solutions in terms of nilpotent elements of the Clifford algebra. These nilpotent elements can be regarded as annihi- lation operators for a fermion. We point out how the nilpotents decompose into Majorana operators and discuss how this point of view is related to the braiding of Majorana Fermions and to the original work of Majorana on the Dirac equation. We also discuss how this work is related to the work of Peter Rowlands and how it is related to the author’s program for elucidating discrete physics in terms of non-commutative algebra.

Abstract: A new approach to the famous Newman’s Diamond Lemma is developed. This approach promises an essential progress in the theory of prime decompositions of geometric objects.

Abstract: In this presentation we investigate the infinite family of circulant graphs C_n(s₁,s₂, … ,s_k). We present an explicit formula for the number of spanning trees, rooted spanning forests and the Kirchhoff index. Then we investigate arithmetical and asymptotic properties of the obtained numbers. All formulas are given in terms of the Chebyshev polynomials. We start with some basic definitions.

Abstract: Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. We investigate residual finiteness of quandles and present some recent observations about residual finiteness of free quandles and linkquandles.

Abstract: We consider the generalised Sieradski group S(2n,7,2), n≥1. We prove that n-cyclic presentations of their groups are geometric, i.e., correspond to the spines of closed connected orientable 3-manifolds. These manifolds M(2n,7,2) are the n-fold cyclic coverings of the lens space L(7,1).

Abstract: A compact hyperbolic tetrahedron T is a convex hull of four points in the hyperbolic space. It is well known that T is uniquely defined up to isometry either by the set of its dihedral angles or the set of its edge lengths. A Gram matrix G(T) of tetrahedron T consists of cosines of its dihedral angles, while an edge matrix E(T) is formed by hyperbolic cosines of its edge lengths. In 1907 Italian mathematician G. Sforza found a formula for the volume of a compact hyperbolic tetrahedron T in terms of its Gram matrix cofactors. The new proof of the Sforza’s formula was recently given by A. Mednykh and N. Abrosimov (2014). At the same time, the Sforza’s formula has some drawback associated with the need to choose the analytical branch of a multivalued function under the integral. In the present work, we obtain an exact formula for the volume of T in terms of its edge matrix cofactors. Our formula is devoid of the above disadvantage since under the integral there is an unambiguous function. Also, this is a first known formula for the volume of a compact hyperbolic tetrahedron of general type in terms of its edge lengths.