Abstract: In this talk we propose an alternative generalization of the quaternionic Fueter Theorem that associates to any holomorphic function a single axially Fueter-regular function. Given any slice-regular function, we prove a decomposition with components in the kernel of the third order operator ∂¯∆, where ∂¯ is the Cauchy-Riemann operator and ∆ is the corresponding Laplacian on an hypercomplex subspace. This decomposition defines a real-linear map that associates to any slice-regular function a (m-1)/2-tuple of axially monogenic functions ∂¯∆-Fueter mapping. Here m+1 is the dimension of the space. We show that, differently from higher powers of the ∆ operator, the ∂¯∆-Fueter mapping has always a small kernel.

Abstract: In this talk we consider homomorphisms acting on Clifford algebra valued monogenic functions. To study and to characterize them we introduce the notion of the proximate order for the set of entire monogenic functions.

As main application we characterize infinite order differential operators that act continuously on spaces of monogenic entire functions. These in turn serve as crucial tools to solve the Cauchy problem for the Schrödinger equation with superoscillatory initial data. This is joint work with F. Colombo, S. Pinton and I. Sabadini. 

Abstract: Let A be one of the following algebras:  R2 ≅ Η (the real algebra of quaternions) or  the Clifford algebra  R3 ≅ Η ⊕ H. For the algebra A, the automorphism group Aut(A) and its invariants are well known.

In this talk we will describe the invariants of the automorphism group of the algebra of slice regular functions over A. Time permitting I will present also the octonionic case.

Abstract: In this talk we explore some algebraic, combinatoric and analytic properties of the class of numerical sequences determined by hybrid numbers whose components are taken from special integer sequences, such as Fibonacci, Lucas, Pell and Jacobsthal. In addition, we extend the study from entire sequences to complex sequences through the Difference Equation and associated generalized Fibonacci fundamental system.  The combinatorial and analytical properties of these identities are established, and applications to the generalized Fibonacci and Pell numbers are studied. Furthermore, some special cases and examples are provided. 

Abstract:  I will describe some extensions of the classical Segal-Bargman coherent state transform to the realm of Clifford analysis. For context, I will also describe the use of generalized coherent state transforms in geometric quantization.

This is work in collaboration with Pei Dang, Will Kirwin, José Mourão, and Tao Qian.

Abstract: I consider the definition of module over non-commutative associative D-algebra A. Free  A-module is A-module that has a basis. I considered the structure of homomorphism of free A-module. If D-algebra A is free D-algebra, then free A-module V also is free D-module. I considered relation between coordinates of homomorphism of A-module V and homomorphism of D-module V.

If D-algebra A is non-associative, then in general we cannot introduce the structure of A-module. However, this rule has exceptions. If D-algebra A is Lie algebra, then there exists the structure of A-module. If D-algebra A is non-associative, then we cannot introduce coordinates of homomorphism of A-module as a matrix of A-numbers. I consider a possible solution to this problem.

Abstract:  We recently gave a new description of the exceptional Lie algebra e8 in terms of generalized 3x3 matrices over the tensor product of two copies of the octonions.  When decomposing e8 over its subalgebras e6 and e7, we obtain not only a description of their adjoint representations, but also of their minimal representations.  Representation theory then guarantees that we can realize the action of e6 and e7 on their minimal representations using (generalized) matrix commutators in e8.  We show here how to construct these actions explicitly and verify that they agree with the standard form of these actions given by Freudenthal in terms of Albert algebras.  Along the way, we show how to implement standard operations on the Albert algebra, such as the determinant, using commutators in e8.  Our formalism naturally extends from the Lie algebras to the corresponding Lie groups.

(Joint work with Corinne Manogue)

Abstract: We recently showed how to interpret the elements of the exceptional Lie algebra e8(-24) as objects in the Standard Model, including lepton and quark spinors with the usual properties, the S1tandard Model Lie algebra  su(3) ⊕ su(2) ⊕ u(1), and the Lorentz Lie algebra so(3,1).  Here, we emphasize the physical interpretation of this description of e8, including new results about color symmetry.

(Joint work with Tevian Dray and Robert Wilson)