Schedule

The theory of slice regular functions of one quaternions variable, launched by Gentili and Struppa in 2006, is a successful analog of the theory of holomorphic functions of one complex variable. It quickly developed into a full-fledged theory, with a focus on quaternionic domains that are symmetric with respect to the real axis. This focus was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric slice domains. Recent work in the theory concerns slice regularity over domains that are not axially symmetric. The talk will outline the motivation for these recent developments and the basic tools for the study of regularity over slice domains that are not symmetric, such as the Local Representation Formula. Then it will describe the theory in the non-symmetric setting, with its new interesting phenomena.

Algebras of Hermitian $4 \times 4$ matrices over octonions appear in M-theory. Motivated by this, we consider algebras of (skew-)Hermitian $n \times n$ matrices over octonions for $n>3$. Unlike the classical case $n=3$, they are no longer Jordan algebras. We will discuss structural properties of such algebras, and pose some open questions about them.

Octonion algebra as a fundament gives a way for construct two simple Malcev algebras. Namely, 7-dimensional simple non-Lie algebra M7 and ternary 8-dimensional simple Malcev algebra M8. We will talk about derivations, local derivations, and generalized derivations of these algebras.

The purpose of this talk is to present the main results on the zeros of quaternionic and coquaternionic polynomials, emphasizing the common features and the most important differences between the zero-sets of these two types of polynomials. We aim to give a historical perspective, showing how, in the case of quaternions, some of the most important results are already present in the pioneering works of Niven [3], Gordon and Motzkin [2] and Beck [1] and are lately rediscovered by other authors. We also give a brief overview of different methods available to compute the zeros of these polynomials.
This is joint work with: M. I. Falcão, F. Miranda and R. Severino
[1] B. Beck. Sur les equations polynomiales dans les quaternions. Enseign. Math., 25(3-4):193{201, 1979.
[2] B. Gordon and T. Motzkin. On the zeros of polynomials over division rings. Trans. Amer. Math. Soc., 116:218{226, 1965.
[3] I. Niven. Equations in quaternions. Amer. Math. Monthly, 48:654{661, 1941.

The fractional Hilbert operators $\mathcal{H}^\alpha$ and $H^\alpha$ are interrelated by $$H^\alpha=e^{-i\frac{\pi\alpha}{2}}\mathcal{H}^\alpha$$ are associated to the Fourier symbols $h_\alpha(\xi)$ and $e^{i\frac{\pi\alpha}{2}} h_\alpha(\xi)$, with $$h_\alpha(\xi)=e^{-i\frac{\pi\alpha}{2}}\left( \cos\left(\alpha\frac{\pi}{2}\right)+\frac{\xi}{|\xi|}\sin\left(\alpha\frac{\pi}{2}\right)\right).$$We want to consider the Factional Dirac operator $ D^{\alpha, \theta} = |D|^{\alpha}H^{\theta} $ with Fourier symbol $|\xi|^{\alpha}\exp(-i\tfrac{\pi}{2}\theta)\left(\cos(\tfrac{\pi}{2}\theta) + \frac{\xi}{|\xi|}\sin(\tfrac{\pi}{2}\theta)\right) = |\xi|^{\alpha}\exp(-i\tfrac{\pi}{2}\theta)\exp(\tfrac{\pi}{2}\theta \frac{\xi}{|\xi|})$.
The Cauchy problem $$ \left\{ \begin{array}{ccl} i\frac{\partial u}{\partial t} (x,t) & = & D^{\alpha, \theta}_x u(x,t) \\ u(x,0) & = & f(x), \quad x\in\mathbb{R}^d, t\geq 0. \end{array} \right. $$ has the formal solution $$ u(x,t) = \int_{\mathbb{R}^d} e^{it |\xi|^{\alpha} h_{\theta}(\xi) } \hat{f}(\xi) e^{i\xi \cdot x } d\xi, $$ where $$ h_{\theta}(\xi) = e^{-i\tfrac{\pi}{2}\theta} \exp\left(\tfrac{\pi}{2}\theta \frac{\xi}{|\xi|}\right) . $$ Therefore we consider the Dirac operator in Geldfand-Shilov spaces to obtain appropriate mapping properties of the fractional Dirac operator.

Paley-Wiener type theorems roughly says that a Fourier type transform provides a bijection between square integrable functions with compact support contained in $\overline{B(0,R)}$, and analytic type functions of exponential growth not exceeding $R>0$. An hypercomplex version of this result obtained by Kou and Qian (2002) [Journal of Functional Analysis, 189(1), 227-241], as well as the higher dimensional version formely obtained by Helgason (1973) [Annals of Mathematics, 98(3), 451-479], makes use of series expansions involving ultraspherical polynomials. Guided by the role that analytic type functions are linked with solutions of Cauchy problems, we exploit a new characterization involving the semigroup encoded by the null solutions of the space-fractional Dirac operator $\partial_{x_0}+D^\alpha$ on $\mathbb{R} \times \mathbb{R}^n$. (This is a joint work with S. Bernstein).

During the last decades, quaternion Fourier transforms (QFT) have been deeply investigated and found applications to color image processing, nuclear magnetic resonance imaging, speech recognition, among others. The two most well-known forms are the two-sided QFT and the right-sided QFT. Several properties, uncertainty principles, time-frequency distributions were studied for these QFT by several authors. In this talk, we present the hyperbolic counterpart of the QFT. We show their main properties, inversion formula, and Parseval's theorem. Concerning the uncertainty principles, we show a sharp Pitt's inequality for the two-sided QFT that allows deriving a logarithmic uncertainty principle, and Weyl's-Heisenberg uncertainty principle in our context. Donoho–Stark’s uncertainty principle and Benedick's qualitative uncertainty principle are also given together with a hyperbolic Poisson summation formula. These results depend heavily on the properties of gyrogroups that we will introduce and explain. In the limiting case, we can recover all the results of the QFT in the Euclidean case.

We discuss recent work on the design of universal single-qubit gate sets for quantum computing. Using quaternionically-derived "super golden gates," we connect the problem of efficient approximate synthesis of given gates to arbitrary precision in quantum hardware design to "icosahedral gates" constructed using the symmetries of the icosahedron, which enjoy a form of optimality. This is joint work with Zachary Stier.

We discuss recent algorithmic work in the design of universal single-qubit gate sets for quantum computing. Using the quaternionically-derived "super golden gates," we connect the problem of efficient approximate synthesis of given gates to arbitrary precision in quantum hardware design to "icosahedral gates" constructed using the symmetries of the icosahedron, which enjoy a form of optimality. Joint work with Terrence Blackman.