https://peertube.wtf/videos/watch/cc024c3a-2917-4886-9c4c-28ba60187bd6 Abstract: In order to study a certain 3-dimensional analogue of the Mandelbrot set, I needed a notion of holomorphy for the corresponding 3-dimensional real algebra. While this motivation (to generalize the basic elements of complex dynamics) is analytic, the focus of this talk will be the connection between holomorphic functions over a real algebra and first-order properties of the algebra. It turns out that, under the assumption of a single (left or right) divisible element, the holomorphic functions over an arbitrary real algebra form a vector space. Properties such as the Product Rule and Chain Rule are equivalent to purely algebraic properties of the underlying real algebra. When our real algebra has a multiplicative identity element, the Product Rule is precisely equivalent to being both commutative and associative. I will present proofs of some of these statements, which are quite succinct once the proper groundwork has been done. PALS: https://math.colorado.edu/algebralogic/index.html Slides: https://aten.cool/documents/slides/aten_PALS_2026.pdf Paper: https://aten.cool/documents/papers/A%20rock-paper-scissors%20Mandelbrot%20set%20[2026].pdf Wed, 14 Oct 2026 22:53:02 GMT https://validator.w3.org/feed/docs/rss2.html PeerTube - https://peertube.wtf https://peertube.wtf/lazy-static/avatars/45856119-df14-4b5f-bff8-4807ae2cfbd2.png https://peertube.wtf/videos/watch/cc024c3a-2917-4886-9c4c-28ba60187bd6 All rights reserved, unless otherwise specified in the terms specified at https://peertube.wtf/about and potential licenses granted by each content's rightholder.