Neural nets, one of the oldest architectures for AI programming, are loosely based on biological neurons and their properties. Recent work on language applications has made the AI code closer to biological reality in several ways. This commentary examines this convergence and, in light of what is known of neocortical structure, addresses the question of whether ``general AI'' looks attainable with these tools.

For many years, I have tried to come to a deeper understanding of what consciousness really means. This paper is a series of meditations about some aspects of this puzzle. First, I look at neuroscientists quest to localize consciousness in the brain and find wildly different conclusions. Second, believing that emotions are an essential component of consciousness, I explore some ideas of those who seek a theory of emotions. Third, I look at the connection of consciousness and time, especially the NOW. Fourthly, I try to lay out the pros and cons of which, if any, animals have consciousness. And finally, I discuss whether consciousness can find a home in silicon, in an “AI.”

Here I describe the history of my interactions with Ulf, a close friend and inspiring example, and sketch some of the basics of his "Pattern Theory".

Multiple studies have addressed the question of whether ancient eclipse predictions correctly predicted known past eclipses. My approach instead is to generate thousands of synthetic eclipses, starting from random choices for the longitude of the conjunction, node and perigee, then comparing the magnitude of resulting eclipses with ancient and modern predictions. Writing code for ancient algorithms requires a detailed analysis of the text, not found in most commentaries. Working with Qu Anjing, Jayant Shah and Mark Schiefsky, we hope to do this with the Almagest, multiple Indian siddhantas and Chinese calendars. This paper will describe the results for the Yuan dynasty Shoushili. These raise the question of how the writers of this calendar understood its remarkably accurate algorithms for lunar parallax and whether and when secret geometric models were known to Chinese astronomers.

At VillecunStarting in 1973, Alexander Grothendieck lived in Villecun, a hamlet near Lodève (about 60 km from Montpellier), in an old and shabby house.The house lacked comfort, but, as he said, it had a soul.I used to go there to do mathematics or simply to visit.Evenings were lit with an old oil lamp, and there was goat's milk and locally produced organic food, which Alexander used to eat with chopsticks, a habit he acquired in Vietnam.He was mainly vegetarian.His house was open to everyone: students, ecologist pals, and sometimes a Buddhist monk with his prayerdrum.

Mathematician who rebuilt algebraic geometry.

Klaus was named one of Springer's directors, and he hired Alice Merker, who had earned degrees from Rochester and Chicago, to be a mathematics

Several generations of students of algebraic geometry have learned the subject from David Mumford's fabled Red Book containing notes of his lectures at Harvard University. Their genesis and evolution are described in the preface as: Initially notes to the course were mimeographed and bound and sold by the Harvard math department with a red cover. These old notes were picked up by Springer and are now sold as the Red book of Varieties and Schemes. However, every time I taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes...This book contains what Mumford had then intended to be Volume II. It covers the material in the Red Book in more depth with several more topics added. The notes have been brought to the present form in collaboration with T. Oda.

I want to talk about how Grothendieck’s revolution profoundly affected my own understanding of algebraic geometry. But to do that, I need to reconstruct for the reader the mathematical environment in which I grew up. When I started studying algebraic geometry around 1956, the Italian school was no longer active. André Weil and Oscar Zariski were carrying the ball and had together pioneered the extension of algebraic geometry to characteristic p. Weil was motivated by the idea of creating a merger of algebraic geometry with number theory to solve, e.g. Mordell’s conjecture. Zariski was motivated by the need to make the work of the Italian school rigorous by using the new methods of commutative algebra. Everyone realized that the field needed better foundations to handle these new ideas, in which the explicit geometry of complex varieties was replaced by an abstract geometry based on algebra and number theory. Both Weil and Zariski cobbled together some tentative definitions to make discussions and papers possible, both written as books for the AMS colloquium series, though only Weil’s was published. But their “Foundations” did not have the feeling of inevitability that one associated, for example, to Bourbaki and his treatise and were never widely used.

AbstractWe study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

IT wAs a stimulating experience to know and collaborate with C. P. Ramanujam. He loved mathematics and he was always ready to take up a new thread or to pursue an old one with infectious enthusiasm. He was equally ready to discuss a problem with a first year student or a colleague, to work through an elementary point or to puzzle over a deep problem. On the other hand, he had very high standards. He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem on any given topic. He wanted mathematics to be beautiful and to be clear and simple. He was sometimes tormented by the difficulty of these high standards, but, in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuinely original stamp.

We consider the groups Diff _ℬ (ℝ ^n) , Diff _H^∞(ℝ ^n) , and Diff _𝒮 (ℝ ^n) of smooth diffeomorphisms on ℝ ^n which differ from the identity by a function which is in either ℬ (bounded in all derivatives), H^∞ = ⋂ _k≥ 0H^k , or 𝒮 (rapidly decreasing). We show that all these groups are smooth regular Lie groups.

Given a finite dimensional manifold N , the group Diff S (N ) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M, N ) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dim M < dim N .For a right invariant weak Riemannian metric on Diff S (N ) induced by a quite general operator L : X S (N ) → Γ(T * N ⊗ vol(N )), we consider the induced weak Riemannian metric on B(M, N ) and we compute its geodesics and sectional curvature.For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O'Neill's formula very transparent, and we use it finally to compute sectional curvature on B(M, N ).

We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,\eta \ge 0$. When $\varepsilon=\eta=0$ we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$ or, if $\varepsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ \|v\|_{\varepsilon,\eta} = \int_{\mathbb R^n} dx $$ where $L_{\varepsilon,\eta} = (I-\tfrac{\eta^2}{p} \triangle)^p \circ (I-\tfrac1{\varepsilon^2} \nabla \circ \div)$. All geodesic equations are locally well-posed, and the $L_{\varepsilon,\eta}$-equation admits solutions for all time if $\eta>0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to "vortex-solitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids.