On the number and location of short geodesics in moduli space

A closed Teichmuller geodesic in the moduli space M_g of Riemann surfaces of genus g is called L-short if it has length at most L/g. We show that, for any L > 0, there exist e_2 > e_1 > 0, independent of g, so that the L-short geodesics in M_g all lie in the intersection of the e_1-thick part and the e_2-thin part. We also estimate the number of L-short geodesics in M_g, bounding this from above and below by polynomials in g whose degrees depend on L and tend to infinity as L does.Comment: 23 pages, 1 figur

Should machines be tools or tool-users? Clarifying motivations and assumptions in the quest for superintelligence

Much of the basic non-technical vocabulary of artificial intelligence is surprisingly ambiguous. Some key terms with unclear meanings include intelligence, embodiment, simulation, mind, consciousness, perception, value, goal, agent, knowledge, belief, optimality, friendliness, containment, machine and thinking. Much of this vocabulary is naively borrowed from the realm of conscious human experience to apply to a theoretical notion of “mind-in-general” based on computation. However, if there is indeed a threshold between mechanical tool and autonomous agent (and a tipping point for singularity), projecting human conscious-level notions into the operations of computers creates confusion and makes it harder to identify the nature and location of that threshold. There is confusion, in particular, about how—and even whether—various capabilities deemed intelligent relate to human consciousness. This suggests that insufficient thought has been given to very fundamental concepts—a dangerous state of affairs, given the intrinsic power of the technology. It also suggests that research in the area of artificial general intelligence may unwittingly be (mis)guided by unconscious motivations and assumptions. While it might be inconsequential if philosophers get it wrong (or fail to agree on what is right), it could be devastating if AI developers, corporations, and governments follow suit. It therefore seems worthwhile to try to clarify some fundamental notions

Art and the Unknown

Abstract: The purpose of this essay is to explore the nature and role of art as a human phenomenon from a broadly cognitive perspective. Like science and religion, art serves to mediate the unknown, at once to embrace and to defend against the fundamental mystery of existence. Thus, it may challenge the status quo while generally serving to maintain it. Art tracks the individuation of subjectivity, serving the pleasure principle, yet is appropriated by the collective’s commitment to the reality principle. While science and religion close in on serious answers to fundamental questions, art opens up possibilities toward playfulness, uselessness, imagination, and arbitrary whim. Though art has no unifying definition, meaning, or intent through time and across cultures, it remains important to people, both to do and to enjoy. It serves to counterbalance the naive realism of science, the “rationality” of modern society, and the literalism of text-based religion. While its allegiance is divided, its most worthy intent is to aid us to confront and negotiate the great mystery revealed to us in consciousness

Abstract commensurators of braid groups

Let B_n be the braid group on n strands, with n at least 4, and let Mod(S) be the extended mapping class group of the sphere with n+1 punctures. We show that the abstract commensurator of B_n is isomorphic to a semidirect product of Mod(S) with a group we refer to as the transvection subgroup, Tv(B_n). We also show that Tv(B_n) is itself isomorphic to a semidirect product of an infinite dimensional rational vector space with the multiplicative group of nonzero rational numbers.Comment: 10 page