Abstract:
Groups $\Pi_k(X;\sigma)$ of "flagged homotopies" are introduced of which the usual (abelian for $k>1$) homotopy groups $\pi_k(X;p)$ is the limit case for flags $\sigma$ contracted to a point $p$. Calculus of exterior forms with values in algebra $A$ is developped of which the limit cases are differential forms calculus (for $A=\bb R$) and gauge theory (for 1-forms). Moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in $A$) is introduced with elements giving representations of $\Pi_k$ in $G=exp A$.

Abstract:
This paper continues our investigation into the question of when a homotopy $\omega = \{\omega_t\}_{t \in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, \omega_0)) \cong K_*(C^*(\mathcal{G}, \omega_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_\Lambda$ is the infinite path groupoid associated to a row-finite higher-rank graph $\Lambda$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $\Lambda$, then $K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_0})) \cong K_*(C^*(\mathcal{G}_\Lambda, \sigma_{c_1})),$ where $\sigma_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_\Lambda$ associated to the 2-cocycle $c_t$ on $\Lambda$. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semi-continuous $C^*$-bundle.

Abstract:
We study the homological algebra of an R = Q/I module M using A-infinity structures on Q-projective resolutions of R and M. We use these higher homotopies to construct an R-projective bar resolution of M, Q-projective resolutions for all R-syzygies of M, and describe the differentials in the Avramov spectral sequence for M. These techniques apply particularly well to Golod modules over local rings. We characterize R-modules that are Golod over Q as those with minimal A-infinity structures. This gives a construction of the minimal resolution of every module over a Golod ring, and it also follows that if the inequality traditionally used to define Golod modules is an equality in the first dim Q+1 degrees, then the module is Golod, where no bound was previously known. We also relate A-infinity structures on resolutions to Avramov's obstructions to the existence of a dg-module structure. Along the way we give new, shorter, proofs of several classical results about Golod modules.

Abstract:
For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.

Abstract:
An anecdotal account of the author's role in the origins of lattice gauge theory, prepared for delivery on the thirtieth anniversary of the publication of "Confinement of Quarks" [Phys. Rev. D10 (1974) 2445].

Abstract:
Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra-the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra-and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the "space of leaves" and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.

Abstract:
On the basis of an arbitrarily non-Euclidian geometrical "thought experiment" involving the cross-species transmission of simian foamy virus (sfv) from a non-primate species Xy to Homo sapiens (Hs), initially excluding all social factors, the following was derived. At the port of exit from Xy (where the species barrier, SB, is defined by the Index of Origin, IO), sfv shedding is (1) enhanced by two transmitting tensors (Tt), (i) virus-specific immunity (VSI) and (ii) evolutionary defenses such as APOBEC, RNA interference pathways, and (when present) expedited therapeutics (denoted e2D); and (2) opposed by the five accepting scalars (At): (a) genomic integration hot spots, gIHS, (b) nuclear envelope transit (NMt) vectors, (c) virus-specific cellular biochemistry, VSCB, (d) virus-specific cellular receptor repertoire, VSCR, and (e) pH-mediated cell membrane transit, (↓pH CMat). Assuming As and Tt to be independent variables, IO = Tt/As. The same forces acting in an opposing manner determine SB at the port of sfv entry (defined here by the Index of Entry, IE = As/Tt). Overall, If sfv encounters no unforeseen effects on transit between Xy and Hs, then the square root of the combined index of sfv transmissibility (√|RTI|) is proportional to the product IO* IE (or ~Vm* Ha* ∑Tt*∑As*Ω), where Ω is the retrovirological constant and ∑ is a function of the ratio Tt/As or As/Tt for sfv transmission from Xy to Hs.I present a mathematical formalism encapsulating the general theory of the origins of retroviruses. It summarizes the choreography for the intertwined interplay of factors influencing the probability of retroviral cross-species transmission: Vm, Ha, Tt, As, and Ω.The order Retroviridae constitutes a collection of non-icosahedral, enveloped viruses with two copies of a single-stranded RNA genome [1-5]. Retroviruses are known to infect avians [1] and murine [2], non-primate [3] and primate [4,5] mammals. Viruses of the order Retroviridae are unique in the sense that they

Abstract:
The string theory introduced in early 1971 by Ramond, Neveu, and myself has two-dimensional world-sheet supersymmetry. This theory, developed at about the same time that Golfand and Likhtman constructed the four-dimensional super-Poincar\'e algebra, motivated Wess and Zumino to construct supersymmetric field theories in four dimensions. Gliozzi, Scherk, and Olive conjectured the spacetime supersymmetry of the string theory in 1976, a fact that was proved five years later by Green and myself.

Abstract:
Intelligence research appears to have overwhelmingly endorsed a superordinate (higher-order model) conceptualization of g, in comparison to the relatively less well-known breadth conceptualization of g, as represented by the direct hierarchical model. In this paper, several similarities and distinctions between the indirect and direct hierarchical models are delineated. Based on the re-analysis of five correlation matrices, it was demonstrated via CFA that the conventional conception of g as a higher-order superordinate factor was likely not as plausible as a first-order breadth factor. The results are discussed in light of theoretical advantages of conceptualizing g as a first-order factor. Further, because the associations between group-factors and g are constrained to zero within a direct hierarchical model, previous observations of isomorphic associations between a lower-order group factor and g are questioned.

Abstract:
This paper continues the author's program to investigate the question of when a homotopy of 2-cocycles $\Omega = \{\omega_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, \omega_0)) \cong K_*(C^*(\mathcal{G}, \omega_1)).$ Building on our earlier work, we show that if $\pi: \mathcal{G} \to M$ is a locally trivial bundle of amenable groups over a locally compact Hausdorff space $M$, a homotopy $\Omega = \{\omega_t\}_{t \in [0,1]}$ of 2-cocycles on $\mathcal{G} $ gives rise to an isomorphism $K_*(C^*(\mathcal{G}, \omega_0)) \cong K_*(C^*(\mathcal{G}, \omega_1)).$