The backward {\lambda}-Lemma and Morse filtrations

Consider the infinite dimensional hyperbolic dynamical system provided by the (forward) heat semi-flow on the loop space of a closed Riemannian manifold M. We use the recently discovered backward {\lambda}-Lemma and elements of Conley theory to construct a Morse filtration of the loop space whose cellular filtration complex represents the Morse complex associated to the downward L2-gradient of the classical action functional. This paper is a survey. Details and proofs will be given in [6].Comment: Conference proceedings, 9 pages, 5 figures. v2: typos corrected, minor modification

On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds

AbstractThe first section of this paper will characterize those cobordism classes in the Thom cobordism ring N∗ and Ω∗ which contain complex manifolds†. The second section attempts to characterize those classes in N∗ which contain spin manifolds†. The attempt succeeds only through dimension 23

Quantum Mechanics as a Gauge Theory of Metaplectic Spinor Fields

A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role.Comment: 61 pages, late

Non-factorial nodal complete intersection threefolds

We give a bound on the minimal number of singularities of a nodal projective complete intersection threefold which contains a smooth complete intersection surface that is not a Cartier divisor

Adelic Integrable Systems

Incorporating the zonal spherical function (zsf) problems on real and $p$-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we find a wide class of integrable evolutions which respect the number-theoretic properties of the zsf problem. This means that at {\it all} times these real and $p$-adic systems can be unified into an adelic system with an $S$-matrix which involves (Dirichlet, Langlands, Shimura...) L-functions.Comment: 23 pages, uses plain TE

Topological phonon modes in filamentous structures

Topological phonon modes are robust vibrations localized at the edges of special structures. Their existence is determined by the bulk properties of the structures and, as such, the topological phonon modes are stable to changes occurring at the edges. The first class of topological phonons was recently found in 2-dimensional structures similar to that of Microtubules. The present work introduces another class of topological phonons, this time occurring in quasi one-dimensional filamentous structures with inversion symmetry. The phenomenon is exemplified using a structure inspired from that of actin Microfilaments, present in most live cells. The system discussed here is probably the simplest structure that supports topological phonon modes, a fact that allows detailed analysis in both time and frequency domains. We advance the hypothesis that the topological phonon modes are ubiquitous in the biological world and that living organisms make use of them during various processes.Comment: accepted for publication (Phys. Rev. E

A two-cocycle on the group of symplectic diffeomorphisms

We investigate a two-cocycle on the group of symplectic diffeomorphisms of an exact symplectic manifolds defined by Ismagilov, Losik, and Michor and investigate its properties. We provide both vanishing and non-vanishing results and applications to foliated symplectic bundles and to Hamiltonian actions of finitely generated groups.Comment: 16 pages, no figure

Compactification, topology change and surgery theory

We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any topology change in dimensions $\geq 5$ may be achieved via a causally continuous cobordism. This extends the known result for 4 dimensions. Therefore, there is no selection rule for compactification at the level of causal continuity. Theorems from surgery theory and handle theory are seen to be very relevant for understanding topology change in higher dimensions. Compactification via parallelisable cobordisms is particularly amenable to study with these tools.Comment: 1+19 pages. LaTeX. 9 associated eps files. Discussion of disconnected case adde

Twisted Elliptic Genera of N=2 SCFTs in Two Dimensions

The elliptic genera of two-dimensional N=2 superconformal field theories can be twisted by the action of the integral Heisenberg group if their U(1) charges are fractional. The basic properties of the resulting twisted elliptic genera and the associated twisted Witten indices are investigated with due attention to their behaviors in orbifoldization. Our findings are illustrated by and applied to several concrete examples. We give a better understanding of the duality phenomenon observed long before for certain Landau-Ginzburg models. We revisit and prove an old conjecture of Witten which states that every ADE Landau-Ginzburg model and the corresponding minimal model share the same elliptic genus. Mathematically, we establish ADE generalizations of the quintuple product identity.Comment: 28 pages; v2 refs adde

Chaos synchronization in gap-junction-coupled neurons

Depending on temperature the modified Hodgkin-Huxley (MHH) equations exhibit a variety of dynamical behavior including intrinsic chaotic firing. We analyze synchronization in a large ensemble of MHH neurons that are interconnected with gap junctions. By evaluating tangential Lyapunov exponents we clarify whether synchronous state of neurons is chaotic or periodic. Then, we evaluate transversal Lyapunov exponents to elucidate if this synchronous state is stable against infinitesimal perturbations. Our analysis elucidates that with weak gap junctions, stability of synchronization of MHH neurons shows rather complicated change with temperature. We, however, find that with strong gap junctions, synchronous state is stable over the wide range of temperature irrespective of whether synchronous state is chaotic or periodic. It turns out that strong gap junctions realize the robust synchronization mechanism, which well explains synchronization in interneurons in the real nervous system.Comment: Accepted for publication in Phys. Rev.