Hopf invariants, Toda brackets and the reduced diagonal map

That there is a close connection between crude (or suspended) Hopf invariants and the reduced diagonal map has long been known. In this paper we build on a classical result of Boardman and Steer (1967) to explore some further relationships in terms of conic structures. In a related result we show that certain Toda brackets contain crude Hopf invariants as elements. In particular, using this latter observation, detection of crude Hopf invariants can be carried out.     

On the Cohomology of Categories, Universal Toda Brackets and Homotopy Pairs

A category of homotopy pairs is characterised by a cohomology class which generalizes the notion of Toda bracket. Explicit computations of such cohomology classes are described.     

Local duality for structured ring spectra

We use the abstract framework constructed in our earlier paper [8] to study local duality for Noetherian ${\mathbb{E}}_{\infty }$-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.     

Semidiscrete Toda lattices

We study integrable cutoff constraints for a semidiscrete Toda lattice. We construct a Lax representation for a semidiscrete analogue of lattices corresponding to simple Lie algebras of the C series. We introduce nonlocal variables in terms of which the symmetries of the infinite semidiscrete lattice can be expressed, and we classify cutoff constraints of a certain form compatible with the symmetries of the infinite lattice.     

Extended Toda Lattice

We introduce nonlocal flows that commute with those of the classical Toda hierarchy. We define a logarithm of the difference Lax operator and use it to obtain a Lax representation of the new flows.     

Witt vectors and equivariant ring spectra applied to cobordism

Given a finite group G we show that Dress and Siebeneicher'sring of G-typical Witt vectors on the Lazard ring, that is,on the polynomial ring on countably many indeterminates overthe integers, embeds as a subring of the unitary cobordism ringof G-manifolds. We also show that the ring of G-typical Wittvectors on the Lazard ring embeds as a subring of the ring ofhomotopy groups of the G-fixed point spectrum of the spectrumMU representing cobordism. The above results are derived byexploiting the interaction between restriction, additive transferand multiplicative transfer. This interaction is described bytwo Mackey functors satisfying a distributivity relation encodedin a formalism developed by Tambara.     

Realizability of algebraic Galois extensions by strictly commutative ring spectra

We discuss some of the basic ideas of Galois theory for commutative -algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions.

We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative -algebras. Examples such as the complex -theory spectrum as a -algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension.

     

The ghost and weak dimensions of rings and ring spectra

The primary object of this paper is to prove the conjecture of [HL09a] explaining how to recover the weak dimension of a ring from its derived category. In the process, we develop a theory of weak dimension, which we call ghost dimension, for the generalized rings, known as ring spectra, that arise in algebraic topology.     

Quantum relativistic Toda chains

q-Deformations of the Toda chains related to infinite series of Lie algebras and to their affine analogs are described. The corresponding 2×2 L-operators satisfying the standard quadratic algebra of XXZ-type are constructed. In the quantum case, the procedure of separation of variables is described. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 71–84. Translated by B. M. Bekker.     

Quantum generalized Toda system

We construct a ??spectral curve?? for the generalized Toda system, which allows efficiently finding its quantization. In turn, the quantization is realized using the technique of the quantum characteristic polynomial for the Gaudin system and an appropriate Alder-Kostant-Symes reduction. We also discuss some relations of this result to the recent consideration of the Drinfeld Zastava space, the monopole space, and corresponding symmetries of the Borel Yangian.     

Matrix semigroups whose ring commutators have real spectra are realizable

We introduce the numerical spectrum \(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.     

Separability of the Toda Lattice

We study integrability of the nonperiodic, finite-dimensional Toda Lattice (TL) from the point of view of the Hamilton-Jacobi (HJ) theory of separation of variables. Known to be completely integrable in the sense of Arnol'd-Liouville, the system nonetheless cannot be integrated by the HJ method in the ‘original’ generalized physical position-momenta coordinates. The intrinsic (coordinate-free) characterization method by Benenti provides a tool to investigate the separability of the TL system independently of a local system of coordinates.     

The Toda rarefaction problem

In the Toda shock problem (see [7], [11], [8], and also [3]) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas. We make use of the Riemann-Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 2 |a| of the driving particle is sufficiently large (|a| > 1), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops. © 1996 John Wiley & Sons, Inc.     

Variational Analysis of Toda Systems

The author surveys some recent progress on the Toda system on a two-dimensional surface Σ,arising in models from self-dual non-abelian Chern-Simons vortices,as well as in differential geometry.In particular,its variational structure is analysed,and the role of the topological join of the barycentric sets of Σ is shown.     

Pfaffianization of the two-dimensional Toda lattice

Pfaffianization procedure due to Hirota and Ohta is applied to the two-dimensional Toda lattice. As a result, a Pfaffianized version of the two-dimensional Toda lattice is found.