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We are presenting proofs of fundamental results related to homotopy idempotents, proofs that are sufficiently simple so that even the author can understand them. The first one is that homotopy idempotents in the category of pointed…
The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a…
We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m, m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a…
The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of…
Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial…
The project of Greenlees et al. on understanding rational G-spectra in terms of algebraic categories has had many successes, classifying rational G-spectra for finite groups, SO(2), O(2), SO(3), free and cofree G-spectra as well as rational…
Let \(\T\) be a commutative ternary \(\Gm\)-semiring in the sense of the triadic, \(\Gm\)-parametrized multiplication \(\{a,b,c\}_{\gamma}\). Building on the affine \(\Gm\)-spectrum \(\SpecG(\T)\), the structure sheaf, and the equivalence…
We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The…
In this work more questions arise than answers given, for which of course we do not apologize. The core of this paper is concerned with the construction of a ``constant'' t-structure on the bounded derived category of coherent sheaves…
In this paper we study a model structure on a category of schemes with a group action and the resulting unstable and stable equivariant motivic homotopy theories. The new model structure introduced here samples a comparison to the one by…
We study the $\mathbb{F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $\mathbb{C}$-motivic and classical stable homotopy groups.
An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an…
We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and…
Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer--Furuta invariants in the context of parametrised stable homotopy theory, leading to a definition of…
We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on…
Let $G$ be a connected reductive group over $\mathbb{Q}$. In this paper, we will stabilize the local trace formula, in particular, we construct the explicit form of the spectral side of stable local trace formula in the Archimedean case,…
In this paper, we compute the BP-cohomology of complex projective Stiefel manifolds. The method involves the homotopy fixed point spectral sequence, and works for complex oriented cohomology theories. We also use these calculations and…
A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…
We use the structure of the homotopy groups of the connective spectrum tmf of topological modular forms and the elliptic and Adams-Novikov spectral sequences to compute the homotopy groups of the non-connective version, Tmf, of that…
One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E-localisation of this model category. We study the…