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Homotopy-based methods in water engineering
\"Exploring the concept of homotopy from topology, different kinds of homotopy-based methods have been proposed for analytically solving nonlinear differential equations, given by approximate series solutions. Homotopy-Based Methods in Water Engineering attempts to present the wide applicability of these methods to water engineering problems. It solves all kinds of nonlinear equations, namely algebraic/transcendental equations, ordinary differential equations (ODEs), systems of ODEs, partial differential equations (PDEs), system of PDEs, and integro-differential equations using the homotopy-based methods\"-- Provided by publisher.
BOOK
Goodwillie Approximations to Higher Categories
We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra.
This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a
tower for a large class of
eBook
Stable Stems
We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over
We
also compute the motivic stable homotopy groups of the cofiber of the motivic element
eBook
Comparison of Relatively Unipotent Log de Rham Fundamental Groups
In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain
proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we
give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a
corollary we give a purely algebraic proof to the transcendental part of Andreatta–Iovita–Kim’s article: obtaining in this way a
complete algebraic criterion for good reduction for curves.
eBook
Stable homotopy groups of spheres
We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classifysmooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.
Journal Article
Motivic and real étale stable homotopy theory
Let
$S$
be a Noetherian scheme of finite dimension and denote by
$\\unicode[STIX]{x1D70C}\\in [\\unicode[STIX]{x1D7D9},\\mathbb{G}_{m}]_{\\mathbf{SH}(S)}$
the (additive inverse of the) morphism corresponding to
$-1\\in {\\mathcal{O}}^{\\times }(S)$
. Here
$\\mathbf{SH}(S)$
denotes the motivic stable homotopy category. We show that the category obtained by inverting
$\\unicode[STIX]{x1D70C}$
in
$\\mathbf{SH}(S)$
is canonically equivalent to the (simplicial) local stable homotopy category of the site
$S_{\\text{r}\\acute{\\text{e}}\\text{t}}$
, by which we mean the small real étale site of
$S$
, comprised of étale schemes over
$S$
with the real étale topology. One immediate application is that
$\\mathbf{SH}(\\mathbb{R})[\\unicode[STIX]{x1D70C}^{-1}]$
is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the
$\\unicode[STIX]{x1D70C}$
-local sphere (over
$\\mathbb{R}$
). As further applications we show that
$D_{\\mathbb{A}^{1}}(k,\\mathbb{Z}[1/2])^{-}\\simeq \\mathbf{DM}_{W}(k)[1/2]$
(improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that
$\\text{}\\underline{\\unicode[STIX]{x1D70B}}_{i}(\\unicode[STIX]{x1D7D9}[1/\\unicode[STIX]{x1D702},1/2])=0$
for
$i=1,2$
and establish some new rigidity results.
Journal Article
Proper Equivariant Stable Homotopy Theory
This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’
alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with
compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller
isomorphisms, and the resulting equivariant cohomology theories support the analog of an
Our
model for genuine proper
An important special case of our theory are
infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group
theory; for example, the
eBook
An arithmetic count of the lines on a smooth cubic surface
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\\mathbf {C}}$ there are $27$ lines, and over ${\\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\\operatorname {GW}(k)$ of $k$,
\\[ \\sum_{\\text{lines}} \\operatorname{Tr}_{L/k} \\langle \\alpha \\rangle = 15 \\cdot \\langle 1 \\rangle + 12 \\cdot \\langle -1 \\rangle, \\]
where $\\operatorname {Tr}_{L/k}$ denotes the trace $\\operatorname {GW}(L) \\to \\operatorname {GW}(k)$. Taking the rank and signature recovers the results over ${\\mathbf {C}}$ and ${\\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in $\\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
Journal Article
THE TOM DIECK SPLITTING THEOREM IN EQUIVARIANT MOTIVIC HOMOTOPY THEORY
We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck’s splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed-point functors and the motivic Adams isomorphism.
Journal Article