De Rham cohomology | EPFL Graph Search

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincar'e polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on

\mathbb{P}^1

. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schr"oer. We finish with constructing natural complete hyperk"ahler metrics on them, which in the

4

-dimensional cases are expected to be of type ALF.

2018

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincaré polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on

\blackbb{P}^1

. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the

4

-dimensional cases are expected to be of type ALF.

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the

p

-adic pushforward of the Haar measure under a hypertoric moment map

\mu

. This leads to an explicit formula for the Igusa zeta function

\FI_\mu(s)

\mu

, and in particular to a small set of candidate poles for

\FI_\mu(s)

. We also study various properties of the residue at the largest pole of

\FI_\mu(s)

. Finally, if

\mu

is constructed out of a quiver

\Gamma

we give a conjectural description of this residue in terms of indecomposable representations of

\Gamma

over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or

p

-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.

EPFL2017

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

\mathbb{P}^1

4

-dimensional cases are expected to be of type ALF.

2018

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

\blackbb{P}^1

. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the

4

-dimensional cases are expected to be of type ALF.

Motivic and p-adic Localization Phenomena

Dimitri Stelio Wyss

p

-adic pushforward of the Haar measure under a hypertoric moment map

\mu

. This leads to an explicit formula for the Igusa zeta function

\FI_\mu(s)

\mu

, and in particular to a small set of candidate poles for

\FI_\mu(s)

. We also study various properties of the residue at the largest pole of

\FI_\mu(s)

. Finally, if

\mu

is constructed out of a quiver

\Gamma

we give a conjectural description of this residue in terms of indecomposable representations of

\Gamma

over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or

p

EPFL2017