Project acronym 2-3-AUT
Project Surfaces, 3-manifolds and automorphism groups
Researcher (PI) Nathalie Wahl
Host Institution (HI) KOBENHAVNS UNIVERSITET
Call Details Starting Grant (StG), PE1, ERC-2009-StG
Summary The scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\'e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.
Summary
The scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\'e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.
Max ERC Funding
724 992 €
Duration
Start date: 2009-11-01, End date: 2014-10-31
Project acronym UniEqTURB
Project Universal Equilibrium and Beyond - Challenging the Richardson-Kolmogorov Paradigm
Researcher (PI) Clara VELTE
Host Institution (HI) DANMARKS TEKNISKE UNIVERSITET
Call Details Starting Grant (StG), PE8, ERC-2018-STG
Summary Turbulence is at a crossroads: The old, established ideas of Richardson and Kolmogorov have with accumulating evidence come under renewed scrutiny, especially in non-stationary and non-equilibrium flows. Many in the community seek new and more accurate ways to describe turbulence. This is a time of re-evaluation and opportunity!
The assumed statistical equilibrium of the smallest and intermediate scales is identified as the main cause of the potentially erroneous deductions. This problem was not previously noticed because experiments that confirmed the previous theories were all in statistical equilibrium. And those experiments and theories which disagreed were labelled ‘anomalous’, no matter how carefully performed or argued.
The proposed theory-intensive approach will therefore specifically use non-equilibrium and statistically non-stationary flows to:
1. Investigate the underlying mechanisms determining the level of dissipation
2. Quantify the resulting effects on the balance equations of central importance
3. Test the results against the established, as well as competing, theories
I will use stationary and accelerating jets well-suited for studying the non-linear interactions and quantifying departures to the assumed equilibrium and the non-stationary dissipation. The feasibility is demonstrated with preliminary results. The databases which will be established should contribute substantially to settling the long-lived ultimate question of turbulence: what are the true underlying mechanisms that set the level of dissipation.
The results will be ground breaking scientifically and economically. The impact for engineering applications is extensive, since Kolmogorov-based turbulence models are routinely used, and since developing flows constitute the rule rather than the exception in the majority of engineering applications. The potential economic consequences for e.g. transportation, climate predictions and power extraction are impossible to underestimate.
Summary
Turbulence is at a crossroads: The old, established ideas of Richardson and Kolmogorov have with accumulating evidence come under renewed scrutiny, especially in non-stationary and non-equilibrium flows. Many in the community seek new and more accurate ways to describe turbulence. This is a time of re-evaluation and opportunity!
The assumed statistical equilibrium of the smallest and intermediate scales is identified as the main cause of the potentially erroneous deductions. This problem was not previously noticed because experiments that confirmed the previous theories were all in statistical equilibrium. And those experiments and theories which disagreed were labelled ‘anomalous’, no matter how carefully performed or argued.
The proposed theory-intensive approach will therefore specifically use non-equilibrium and statistically non-stationary flows to:
1. Investigate the underlying mechanisms determining the level of dissipation
2. Quantify the resulting effects on the balance equations of central importance
3. Test the results against the established, as well as competing, theories
I will use stationary and accelerating jets well-suited for studying the non-linear interactions and quantifying departures to the assumed equilibrium and the non-stationary dissipation. The feasibility is demonstrated with preliminary results. The databases which will be established should contribute substantially to settling the long-lived ultimate question of turbulence: what are the true underlying mechanisms that set the level of dissipation.
The results will be ground breaking scientifically and economically. The impact for engineering applications is extensive, since Kolmogorov-based turbulence models are routinely used, and since developing flows constitute the rule rather than the exception in the majority of engineering applications. The potential economic consequences for e.g. transportation, climate predictions and power extraction are impossible to underestimate.
Max ERC Funding
1 499 036 €
Duration
Start date: 2019-04-01, End date: 2024-03-31
