Skip to main content
MathOverflow

Return to Question

Just trying to give more context in which the question came to my mind.
Source Link
DGH
DGH
  • 21
  • 5

In the recent times I have heard a lot about the following:

  1. The Atiyah-Singer Index theorem
  2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications.

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.

Edit: With my thin background in geometry and topology I cannot make the question more precise or add more examples but I can tell why I would like to know such programs. After I heard these two theorems and their applications I thought every mathematician should know a little about these results even if they don't try to completely understand its proof. For example, Hamilton's 1982 result mentioned by PVAL is something I want to read about. Thanks for many good answers. Again, I emphasize that I have a very thin background in these things.

In the recent times I have heard a lot about the following:

  1. The Atiyah-Singer Index theorem
  2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications.

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.

In the recent times I have heard a lot about the following:

  1. The Atiyah-Singer Index theorem
  2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications.

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.

Edit: With my thin background in geometry and topology I cannot make the question more precise or add more examples but I can tell why I would like to know such programs. After I heard these two theorems and their applications I thought every mathematician should know a little about these results even if they don't try to completely understand its proof. For example, Hamilton's 1982 result mentioned by PVAL is something I want to read about. Thanks for many good answers. Again, I emphasize that I have a very thin background in these things.

Post Closed as "Needs more focus" by YCor, Jan-Christoph Schlage-Puchta, Stefan Kohl♦, Yemon Choi, Franz Lemmermeyer
Post Made Community Wiki by Todd Trimble
Source Link
DGH
DGH
  • 21
  • 5

Theorems that led to very successful research programs in Geometry and Topology

In the recent times I have heard a lot about the following:

  1. The Atiyah-Singer Index theorem
  2. H-principle of Gromov ( and others )

It seems to me that these results led to decades of successful research in terms of improving the result, giving a simpler proof or new applications.

This motivates me to ask: What are the other such results which led to very successful research programs? I would add Bott Periodicity also.