I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not correct.
That is why I am looking for a list of new big results in algebraic topology and homotopy theory that shocked the mathematical community since the last 10 years.
Ideally, it will be nice to have a very small introduction and references for each answer.
Thank you in advance.
No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have lowered the technical prerequisites) and has more power to touch an even wider array of other fields, via $\infty$-categories. Below are a few highlights from recent years.
2023: advances in algebraic $K$-theory allow for the disproof of the telescope conjecture.
2022: advances in algebraic $K$-theory allow for a proof of the redshift conjeture.
2017: Lurie's Higher Algebra, building on Higher Topos Theory, resolving the Cobordism Hypothesis.
2016: publication of work of Hill, Hopkins, Ravenel, using equivariant homotopy theory to resolve the Kervaire Invariant One problem (open since the 1960s).
2005-present: homotopy type theory
For more, see:
If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using Morava $K$-theory? Morava $K$-theory has been around since the ‘70’s or so and has become a central part of our understanding of stable homotopy theory via the chromatic picture, but it was always notoriously hard to find direct geometric applications of this stuff. Not only did Abouzaid and Blumberg find a geometric application, but the application they found resolved a longstanding open conjecture in symplectic geometry.