In this work, we propose a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ${\mathbb{R}}^{1+3}$. A crucial ingredient is the associated Yang–Mills heat flow. As this approach avoids the drawbacks of previous approaches, it is expected to be more robust and easily adaptable to other settings. Building on the author’s previous results, we prove, as the first application of our approach, finite energy global well-posedness of the Yang–Mills equations on ${\mathbb{R}}^{1+3}$. This is a classical result first proved by Klainerman and Machedon using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck’s lemma and hence does not involve localization in space-time.

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori.

We show that an analytic invariant torus ${\mathcal{T}}_0$ with Diophantine frequency ${\omega }_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_0$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_0$ is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency ${\omega }_0$.

For frequency vectors ${\omega }_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov nondegeneracy condition at ${\mathcal{T}}_0$, then ${\mathcal{T}}_0$ is accumulated by KAM tori of positive total measure.

In four degrees of freedom or more, we construct for any ${\omega }_0\in {\mathbb{R}}^d$, $C^{\infty }$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_0$ with frequency ${\omega }_0$ that is not accumulated by a positive measure of invariant tori.

The classical Faber–Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet Laplacian among sets with given volume. In this article we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and by Bhattacharya and Weitsman. More generally, the result applies to every optimal Poincaré–Sobolev constant for the embeddings $W_0^{1,2}(\Omega )\hookrightarrow L^q(\Omega )$.

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about:

• the cohomology of the configuration space of $n$ distinct ordered points on an arbitrary (connected, oriented) manifold;

• the diagonal coinvariant algebra on $r$ sets of $n$ variables;

• the cohomology and tautological ring of the moduli space of $n$-pointed curves;

• the space of polynomials on rank varieties of $n\times n$ matrices;

• the subalgebra of the cohomology of the genus $n$ Torelli group generated by $H^1$;

and more. The symmetric group $S_n$ acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for $n$ large enough, by a polynomial in the cycle-counting functions that is independent of $n$. In particular, the dimension is eventually a polynomial in $n$. In this framework, representation stability (in the sense of Church–Farb) for a sequence of $S_n$-representations is converted to a finite generation property for a single FI-module.