Journal of the American Mathematical Society

We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.

This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials,r−(p−1)r^{-(p-1)}withp>2p>2, for initial perturbations of the Maxwellian equilibrium states, as announced in an earlier paper by the authors. We more generally cover collision kernels with parameterss∈(0,1)s\in (0,1)andγ\gammasatisfyingγ>−n\gamma > -nin arbitrary dimensionsTn×Rn\mathbb {T}^n \times \mathbb {R}^nwithn≥2n\ge 2. Moreover, we prove rapid convergence as predicted by the celebrated BoltzmannHH-theorem. Whenγ≥−2s\gamma \ge -2s, we have exponential time decay to the Maxwellian equilibrium states. Whenγ>−2s\gamma >-2s, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only whenγ≥−2s\gamma \ge -2s, as conjectured by Mouhot and Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.

We compute the characters of the finite dimensional irreducible representations of the Lie superalgebragl(m|n)\mathfrak {gl}(m|n), and determineExt{\operatorname {Ext}}’s between simple modules in the category of finite dimensional representations. We formulate conjectures for the analogous results in categoryO\mathcal O. The combinatorics parallels the combinatorics of certain canonical bases over the Lie algebragl(∞)\mathfrak {gl}(\infty ).

A Kakeya set is a subset of F n \mathbb {F}^n , where F \mathbb {F} is a finite field of q q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C n ⋅ q n C_{n} \cdot q^{n} , where C n C_{n} depends only on n n . This answers a question of Wolff.

If S S is a smooth compact surface in R 3 \mathbb {R}^3 with strictly positive second fundamental form, and E S E_S is the corresponding extension operator, then we prove that for all p > 3.25 p > 3.25 , ‖ E S f ‖ L p ( R 3 ) ≤ C ( p , S ) ‖ f ‖ L ∞ ( S ) \| E_S f\|_{L^p(\mathbb {R}^3)} \le C(p,S) \| f \|_{L^\infty (S)} . The proof uses polynomial partitioning arguments from incidence geometry.

We prove that the partial C 0 C^0 -estimate holds for metrics along Aubin’s continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.