Journal of the American Mathematical Society
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Combinatorial Laplacians of matroid complexes
Journal of the American Mathematical Society - Tập 13 Số 1 - Trang 129-148
We combinatorially interpret the spectra of discrete Laplace operators from the
boundary maps in the simplicial complex of independent sets of a matroid. The
interpretation follows from a surprising orthogonal decomposition of the
simplicial chain groups. This decomposition is in general finer than the
spectral decomposition. As a consequence, the spectra are integral. One
corollary to our combina...
Polylogarithms, regulators, and Arakelov motivic complexes
Journal of the American Mathematical Society - Tập 18 Số 1 - Trang 1-60
We construct an explicit regulator map from the weight n n Bloch higher Chow
group complex to the weight n n Deligne complex of a regular projective complex
algebraic variety X X . We define the weight n n Arakelov motivic complex as the
cone of this map shifted by one. Its last cohomology group is (a version of) the
Arakelov Chow group defined by H. Gillet and C. Soulé. We relate the
Grassmannian...
The Vlasov-Poisson-Landau system in a periodic box
Journal of the American Mathematical Society - Tập 25 Số 3 - Trang 759-812 - 2012
A bilinear estimate with applications to the KdV equation
Journal of the American Mathematical Society - Tập 9 Số 2 - Trang 573-603
The honeycomb model of $GL_n(\mathbb C)$ tensor products I: Proof of the saturation conjecture
Journal of the American Mathematical Society - Tập 12 Số 4 - Trang 1055-1090 - 1999
Global classical solutions of the Boltzmann equation without angular cut-off
Journal of the American Mathematical Society - Tập 24 Số 3 - Trang 771-847
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,...
New extremal problems for the Riemannian recognition program via Alexandrov geometry
Journal of the American Mathematical Society - Tập 8 Số 1 - Trang 1-28
Non-commutative circuits and the sum-of-squares problem
Journal of the American Mathematical Society - Tập 24 Số 3 - Trang 871-898
We initiate a direction for proving lower bounds on the size of non-commutative
arithmetic circuits. This direction is based on a connection between lower
bounds on the size of non-commutative arithmetic circuits and a problem about
commutative degree-four polynomials, the classical sum-of-squares problem: find
the smallest n n such that there exists an identity ( 0.1 ) ( x 1 2 + x 2 2 + ⋯
+ x k 2...
A restriction estimate using polynomial partitioning
Journal of the American Mathematical Society - Tập 29 Số 2 - Trang 371-413
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....
On the size of Kakeya sets in finite fields
Journal of the American Mathematical Society - Tập 22 Số 4 - Trang 1093-1097
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....
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