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作者：Cohn, Henry; Kumar, Abhinav; Miller, Stephen D.; Radchenko, Danylo; Viazovska, Maryna
摘要：We prove that the E8 root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions eight and twenty-four, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently sh...
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作者：Hausmann, Markus
摘要：For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian 2-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, ...
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作者：Milman, Emanuel; Neeman, Joe
摘要：We establish the Gaussian Multi-Bubble Conjecture: the least Gaussianweighted perimeter way to decompose Rn into q cells of prescribed (positive) Gaussian measure when 2 < q < n+ 1, is to use a simplicial cluster, obtained from the Voronoi cells of q equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to nullsets). In particular, the case q = 3 confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter wa...
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作者：Bachmann, Tom; Kong, Hana Jia; Wang, Guozhen; Xu, Zhouli
摘要：We define the Chow t-structure on the infinity-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)(c(sic)) when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)(cell,c(sic)) as the category of even graded MU2*MU-comodules. Furthermore, we show that the co-category of modules over the Chow truncated sphere spectrum 1(c=0) is algebraic. Our results generalize th...
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作者：Aizenman, Michael; Duminil-Copin, Hugo
摘要：We prove that the scaling limits of spin fluctuations in four-dimensional Isingtype models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the lambda phi(4) fields over R-4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models random current representation, in which the correlation functions deviation from Wicks law is expressed in terms...
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作者：Hugelmeyer, Cole
摘要：We prove that for every smooth Jordan curve gamma, if X is the set of all r is an element of [0, 1] so that there is an inscribed rectangle in gamma of aspect ratio tan(r . pi/4), then the Lebesgue measure of X is at least 1/3. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in R x RP3. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in ...
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作者：Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp
摘要：Consider a smooth, geometrically irreducible, projective curve of genus g >= 2 defined over a number field of degree d >= 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the MordellWeil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in g and d, for the number of geometric torsion points ...
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作者：Jiang, Wenshuai; Naber, Aaron
摘要：Consider a Riemannian manifold with bounded Ricci curvature vertical bar Ric vertical bar <= n - 1 and the noncollapsing lower volume bound Vol vertical bar(B-1 (p)) > v > 0. The first main result of this paper is to prove that we have the L-2 curvature bound f(B1(p)) vertical bar Rm vertical bar(2)(x) dx < C(n, v), which proves the L-2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if (M-j(n),d(j), p(j)) -> (X,d,p) is a GH-li...
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作者：Burungale, Ashay A.; Kobayashi, Shinichi; Ota, Kazuto
摘要：We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic Z(p)-extension of the unramified quadratic extension of Q(p) for p >= 5 a prime. Rubin's conjecture underlies Iwasawa theory of the anticyclotomic deformation of a CM elliptic curve over the CM field at primes p of good supersingular reduction, notably the Iwasawa main conjecture in terms of the p-adic L -function. As a consequence, we prove an inequality in the p-adic Birch and Swinnerton-Dyer con...
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作者：Kaluba, Marek; Kielak, Dawid; Nowak, Piotr W.
摘要：We prove that Aut(F-n) has Kazhdan's property (T) for every n >= 6. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for n >= 5. We also provide explicit lower bounds for the Kazhdan constants of SAut(F-n) (with n >= 6) and of SLn (Z) (with n >= 3) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever n > 6.
