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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.
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作者：Jiang, Zilin; Tidor, Jonathan; Yao, Yuan; Zhang, Shengtong; Zhao, Yufei
摘要：Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0 < alpha < 1. Let N-alpha(d) denote the maximum number of lines through the origin in R-d with pairwise common angle arccos alpha. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1 - alpha)/(2 alpha). If k < infinity...
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作者：Greene, Joshua Evan; Lobb, Andrew
摘要：For every smooth Jordan curve gamma and rectangle R in the Euclidean plane, we show that there exists a rectangle similar to R whose vertices lie on gamma. The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smooth Lagrangian embedding in C-2.
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作者：Jaye, Benjamin; Tolsa, Xavier; Villa, Michele
摘要：In this paper we provide a proof of the Carleson epsilon(2)-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson epsilon(2)-square function.
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作者：Gardam, Giles
摘要：The unit conjecture, commonly attributed to Kaplansky, predicts that if K is a field and G is a torsion-free group, then the only units of the group ring K[G] are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
