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作者：Mok, Ngaiming; Pila, Jonathan; Tsimerman, Jacob
摘要：We prove the Ax-Schanuel theorem for a general (pure) Shimura variety. A basic version of the theorem concerns the transcendence of the uniformization map from a bounded Hermitian symmetric space to a Shimura variety. We then prove a version of the theorem with derivatives in the setting of jet spaces, and finally a version in the setting of differential fields. Our method of proof builds on previous work, combined with a new approach that uses higher-order contact conditions to place varietie...
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作者：Hyde, James
摘要：A left-order on a group G is a total order < on G such that for any f, g and h in G we have f < g double left right arrow hf < hg. We construct a finitely generated subgroup H of Homeo(I-2; delta I-2), the group of those homeomorphisms of the disc that fix the boundary pointwise, and show H does not admit a left-order. Since any left-order on Homeo(I-2; delta I-2) would restrict to a left-order on H, this shows that Homeo(I-2; delta I-2) does not admit a left-order. Since Homeo(I; delta I) adm...
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作者：Yun, Zhiwei; Zhang, Wei
摘要：For arithmetic applications, we extend and refine our previously published results to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension F'/F of function fields over a finite field in odd characteristic, and a finite set of places Sigma of F that are unramified in F', we define a collection of Heegner-Drinfeld cycles on the moduli stack of PGL(2)-Shtukas with r-modifications and Iwahori level structures at places of Sigma. For a cuspidal automorphic rep...
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作者：Shitov, Yaroslav
摘要：The chromatic number of G x H can be less than the minimum of the chromatic numbers of finite simple graphs G and H.
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作者：Goldstern, Martin; Kellner, Jakob; Shelah, Saharon
摘要：Assuming four strongly compact cardinals, it is consistent that all entries in Cichon's diagram (apart from add(M) and cof(M), whose values are determined by the others) are pairwise different; more specifically, aleph(1) < add(N) < cov(N) < b < non(M) < cov(M) < partial derivative < non(N) < cof(N) < 2(aleph 0).
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作者：Varju, Peter P.
摘要：The Bernoulli convolution nu(lambda) with parameter lambda is an element of (0, 1) is the probability measure supported on R that is the law of the random variable Sigma +/-lambda(n), where the +/- are independent fair coin-tosses. We prove that dim nu(lambda) = 1 for all transcendental lambda is an element of (1/2, 1).
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作者：Moreira, Carlos Gustavo
摘要：We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and we may assume any real value in the interval [0, 1].
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作者：Tsimerman, Jacob
摘要：We give a proof of the Andre-Oort conjecture for A(g) - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven averaged version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The Andre-Oort conjecture then follows from previous work of Pila and the author.
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作者：Dasgupta, Samit; Kakde, Mahesh; Ventullo, Kevin
摘要：In 1980, Gross conjectured a formula for the expected leading term at s = 0 of the Deligne-Ribet p-adic L-function associated to a totally even character psi of a totally real field F. The conjecture states that after scaling by L(psi omega(-1), 0), this value is equal to a p-adic regulator of units in the abelian extension of F cut out by psi omega(-1). In this paper, we prove Gross's conjecture.
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作者：Adiprasito, Karim; Huh, June; Katz, Eric
摘要：We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.
