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作者：McMullen, Curtis T.
作者单位：Harvard University
摘要：In this paper we introduce a new height on P-1 (K) associated to an Abelian variety with real multiplication by K, and use it to study non-arithmetic triangle groups, Teichintiller curves, and billiards in lattice polygons. Complementary results on matrix coefficients and measures are obtained using modular symbols. In particular, we show the matrix entries m of the classical Hecke group Delta(2, 5, infinity) are constrained by the condition that -gamma(-2)(m'/m) lies in a countable, closed se...
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作者：Bhargava, Manjul; Shankar, Arul; Wang, Xiaoheng
作者单位：Princeton University; University of Toronto; University of Waterloo
摘要：We determine the density of monic integer polynomials of given degree n > 1 that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f (x), such that f (x) is irreducible and Z[x]/(f (x)) is the ring of integers in its fraction field, is positive, and is in fact given by zeta (2)(-1). It also follows from our methods that there are >> X1/2+1/n monogenic...
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作者：Buzzi, Jerome; Crovisier, Sylvain; Sarig, Omri
作者单位：Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Weizmann Institute of Science
摘要：We study the entropy and Lyapunov exponents of invariant measures mu for smooth surface diffeomorphisms f, as functions of (f, mu). The main result is an inequality relating the discontinuities of these functions. One consequence is that for a C-infinity surface diffeomorphism, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semicontinuity of the Hausdorff dimension on the set of...
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作者：Mark, Thomas E.; Tosun, Bulent
作者单位：University of Virginia; University of Alabama System; University of Alabama Tuscaloosa
摘要：We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic R-4. Using an obstruction derived from Heegaard Hoer homology we prove that no Brieskorn homology sphere admits a contact type embedding in R-4 , a result that has bearing on conjectures of Gompf and Kollar. This implies in particular that no rationally convex domain in C-2 has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of...
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作者：Schmieding, Scott
作者单位：University of Denver
摘要：For a homeomorphism T : X -> X of a compact metric space X, the stabilized automorphism group Aut((infinity))( T) consists of all self-homeomorphisms of X which commute with some power of T. Motivated by the study of these groups in the context of shifts of finite type, we introduce a certain entropy for groups called local P entropy. We show that when (X, T) is a non-trivial mixing shift of finite type, the local P entropy of the group Aut((infinity))(T) is determined by the topological entro...
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作者：Zhou, Yang
作者单位：Fudan University
摘要：In this paper, we prove a wall-crossing formula for epsilon-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that stability conditions in adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small I-function. Using this we prove that the...
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作者：Rudenko, Daniil
作者单位：University of Chicago
摘要：We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron as values of period maps for the corresponding surface. As a corollary we show that the cross-ratio of the exponents of the solid angles of a tetrahedron is equal to the cross-ratio of the exponents of the perimeters of its faces. The Reg...
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作者：Liu, Ruochuan; Wang, Guozhen
作者单位：Peking University; Fudan University
摘要：A Correction to this paper has been published: 10.1007/s00222-022-01134-9
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作者：Delecroix, Vincent; Goujard, Elise; Zograf, Peter; Zorich, Anton
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite de Bordeaux; Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Universite de Bordeaux; Centre National de la Recherche Scientifique (CNRS); Inria; Universite Paris Cite
摘要：We study the combinatorial geometry of a random closed multi-curve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components , gamma(1), ...,gamma(k) of a random multicurve m(1)gamma(1) + ... + m(k)gamma(k) represent linearly independent homology cycles with asymptotic probability 1 and that all its weights m(i); are equal to 1 with asymptotic probability root 2/2. We prove analogous properties for random square-tiled surfaces. In p...
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作者：Li, Chao; Zhang, Wei
作者单位：Columbia University; Massachusetts Institute of Technology (MIT)
摘要：We formulate and prove a local arithmetic Siegel-Weil formula for GSpin Rapoport-Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport-Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel-Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place o...
