Zenzeli

https://www.maths.ed.ac.uk/jcook/

математический самиздатовский зин с problem solving математикой и физикой но прикольной
круто что кто-то такие вещи сохраняет
среди авторов Марк Кисин например, который тогда был школьником в австралии видимо и
Теренс Тао, который наверное ходил в детский сад
ну и в целом самиздатовский вайб очень очаровательный

Background to publication of JCMN

The James Cook Mathematical Notes were started by Basil Rennie in 1975, when he was Head of the Department of Mathematics at the James Cook University of North Queensland, in Townsville, Australia. He was the editor and publisher, as well as the author of a great many of the articles. After his retirement in 1985, he and his wife Barbara moved back to Adelaide, South Australia, where, with Barbara's help, he continued to produce the journal until his death in 1996. The journal then ceased publication.

The JCMN is mainly concerned with mathematical problems, and their solution. The journal was fed by a wide network of correspondents throughout the world. Easily the most famous was Paul Erdös, who came to Townsville to see Basil on many occasions. Problems are posed, and then usually, but not always, solved by correspondents in later issues, often in several different ways. Some of these problems turn out to be old chesnuts, but most are new. There are articles on the mathematics of navigation. There are also some historical notes, usually having some connection to James Cook, the man. And then there is the occasional off-beat quotation.

Many of the articles in JCMN have no author's name attached. These were all written by Basil Rennie. But he also had a hand in many of the articles with other names attached! Sometimes, he would turn a correspondent's letter into an article, and attach their name. This process often entailed making non-trivial mathematical contributions to it!

A couple of the issues were edited by other staff members at James Cook University, when Basil Rennie was on leave. While on the staff at James Cook University from 1977 to 1984, I edited one of the issues.

https://arxiv.org/abs/2309.06993

Thurston's theorem and the Nielsen-Thurston classification via Teichmüller's theorem
James Belk, Dan Margalit, Rebecca R. Winarski

We give a unified and self-contained proof of the Nielsen-Thurston classification theorem from the theory of mapping class groups and Thurston's characterization of rational maps from the theory of complex dynamics (plus various extensions of these). Our proof follows Bers' proof of the Nielsen-Thurston classification.

кстати вот есть такое явление

пусть есть комплексная кривая С и положительное голоморфное линейное расслоение L.
возьмем точку p на С и любую маленькую окрестность V точки p.
тогда у L есть степень у которого есть сечение s, все нули которого лежат в V.

Пусть s -- голоморфное сечение с дивизором нулей S.
Рассмотрим отображение Абеля-Якоби из g-ой симметрической степени u: C^(g) \to J(C)
Это голоморфное сюрьективное отображение, следовательно открытое.
Значит u(V^(g)) открытая окрестность u(p)=1, она порождает J(C) и в силу компактности
якобиана есть большое целое число r такое что r*u(V^(g))=J(C), r можно увеличивать например взять deg(S)*r

в частности есть эффективный дивизор D степени g носитель которого сидит в V который отображается в произвольную точку якобиана после растягивания на deg(S)*r:

r*g*u(S)=deg(S)*r*u(D)

значит deg(S)*r*m - r*g*S=(f) главный дивизор

f*s^{r*g} это сечение L^{r*g} у которого дивизор нулей это deg(S)*r*D то есть носитель сидит в V.


----

ну или можно сказать еще что для любой точки p на кривой и ее маленькой окрестности V есть большое число n
такое что любое линейное расслоение степени больше n имеет сечение, нули которого сконцентрированы в V.
можно взять n>r*g

Canonical Heights on Shimura Varieties and the André-Oort Conjecture
Jonathan Pila, Ananth Shankar, Jacob Tsimerman, Hélène Esnault, Michael Groechenig

The main purpose of this work is to prove the André-Oort conjecture in full generality.

Equidistribution in Families of Abelian Varieties and Uniformity
Lars Kühne
Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer g≥2, we prove that the number of torsion points lying on a smooth complex algebraic curve of genus g embedded into its Jacobian is uniformly bounded. Complementing other recent work of Dimitrov, Gao, and Habegger, we obtain a rather uniform version of the Mordell-Lang conjecture as well. In particular, the number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian.

Arnold Conjecture and Morava K-theory

Mohammed Abouzaid, Andrew J. Blumberg


We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic p is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof relies on constructing a homology group associated to each such flow, and comparing it with the homology of the ambient symplectic manifold. The proof does not proceed by constructing a version of Floer's complex with characteristic p coefficients, but uses instead the canonical (stable) complex orientations of moduli spaces of Floer trajectories to construct a version of Floer homology with coefficients in Morava's K-theories, and can thus be seen as an implementation of Cohen, Jones, and Segal's vision for a Floer homotopy theory. The key feature of Morava K-theory that allows the construction to be carried out is the fact that the corresponding homology and cohomology groups of classifying spaces of finite groups satisfy Poincaré duality.

https://arxiv.org/abs/2102.01227

Volumes of definable sets in o-minimal expansions and affine GAGA theorems
Patrick Brosnan
I show that a d-dimensional definable set S⊆Rn in an o-minimal expansion of the ordered field of real numbers satisfies the volume estimate Hd({x∈S:∥x∥

We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be {\em nonunique}. Indeed, we will construct one exotic entire curve f:C→X which produces infinitely many cohomologically different Ahlfors currents. Moreover, concerning Siu's decomposition, for an arbitrary k∈Z+∪{∞}, some of the obtained Ahlfors currents have singular parts supported on k irreducible curves. In addition, they can have {\em nonzero} diffuse parts as well. Lastly, we provide new examples of diffuse Ahlfors currents on the product of two elliptic curves and on P2(C), and we show cohomologically elaborate Ahlfors currents on blow-ups of X.

Cm Semialgebraic Sections Over the Plane
Charles L. Fefferman, Garving K. Luli
In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let H be a semialgebraic bundle with respect to Cmloc(R2,RD). If H has a section, then it has a semialgebraic section.

Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications
Damian Brotbek, Yohan Brunebarbe
We prove a Second Main Theorem type inequality for any log-smooth projective pair (X,D) such that X∖D supports a complex polarized variation of Hodge structures. This can be viewed as a Nevanlinna theoretic analogue of the Arakelov inequalities for variations of Hodge structures due to Deligne, Peters and Jost-Zuo. As an application, we obtain in this context a criterion of hyperbolicity that we use to derive a vast generalization of a well-known hyperbolicity result of Nadel. The first ingredient of our proof is a Second Main Theorem type inequality for any log-smooth projective pair (X,D) such that X∖D supports a metric whose holomorphic sectional curvature is bounded from above by a negative constant. The second ingredient of our proof is an explicit bound on the holomorphic sectional curvature of the Griffiths-Schmid metric constructed from a variation of Hodge structures. As a byproduct of our approach, we also establish a Second Main Theorem type inequality for pairs (X,D) such that X∖D is hyperbolically embedded in X.