( `д´)就不能学学动画版的萌豚，多看看动画片

6/25

昨晚我在日落之后回到了海滩散步，我想看看深夜的月亮是什么样子。我在没过脚踝的水中向着月亮的方向前进，一开始冰冷的海水和海风让我无法忍受，但是走着走着我就习惯了环境的温度。我和Louis边走边通话，我们聊起了我的旅程，我关于人生有限性的思考，我最近读的书。我时不时看向月亮，推算此时在摩洛哥的海岸，如果你也在沙滩上的话，会以什么角度看到什么样的月亮。海边的月亮真的很有魔力，周围没有恼人的人造光源，没有喧噪的汽车噪音，我就站在水中，感受海浪冲刷小腿，沙子裹住脚掌，海风拂过因为晒伤而发热的皮肤。月光变得格外明亮和平静，银白的光线在夜空闪耀。我不知道月光原来这么亮，难以想象这来自地球的卫星，难以想象现在的清冷的月光和白天灼热的日光都来自太阳。虽然我无法用双手触及月亮，但是理论上无论是它反射的月光还是本身的引力都能算是一种和我的接触。我其实一直在和月亮握手。

昨晚我又梦到了你，我回到了高中的教室，这次你穿着浅绿色的毛衣，披散着头发坐在我旁边，是我的同桌。一开始我还沉浸在重回高中的惊讶中，打量着周围熟悉又陌生的环境。但是突然我注意到了你，我的同桌。你就坐在那里，低着头写着什么，并没有向我转身。这真的是你吗？你注意到我了吗？我不知道。你是不想理我所以写作业逃避吗？你是注意到我但是不知道如何表现和开口，所以低头写字掩盖你的心理活动吗？总之我在梦里重新遇见你后，一股复杂的情绪涌上我的心头，我渴望重新接近你，我想说出这段时间我的所有思绪和故事，我想看看你的变化。但是千言万语堵在胸口，我不知道从何开始，也不知道是否应该说出口。最后我只是拍了拍你的肩膀，向我的新同桌介绍了我自己：“你好，我是徐嘉余。你能告诉我你的名字吗？”

醒来后我获得了巨大的平静。如果我们未来再次见面，我不会问你为什么提出分手，这段时间发生了什么，你过得如何，未来有什么计划之类的，我会把这些全部抛开，我虽然想知道，但是我主动选择不知道。我会像梦里一样，重新认识新的你。那么这份日记，甚至之前计划的压缩简化版，似乎没必要在一开始就交给你。向一个刚认识的朋友交出这么厚重的日记是不妥当的。我会认真地重新和你交朋友。

Night：

我在沙滩上玩得挺开心，但是总忍不住想你。要是我和你一起下水，在波浪中还牵着手，一起随波逐流，该有多好。要是你和我一起躺在沙滩上，你拿着小铲子把我埋进沙子里，该有多好。

沙滩上有很多穿着比基尼的女生，但是我对她们毫无兴趣，我只是低着头前进，想象着我和你的可能性。

6/26

Morning:

我又梦到了你。我是一个超级英雄，在和朋友打闹着走在拯救世界的道路上。迎面我看到了班里的女生团体走来，你穿着红色衣服格外醒目。但是你的表情是死的，旁边的人都围着你欢笑，但是你的表情宛如一个死人。我们擦肩而过，但是又忍不住双双回头。我们相隔了十米远的距离，沉默地注视着彼此。该说什么呢？我想冲上去抱住你，但是看见你的表情，我放弃了这个念头。我非常心疼你，我不想看到你这样的表情。我们就这么长久地凝望着对方，直到我被人流挤回原来的道路。我没有再回头，我知道你已经选择回来，哪怕情绪依旧很糟，但是你已经决定回来，你已经被朋友围绕。知道这一点就让我满足了，虽然我和你的道路又暂时错开，但是我知道等我拯救完世界，也许我会给你发条信息。

Afternoon:

我真的能在九月之前见到你吗？也许不会了吧，我觉得你不会专门来做实习，defence也可以线上完成。一下子就感觉有点失落，我还是想看看你，哪怕只是远远地确认一眼你的存在。那我该怎么判断我和你对话的时机呢？我见不到你，看不到你的Instagram，不能联系你，不能见到你。那我的沉默要到什么时候才能结束呢？你会来找我吗？

今天我在Cap Blanc Nez散步，边走边思考着。前几天我在沙滩上堆出了你的名字和月亮的图案，坐在你的名字旁边，我想象着这不只是一个名字，而是你本人此刻陪着我。今天在悬崖边上，我觉得沙子做的图案太过脆弱，一阵海风、路人的脚步都能轻易抹去它。我想要找到一个更持久的方式，把这段时间封存进去，在日后和你一起分享。我捡了一块鹅卵石，一个想法冒上心头：这块石头也许已经在岸边存在了几百万年，已经记录了无数时间的痕迹，我可以把它带回家，每天晚上慢慢雕琢，刻出月亮的样子，用脱落的残片配合荧光材料做成星星，再手绘一个夜空的背景，最后全部放入木盒，让你一打开盒子就能看见一片夜空，感受到我投注的生命。无论明年三月你生日的时候我们关系如何，恋人/朋友/不再联系，我都会把它送给你，作为我能给你的最独特的生日礼物。如果那时候我还没给你看我的日记的话，也许我也会附上我的日记。你说过你会珍惜所有人送出的礼物，我也看到了你带着各种首饰，它们都是你的父母和姐姐在你生日时送的礼物，你的书架上也专门有一层放着代表朋友和老师好意的礼物。就连偶像科学家送你的一颗糖你都放在罐子里珍藏着。能在你的收藏里加入我的这份真心吗？

Night：

明天早上就要回巴黎了，我突然有点恐慌。回去了就要面对实习、高温、熟悉的人、熟悉的街道。虽然我知道我不会遇见你，但是我还是有点恐慌。一方面我想回去，我想彻底搞定实习，回到科研生活，但是又有点不想结束这种“隐居”生活。回去后哪怕几乎不可能，但是我还是有可能见到你来补考，这一点点可能性就让我慌张，哪怕我已经想好了要重新认识你，慢慢来，但是如果你不按照我的预想来怎么办，要是你见到我就开始继续说出伤我心的话怎么办。我想逃离这个可能性，但是我也知道我逃不开。我也许有点受伤，我现在极度抗拒阅读和你有关的聊天记录，尤其是伤到我的那些。看到关键词我就条件反射地开始恐慌，会害怕这样的话语再次出现。我不知道我如果当面再次听到会有什么反应，现在想想就已经呼吸急促，非常恐慌，很想逃离。如果见面，或者恢复线上对话，我觉得我会绝对避开父亲、学业、未来有关的话题，只关注当天的生活，拍点偶遇的小猫的照片，月亮，我做的饭，仅此而已。我不止是害怕刺激到你，也是害怕你受刺激后反过来伤害我，所以我期待用最轻柔最简单的日常聊天作为可能的重新认识的开始。

你真的治疗了我的创伤吗？也许是在治愈了一部分的同时割开了另一个。我从未体验过这样的持续性的恐慌。

我拍了很多月亮的照片，我又获得了一丝平静。我也许可以做一个相册，把你不在的日记里的拍到的不同形态的月亮都收藏起来，以后发给你，也许作为重逢时话题的starter。

6/29

Morning:
周末我看了一部电影，黑泽明的《Ikiru》，写了观后感。又写了一篇随笔，探讨人应该怎么生活。我收到了一封来自以前大学后辈的邮件，我给他回了一封随笔性的长信。在周日一天内总共可能写了一万字的内容。今天巴黎的温度终于回到了清爽宜人的区间，我也正式回到了lab，准备把我积攒了一周的干劲彻底释放。我觉得我又进入了一个新的平静区，对你的思念是一切的底色和思维网的中心，从这里出发我以各种方式认真生活，认真和人交流。我觉得我做到了一开始做的承诺，也就是在你不在的时候，努力活成以前从未有过的样子，证明我的等待不是徒劳的精神消耗。事实上这段等待确实可能是我人生中最重要的时间段之一，有太多重要的想法和体验在这段时间出现，它在极大程度上重塑了我的灵魂。等到下次见面的时刻，我想我确实变成了一个需要重新认识的人。

导师给我的评语，看了让人很开心
Jiayu Xu carried out his internship at LPTMS with a high degree of independence and scientific curiosity and with an evident inclination to mathematical rigorousness. Throughout the internship, he demonstrated an ability to work independently, identify relevant research directions, and develop original approaches to the problems under investigation. Rather than relying on my suggestions, he often proposed his own methods of analysis and actively explored the broader scientific literature (including AI) surrounding the specific questions addressed during the project.

The internship of Jiayu Xu focused on path-counting problems, localization phenomena, and spectral methods on rooted trees and related graph structures. Jiayu rapidly acquired the necessary theoretical tools and showed a strong capacity to connect ideas originating from different areas of mathematical physics, probability theory, combinatorics, and spectral analysis. A important aspect of his work is his attempt to go beyond the specific problem trying to look at it from a broader perspective.

One of the questions considered by Jiayu Xu during the internship concerns the study of localization phenomena for Brownian bridges and path-counting observables on rooted trees. Jiayu obtained several original results clarifying the relation between different observables and their associated analytic structures. In particular, his analysis of localization effects and the distinction between open-path, fixed-shell, and closed-loop provides important insights into the problem. These results could constitute the core of a forthcoming research paper after further development and refinement.

In addition to these accomplishments, Jiayu made important progress toward understanding phase transitions on trees with linearly growing vertex degree. He successfully reformulated the problem in terms of Hermite/Krylov chains and investigated the effects of localized perturbations using spectral methods. Although the available internship period did not allow enough time for a complete analysis of all obtained results, the work made during this stage is promising and opens several interesting directions for future research.

As I mentioned already, throughout the internship, Jiayu displayed a clear inclination toward rigorous mathematical reasoning. He has a strong taste for precise definitions, careful derivations, and rigorous arguments. His work was consistently characterized by a high level of attention to detail. At the same time, he showed creativity in formulating new questions and exploring alternative viewpoints when standard approaches proved insufficient.

The quality of Jiayu’s work, combined with his independence and persistence, allowed him to make progress on problems that are of research-level going beyond the pure educational tasks. He approached difficulties constructively and was able to suggest the way towards the final destination even when the conclusions were not immediately accessible. This ability is particularly valuable in theoretical research, where substantial advances often emerge only after extended exploration.

As suggestions for future development, I would encourage Jiayu to complement his rigorous mathematical analyses with a more systematic physical interpretation of the obtained results. In several instances, formal derivations were achieved successfully, but the broader physical implications remain to be clarified. Whenever possible, it would also be beneficial to support analytical findings with numerical checks, which can provide additional intuition and help understand phenomena. Some results obtained during the internship, such as the derivation presented in the final section concerning the transition point for a rank-one perturbation, remain without a definitive conclusion and deserve further investigation.

Overall, Jiayu Xu completed an excellent internship. He demonstrated independence, originality, mathematical talent, and a genuine aptitude for research. The results obtained during this project are of high quality and provide a strong basis for future work. I am confident that he possesses the abilities required to pursue advanced research successfully and I strongly recommend him for future academic career.