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Eric C. Peterson

Eric C. Peterson. I was a Benjamin Peirce fellow in the Harvard math department, but I moved on to work in quantum computing at Rigetti. Before any of that, I was a graduate student at Berkeley. Before that, I was an undergraduate in computer science at Urbana-Champaign. This URL used to host a group research blog, Chromotopy. My mathematical interests are in using algebro-geometric tools to answer questions in algebraic topology, and I have a penchant for computations. ArXiv preprint available,. With Aa...

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Eric C. Peterson | chromotopy.org Reviews

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Eric C. Peterson. I was a Benjamin Peirce fellow in the Harvard math department, but I moved on to work in quantum computing at Rigetti. Before any of that, I was a graduate student at Berkeley. Before that, I was an undergraduate in computer science at Urbana-Champaign. This URL used to host a group research blog, Chromotopy. My mathematical interests are in using algebro-geometric tools to answer questions in algebraic topology, and I have a penchant for computations. ArXiv preprint available,. With Aa...

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Eric C. Peterson | chromotopy.org Reviews

https://chromotopy.org

Eric C. Peterson. I was a Benjamin Peirce fellow in the Harvard math department, but I moved on to work in quantum computing at Rigetti. Before any of that, I was a graduate student at Berkeley. Before that, I was an undergraduate in computer science at Urbana-Champaign. This URL used to host a group research blog, Chromotopy. My mathematical interests are in using algebro-geometric tools to answer questions in algebraic topology, and I have a penchant for computations. ArXiv preprint available,. With Aa...

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chromotopy.org

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Chromotopy

http://www.chromotopy.org/archive.html

May 01: On L-functions and algebraic K-theory. March 13: The K-Theory of Endomorphisms. February 26: Hypothetical abelian varieties from K-theory. November 27: Some Thoughts on Descent and Descent Data. November 16: Truncations of E-theory. October 20: Loop spaces and descent. May 02: Topologized objects in algebraic topology. April 19: Stanley symmetric functions. April 04: Determinantal K-theory. April 01: The (very nice) Bousfield Lattice of Harmonic Spectra. February 18: Reduced words. February 12: J...

2

Chromotopy

http://www.chromotopy.org/k-thy-of-endomorphisms

The K-Theory of Endomorphisms. March 13, 2014, by Matthew Pancia. For the purposes of this post, I am going to assume that you are all familiar with the basics of Algebraic -theory. If you’re not, just treat it as a black box, a gadget which takes in one of :. A (simplicial) ring (spectrum). An augmented (simplicial) bimodule over a ring. A small, stable -category. And spits out a spectrum , such that does the job of storing all Euler characteristics or additive invariants . Where is the tensor algebra o...

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Chromotopy

http://www.chromotopy.org/what-are-witt-vectors

What are Witt vectors? December 18, 2010, by Eric Peterson. The ring of Witt vectors has shown up at multiple points in my reading, and I’ve been pretty dissatisfied with the descriptions of this object. Looking into things more seriously, I’ve found that this is such an immensely complicated construction that no single description is going to do it justice (Hazewinkel tries for 300ish pages! But I’d like to share a couple viewpoints that I find important and/or enlightening. Write for the parenthesized ...

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Chromotopy

http://www.chromotopy.org/hypothetical-abelian-varieties

Hypothetical abelian varieties from K-theory. February 26, 2014, by Eric Peterson. I heard an idea tossed around recently that I’d like to share with you all. I worry that it might be a little half-baked, as I’ve only heard about it recently and so maybe haven’t spent enough time sketching out the edges of it. Maybe writing this will help. Throughout, is a perfect field of positive characteristic . Say that a Dieudonné module is. Dieudonné modules are thrilling because. They are just modules. Whereas -di...

5

Chromotopy

http://www.chromotopy.org/l-functions-and-kthy

On L-functions and algebraic K-theory. May 01, 2014, by Saul Glasman. In this post, I’ll attempt to make some progress towards demystifying the relationship between L-functions and algebraic K-theory, with reference to a very simple example. I’ll say something about L-functions a little lower down, but the only thing I’ll say about K-theory in general is that it’s a way to get a spectrum from, say, a scheme. A conjecture of this form has been proven for the ring of integers of a totally real abelian fiel...

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Fight with Infinity | Wir müssen wissen, wir werden wissen | Page 2

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Wir müssen wissen, wir werden wissen. 概言之一方面我们有代数闭链这个几何对象，另一方面我们有Hodge结构这个分析对象。在较初等的层面上，Hodge理论的 unreasonable effectiveness 已显露无遗对于复代数簇，我们可以用它重新诠释大部分Lefschetz的拓扑理论。一文，它给出了强Lefschetz定理公理 (8) 的分析证明，并叙述了de Rham上同调群的Lefschetz分解这对于理解标准猜想是必须的。 Hodge Theory and Complex Algebraic Geometry. 这提供了证明(W3)的一个思路首先用Hodge理论证明(W3)在高维复射影簇上的类比由Serre完成，再考虑如何将这个证明移植到特征. 作为闭链映射的象，必然有同调类(1,1). 另一方面， Lefschetz. 一个良定义显然要求我们首先证明一个类似(1,1)类-定理的Lefschetz型定理. Standard Conjectures on Algebraic Cycles. Coherent sheaf) ，...

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Bibliography | Climbing Mount Bourbaki

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Here is a (periodically updated) list of books and sources that I have referred to, or plan to in the future, sorted by field:. Representation Theory: A First Course,. By William Fulton and Joseph Harris. Commutative Algebra: With a View Towards Algebraic Geometry,. An introduction to Homological Algebra,. Introduction to Commutative Algebra. By Michael Atiyah and Ian Macdonald. Linear Representations of Finite Groups. Lie Groups and Lie Algebras. Introduction to Lie Algebras and Representation Theory.

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Weil猜想漫谈：幕间 VIII | Fight with Infinity

https://zx31415.wordpress.com/2016/10/20/weil猜想漫谈：幕间-viii

Wir müssen wissen, wir werden wissen. 在进入对Deligne I的讨论前，让我们轻松一下假想现在是60年代而我们是尚未提出标准猜想的 Grothendieck，让我们试着来证明. 弱Lefschetz定理公理 7 立即派上了用场与Poincaré对偶公理 4 相结合，并应用归纳假设，. 成立，由Künneth公式公理 5 ，. 我们提醒读者，我们曾在漫谈 III 第二主题. 基于这个简单的推理，Grothendieck曾做过非常天真的猜想立即被Serre证否了. Leave a Reply Cancel reply. Enter your comment here. Fill in your details below or click an icon to log in:. Email (Address never made public). You are commenting using your WordPress.com account. ( Log Out. Notify me of new comments via email.

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月旦 XI | Fight with Infinity

https://zx31415.wordpress.com/2017/01/01/月旦-xi

Wir müssen wissen, wir werden wissen. For December, 2017. AlphaGo的故事还没有结束有传言说，现今的中国围棋第一人柯洁已开始使用Google提供的AlphaGo程序进行日常训练这同时也是对AlphaGo稳定性的进一步测试在与李世乭的五番棋大战中，程序曾经出现过一次明显的崩溃，直接导致了败局，并将在合适的时候代表人类再次挑战AlphaGo. 通过机器学习技术训练出的机器人杀手已在摩拳擦掌随时准备进入战场，以致联合国下辖的特定常规武器公约缔约方开始严肃考虑禁止此类机器人的应用. 这是美国粒子物理学界寄望于中国的理由他们厌倦了和国会以及他们背后的庸众打交道，企图抄近道。讽刺的是，或许这也会成为这个项目最终无疾而终的理由因为一句话打开的大门也可能因为一句话而关上。因为政府和公众的回答一定是不能不能不能. 同样是一个关于量子系统控制的问题，但难度要高出不少有些人认为难到不可能的程度。还有太多细节需要被梳理，被归纳，from bottom to top. 的形态我们又从十回到了一。

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Weil猜想漫谈 VII：插曲 | Fight with Infinity

https://zx31415.wordpress.com/2016/10/18/weil猜想漫谈-vii：插曲

Wir müssen wissen, wir werden wissen. 本章是漫谈的插曲提及的所有结果都和Weil猜想的证明或者更严格地说，Deligne I 中给出的证明没有直接关联。 Weil除子模去有理等价线性等价关系给出我们熟悉的除子类群/ Picard群. 模去代数等价关系给出有限生成的 Néron Severi群. 上，代数等价、同调等价和数值等价是重合的 (Matsusaka)，这个结论给人以强烈的暗示经典等价关系是否在本质上只有2类呢 Griffiths找出了反例. Conjectures in Arithmetic Geometry. Lectures on Algebraic Cycles. 这是后续许多发展的源头在Voison, Vol.2中详细介绍了和复几何相关的部分。本章引入的概念太少，我们只能努力尝鼎于一脔，考虑经典的Abel-Jacobi定理的一种推广却期望但愿不是奢望读者能够食髓知味，瞥见Chow环与Hodge结构的微妙关系. 在无法用有限对象表出的意义上是大的。 Well, no promise! 160; ↩.

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Pauli矩阵，表示论与Kähler恒等式 | Fight with Infinity

https://zx31415.wordpress.com/2012/04/08/pauli矩阵，表示论与kahler恒等式

Wir müssen wissen, wir werden wissen. Leave a Reply Cancel reply. Enter your comment here. Fill in your details below or click an icon to log in:. Email (Address never made public). You are commenting using your WordPress.com account. ( Log Out. You are commenting using your Twitter account. ( Log Out. You are commenting using your Facebook account. ( Log Out. You are commenting using your Google account. ( Log Out. Notify me of new comments via email. Notify me of new posts via email.

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月旦 IX | Fight with Infinity

https://zx31415.wordpress.com/2016/10/31/月旦-ix

Wir müssen wissen, wir werden wissen. For October, 2016. 老爷子似乎不太满意当前研究几何Langlands纲领的主流方式我称之为 from top to bottom。 From bottom to top ，他重新提出了相当天真的问题. 过这样的图像复几何和辛几何是同一棵树上镜像对称的不同分支，伪全纯曲线和Picard-Lefschetz理论生长在比Floer同调和Fukaya范畴更加靠近根部的地方。那么，同样 from bottom to top ，我们应该从这棵树的根部开始强调复几何和辛几何的统一性据我所知，还没有人发展过这样的辛/复统一理论，即使是从与II型弦论类比的角度。我认为一个极具潜力的大方向是探索经典变换的范畴化. 此外，至少还有一个经典变换值得被范畴化，即Legendre变换无论怎么看，这个变换都应该是理解辛几何的关键工具例如，将这个变换和Morse理论-Lefschetz纤维化的平行性一同加以考察，但奇怪的是似乎从来没有人从范畴化的角度严肃考虑过它。 You are commenting us...

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Weil猜想漫谈 III：第二主题 | Fight with Infinity

https://zx31415.wordpress.com/2016/10/03/weil猜想漫谈-iii：第二主题

Wir müssen wissen, wir werden wissen. 通常在介绍Weil猜想时，第二主题的进入会被推迟到末乐章即Deligne完成证明的最后一步。因此，我们决定在发展第一主题之前，让第二主题先行进入。是精确到常数意义上唯一的权为12的尖点形式. 存在某种相似性，并提出或许可以通过研究假想的 Ramanujan簇证明. 再往后就是乐曲的高潮 Deligne综合了所有这些发展，得到了一个伟大的交互证明，即借助经Langlands推广后的 Rankin技巧证明了Weil猜想，又利用Weil猜想给出了Ramanujan-Petersson猜想的证明。 Automorphic Forms and Representations. 160; ↩. The Riemann Hypothesis over Finite Fiels: From Weil to the Present Day. 160; ↩. Leave a Reply Cancel reply. Enter your comment here. Tsun-Huai Li on 月旦 X.

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从Bott周期性谈起 Ⅳ | Fight with Infinity

https://zx31415.wordpress.com/2012/01/02/从bott周期性谈起-Ⅳ

Wir müssen wissen, wir werden wissen. K-Theory and the Hopf Invariant. 1)外乘幂运算简单地将向量空间的外乘幂运算搬运到向量丛上。 Vector Fields on Spheres. Leave a Reply Cancel reply. Enter your comment here. Fill in your details below or click an icon to log in:. Email (Address never made public). You are commenting using your WordPress.com account. ( Log Out. You are commenting using your Twitter account. ( Log Out. You are commenting using your Facebook account. ( Log Out. Notify me of new comments via email.

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Eric C. Peterson

Eric C. Peterson. I was a Benjamin Peirce fellow in the Harvard math department, but I moved on to work in quantum computing at Rigetti. Before any of that, I was a graduate student at Berkeley. Before that, I was an undergraduate in computer science at Urbana-Champaign. This URL used to host a group research blog, Chromotopy. My mathematical interests are in using algebro-geometric tools to answer questions in algebraic topology, and I have a penchant for computations. ArXiv preprint available,. With Aa...

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