I am a first-year PhD student at the University of Michigan, Ann Arbor. My mathematical interest lies in the intersection of number theory, algebraic geometry, and homotopy theory.

I was an undergraduate student at the University of Chicago from 2020 to 2024. Before that, I spent my high school years in the beautiful city of Maastricht, the Netherlands. Before that, I lived a happy life in Hefei, China.

Here is my cv. You can reach me at: yhsheng at umich dot edu.

Expository writings

Notes on real orientations. pdf
This set of notes is written for a reading course supervised by Prof. Peter May, where I intend to explain in detail the results and argument of the paper Real orientaions of Lubin-Tate spectra by Jeremy Hahn and XiaoLin Danny Shi. It also contains an exposition on the Ando-Blumberg-Gepner-Hopkins-Rezk construction of the Thom spectra functor.

Cofreeness of Lubin-Tate theory. pdf
Final paper written for the 2023 UChicago REU. This article is a detailed survey of Section 3 of the 2020 paper The chromatic Nullstellensatz by Burklund, Schlank, and Yuan. It aims to prove that at the level of algebras, Lubin-Tate theory E is cofree, i.e., right adjoint to some forgetful functor forgetting the structure of power operations. This builds on my previous notes on power operations.

Power operations and HKR character theory. pdf
These are some notes I took while reading Stapleton’s survey paper Lubin-Tate theory, character theory, and power operations. It is about extending the power operations on Morava E-theory to the codomain of the Hopkins-Kuhn-Ravenel character map. Important results by Strickland and Ando-Hopkins-Strickland on the moduli-theoretic interpretation of Lubin-Tate theory is introduced.

Phylogenetic Treespace. pdf
Slides of a participant talk given at the 2023 PCMI Undergraduate Summer School. It is based on the paper Geometry of the Space of Phylogenetic Trees by Billera, Holmes, and Vogtmann.

Complex multiplication of elliptic curves and abelian varieties. pdf
Final paper written for the 2022 UChicago REU. Hilbert’s twelfth problem asks: given an abelian extension of a number field K, what are its generators? When K is an imaginary quadratic field, complex multiplication provides a solution. It turns out that in this case, the generators are closely related to certain invariants attached to an elliptic curve whose endomorphism group is the ring of integers of K.

Complex Multiplication. pdf
Slides of a participant talk given at the 2022 UChicago REU.

On realizing rational and polynomial cohomology rings. pdf
Final paper written for the 2021 UChicago REU. This paper focuses on the following inverse cohomology problem: which graded R-algebras can be realized as the cohomology ring (with coefficients in R) of a topological space? We explain in detail the 1976 paper of Bousfield and Gugenheim, On PL de Rham Theory and Rational Homotopy Type, which solves the problem for R is the ring of rational numbers. The paper uses Quillen’s model categories in an essential way.

Elements of complex K-theory. pdf
Final project for the course Proseminar in Mathematics with Prof. Akhil Mathew. I focused on explaning how topological K-theory is related to singular cohomology via Chern characters, the Atiyah-Hirzebruch spectral sequence, and the Adams operation.

Kähler manifolds and Hodge theory. pdf
Final project for the course Topology and Geometry 2: Differential Topology with Prof. Eduard Looijenga. I wrote about the basics of complex and Kähler manifolds, and outlined a proof of Hodge decomposition assuming the regularity lemma from functional analysis. I also discussed some applications of Hodge theory.

Commutative ring theory. pdf
Class notes for the course Algebra II (commutative algebra)* with Prof. Ngo Bao Chau.

Brownian motion and stochastic calculus. pdf
Class notes for the course Brownian motion and stochastic calculus* with Prof. Greg Lawler.

Miscellaneous

I play piano, organ, and carillon. You can access some of my recordings on Youtube or Bilibilibi. Here are some recent ones:

J. S. Bach: Prelude and Fugue in E Major from WTC Book II (BWV 878)
Gubaidulina: Chaconne
Medtner: Sonata-Reminiscenza from Forgotten Melodies, Op. 38 No. 1
Organ recital (Bach, Brahms, & Messiaen)

I am broadly interested in philosophy (Aristotle, Hegel, and Kierkegaard etc.). I often find connections between music and philosophy, so I write them down when I get a chance. They are mostly in Chinese, although I do plan to write in English in the future. Here are some of them:

精神的归程——贝多芬降A大调第三十一号奏鸣曲Op. 110的演奏与哲学解读
A Hegelian phenomenological interpretation of Beethoven’s piano sonata Op. 110.
追寻钟声的脚步
A Documentation of my trip to Europe where I played on different caillons.
幻想曲式：北德管风琴学派与巴赫
On the topic of Stylus Phantasticus as represented by the north German organ school.
一篇德语翻译练习
A Chinese translation of a paper in German on Bach’s Pièce d’Orgue BWV 572.