The Geometry of Rings and Schemes (Rényi Institute/ELTE)

In the fall semester of 2024 I taught a course on scheme theory; the details were as follows:

  • Title: The Geometry of Rings and Schemes (listed in the ELTE catalogue as "Sémák")
  • Time: Tuesdays 15:30-17:00
  • Location: Kutyás Terem, Rényi Institute
  • Description: An introduction to scheme theory with an emphasis on motivating the core constructions and developing geometric intuition. The main prerequisites are point-set topology and introductory commutative ring theory — basic knowledge of category theory, differential geometry, and more classical approaches to algebraic geometry will also be useful but are not strictly necessary.
  • Reference Texts: Robin Hartshorne's Algebraic Geometry, Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry (July 2024 version), and David Eisenbud and Joe Harris' The Geometry of Schemes.

The lectures were as follows; for each, I've posted typed notes and listed the parts of each reference text roughly corresponding to the content covered:

  • Lecture 1: Introduction (2024/09/10)
    • Hartshorne: Introduction, Section II.2
    • Vakil: Preface, Sections 3.2-5
    • Eisenbud & Harris: Introduction, Sections I.1.1-2
  • Lecture 2: Schemes (2024/09/17)
    • Hartshorne: Section I.1, Sections II.1-2
    • Vakil: Sections 2.1-3, Section 2.5, Sections 3.1-5, Section 4.1, Sections 4.3-4, Section 5.3, Sections 7.1-3
    • Eisenbud & Harris: Section I.1, Sections I.2.3-4, Section I.4, Section II.1.1
  • Lecture 3: Fiber Products and Such (2024/09/24)
    • Hartshorne: Sections II.2-3
    • Vakil: Section 8.3, Sections 9.1-2, Sections 10.1-4
    • Eisenbud & Harris: Sections I.3.1-2, Section III.1.1
  • Lecture 4: Some Properties of Schemes (2024/10/01)
    • Hartshorne: Section I.1, Sections II.2-3
    • Vakil: Section 3.6, Section 4.3, Sections 5.1-3
    • Eisenbud & Harris: Sections I.2.1-2
  • Lecture 5: Dimension Theory and Singular Points (2024/10/08)
    • Hartshorne: Section I.1, Section I.5, Section II.3
    • Vakil: Section 12.1, Section 12.3, Sections 13.1-2
    • Eisenbud & Harris: Section I.2.2, Section II.3.1
    • Bonus: Chapters 8-10 of Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry
  • Lecture 6: The Geometry of Modules (2024/10/15)
    • Hartshorne: N/A
    • Vakil: N/A
    • Eisenbud & Harris: N/A
  • Lecture 7: Quasicoherent Sheaves I (2024/10/22)
    • Hartshorne: Section II.5
    • Vakil: Sections 6.1-4, Sections 14.1-5, Section 17.1
    • Eisenbud & Harris: Section I.2.1, Section I.3.3
  • Lecture 8: Quasicoherent Sheaves II (2024/11/05)
    • Hartshorne: Section II.5
    • Vakil: Sections 6.1-4, Sections 14.1-5, Section 17.1
    • Eisenbud & Harris: Section I.2.1, Section I.3.3
  • Lecture 9: Calculus on Schemes I (2024/11/12)
    • Hartshorne: Section II.8
    • Vakil: Section 13.2, Sections 21.1-2, Section 21.6
    • Eisenbud & Harris: Section V.3.1
    • Bonus: Chapter 16 of Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry
  • Lecture 10: Calculus on Schemes II (2024/11/19)
    • Hartshorne: Section II.8
    • Vakil: Section 13.2, Sections 21.1-2, Section 21.6
    • Eisenbud & Harris: Section V.3.1
    • Bonus: Chapter 16 of Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry
  • Lecture 11: Separated and Proper Maps (2024/11/26)
    • Hartshorne: Section I.6, Section II.4
    • Vakil: Section 11.2, Section 11.4, Section 13.5, Section 13.7
    • Eisenbud & Harris: Section III.1.2
  • Lecture 12: Projectivization I (2024/12/03)
    • Hartshorne: Section II.2, Section II.5, Section II.7
    • Vakil: Section 4.5, Section 7.4, Section 9.3, Section 14.6, Sections 15.1-3, Section 15.7, Sections 16.1-2, Sections 17.2-3
    • Eisenbud & Harris: Section I.2.4, Chapter III
  • Lecture 13: Projectivization II (2024/12/10)
    • Hartshorne: Section II.2, Section II.5, Section II.7
    • Vakil: Section 4.5, Section 7.4, Section 9.3, Section 14.6, Sections 15.1-3, Section 15.7, Sections 16.1-2, Sections 17.2-3
    • Eisenbud & Harris: Section I.2.4, Chapter III
  • Lecture 14: Blowups and Normal Cones (2024/12/17)
    • Hartshorne: Section II.7, Sections III.9-10
    • Vakil: Section 13.6, Chapter 22, Section 24.1, Section 24.5, Section 24.8, Sections 26.1-2
    • Eisenbud & Harris: Section II.3.4, Section IV.2

Derived Categories Mini-Course (Rényi Institute Junior Singularities Seminar)

In the summer of 2024 I gave an informal series of talks on derived categories, with the goal of communicating a high-level perspective on the basic concepts and constructions without getting into any farther into technical details than is necessary. The basic reference materials I based the talks around were as follows:

  • Chapters 1, 2, & 10 of Charles Weibel's An Introduction to Homological Algebra (1994).
  • Chapters 4 & 5 of Laurenţiu Maxim's Intersection Homology and Perverse Sheaves with Applications to Singularities (2019).

Supplementary references include the following:

  • Armand Borel et al.'s Intersection Cohomology (1984).
  • Robin Hartshorne's Algebraic Geometry (1977)
  • Birger Iversen's Cohomology of Sheaves (1986).
  • Masaki Kashiwara and Pierre Schapira's Sheaves on Manifolds (2002).
  • Saunders Mac Lane's Categories for the Working Mathematician (1971).

For each talk, I suggested some possible readings to do in advance, aimed at background I wasn't covering in detail or related topics not discussed during the seminar. Here are the suggestions by talk:

  • Week I: Preliminaries
    • Sections I.3 & I.4 of Mac Lane's book or Sections A.3 & A.4 of Weibel's.
    • Chapter VIII of Mac Lane's book or Section A.4 of Weibel's.
    • (more optional) Section 1.6 of Weibel's book or some subset thereof.
  • Week II: Derived Category and Functors
    • Section 10.3 of Weibel's book.
    • (more optional) Chapter 2 of Weibel's book or some subset thereof.
    • (more optional) Chapter 5 of Weibel's book or some subset thereof.
    • (more optional) Section II.1 of Hartshorne's book and/or Section(s) 4.1ff of Maxim's.
  • Week III: Exact Triangles and Triangulated Categories
    • Section 1.2 of Weibel's book, particularly 1.2.7 and after.
    • Section 1.3 of Weibel's book.
    • Section 1.5 of Weibel's book.

Geometry/Topology SEP (UW-Madison)

In the summers of 2021 and 2023 I taught the Summer Enhancement Program for the University of Wisconsin-Madison's geometry/topology qualifying exams. The 2023 worksheets for both versions of the exam, which are composed primarily of exam problems from prior years, can be found below.