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

This semester I'm teaching a course on scheme theory — for those registering through ELTE, the course title is listed as "Sémák" and the instructor as András Némethi. Additional details are as follows:

  • 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.

For each week, I'll try to point out the parts of each reference text roughly containing the material covered in the lecture. I may also post typed lecture notes, but I reserve the right to stop doing this if it ends up being too much of a hassle - no guarantees!

  • 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.