Math 320 Differential Equations and Linear Algebra at UW-Madison, Spring 2024
Math 222 Calculus and Analytic Geometry II at UW-Madison, Spring 2023
Math 222 Calculus and Analytic Geometry II at UW-Madison, Fall 2022
Math 742 Abstract Algebra at UW-Madison, Spring 2023
Math 764 Algebraic Geometry I at UW-Madison, Fall 2023
I was a graduate peer mentor for the incoming Math PhDs for 2023-2024.
Directed Reading Course Abstract.
The goal of this reading course is to read "Linear Representations of Finite Groups" by J.P. Serre. The general plan is to cover most of chapters 1-5.
Directed Reading Course Abstract.
We shall be reading "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea. The general plan is to cover most of the material in the earlier chapters. A link to an online pdf can be found here.
I think the best way to do a DRP like this is to have someone give a talk each meeting.
Prerequisites: Mathematical maturity. I expect that most people will need a minimal level of background in algebra. The book is quite readable.
Directed Reading Course Abstract.
We shall be reading "Algebraic Curves and Riemann Surfaces" by Rick Miranda. The ambitious goal will be to cover most of Chapters I-VI which has the proof of the Riemann-Roch Theorem and a statement of Serre Duality. The course itself should serve as a gentle introduction to many notions in basic algebraic geometry as well as the deep connections between complex geometry and algebraic geometry. The first chapter covers the basic definitions of Riemann surfaces (as second countable Hausdorff spaces with a complex structure) and standard examples. The following three chapter present examples of Riemann surfaces, the study of functions on these surfaces, and operations that can be done on them such as integration. For integration, we shall discuss integration on surfaces (which is helpful intuition if one wants to study integration on manifolds) and we prove a version of Stoke's Theorem, the Residue Theorem, and utilize homotopy and homology. Afterwards, Chapters V and VI cover the theory of divisors and meromorphic functions which can be used to specify maps or embeddings into projective space and classify these surfaces based on a notion of "genus" (in fact, there are three proven equivalent definitions).
The course itself will barely scratch at the connection between Riemann surfaces and algebraic curves, but it should equip students with enough background to pursue that connection on their own time.
The student should expect approximately 2-4 hours of reading each week on their own time. During our meetings, I shall ask questions in an attempt to provoke student insights and ask that students come to the DRP meetings prepared with some questions.Â
Additional References to have at hand.
For the complex analysis, a great undergraduate level textbook is Complex Variables and Applications by Churchill and Brown (an online version is available). I personally use Conway's Functions of One Complex Variable I or Rudin's Real and Complex Analysis.
For the differentiable manifolds, a good reference textbook is Lee's Introduction to Smooth Manifolds, but a good expository textbook is Tu's An Introduction to Manifolds (which has a section on computing de Rahm cohomology groups for a Riemann surface of genus g).
For the abstract algebra, I personally use either Dummit and Foote's Abstract Algebra 3rd Edition or Lang's Algebra Revised 3rd Edition.
Schedule.
Currently, we meed Tuesdays 4PM - 5PM.
Review Sheet.
Linked below shall be a review which summarize the main ideas, heuristics, and techniques of each chapter.
Link to the Review Sheet. Last updated: Never.