Teaching | Keerthi Madapusi

Academic Year 2025–2026

Fall 2025

MATH 3311: Algebra I

View Lectures, Notes & Videos

Lec	Topic	Notes	Video
40	Constructing finite fields using irreducible polynomials
39	Finite fields of order $4$
38	Uniqueness part $II$; fields
37	Uniqueness part of Fundamental Theorem for $p$-groups
36	The existence part of the fundamental theorem for finitely generated abelian groups
35	Finding the Smith Normal Form
34	HW $11$ Problem $9$; the Smith Normal Form
33	Finitely generated abelian groups $III$: reduction to a matrix problem
32	Finitely generated abelian groups $II$: subgroups of $Z^m$
31	Finitely generated abelian groups $I$
30	Semi-direct products $III$; finitely generated groups
29	Semi-direct products $II$
28	Semi-direct products $I$
27	Complements and internal direct products
26	Midterm $2$ debrief
25	Conjugacy classes in $S_n$ and $A_n$; midterm review
24	Odd and even permutations; cycle types and conjugacy
23	The sign homomorphism
22	Recap of Sylow theorems; cycle notation; the sign homomorphism
21	Sylow Theorem $C$
20	Sylow Theorem $B$
19	Proof of Sylow Theorem $A$
18	Subgroups of quotient groups
17	Some applications of the factoring triangle
16	The factoring triangle
15	Quotient groups and quotient homomorphisms
14	Some midterm problems; Orbit-stabilizer $II$
13	Cosets and isomorphisms between group actions
12	Cosets and Lagrange's theorem; review
11	Proof of the orbit-stabilizer formula; (left) cosets
10	Orbits, stabilizers and orbit-stabilizer
9	Two perspectives on group actions; stabilizers and orbits
8	Subgroups; group actions
7	The dihedral and symmetric groups; subgroups
6	Cyclic and abelian groups; $(Z/mZ)^\times$; the dihedral groups
5	Some basic facts about groups and group homomorphisms
4	Groups and group homomorphisms
3	Bijections preserving particular structure
2	Symmetries of the fifth roots of $1$
1	Introduction to algebra

Earlier years

Spring 2023

MATH 8822: The Geometric Satake Equivalence

View Lectures & Videos

Lec	Topic	Video
22	Properties of semi-infinite orbits I
21	The Bruhat-Tits building and semi-infinite orbits for GL_2
20	The monoidal structure on the cohomology functor
19	Musings on the monoidal structure of the cohomology functor
18	The commutativity constraint using fusion
17	The Beilinson-Drinfeld affine Grassmannian
16	Exactness of convolution II
15	Exactness of convolution I
14	Convolution in the Satake category
13	Wrapping up semisimplicity
12	Semisimplicity of the Satake category
11	Towards semisimplicity of the Satake category
10	A haphazard introduction to perverse sheaves
9	Closure relations
8	Moduli description of strata in general
7	Examples of open strata
6	The Schubert stratification
5	Loop groups
4	Ind-projectivity of the affine Grassmannian
3	Projectivity of the classical Grassmannian
2	A quick tour of functor-of-points approach to schemes
1	Introduction

Spring 2022

Derived Algebraic Geometry and Shimura Varieties

Course Description & Syllabus

This graduate course explores the intersection of derived algebraic geometry and the theory of cycles, specifically focusing on their applications to the geometry and cohomology of Shimura varieties.

Key Topics:

Construction of $\infty$-categories and the Yoneda lemma.
The process of animation and animated commutative algebra.
The cotangent complex, deformation theory, and obstruction theory.
Quasi-smooth schemes, subschemes, and the construction of cycle classes.

View Lectures & Videos

Lec	Topic	Video
10	Quasi-smooth subschemes and cycle classes
9	A quasi-smooth scheme of homomorphisms of abelian schemes
8	Derived homomorphisms between abelian schemes
7	Deformations of maps between abelian schemes
6	Obstruction theory and tangent spaces of mapping prestacks
5	The cotangent complex and deformation theory
4	Animated commutative algebra
3	Animation
2	Functor categories and the Yoneda lemma
1	How to produce infinity categories?

Primary References & Reading

[Lur09] J. Lurie, Higher Topos Theory. This is the most complete reference for the basic theory of infinity categories, though of course it's quite the chunky morsel.
[Lur17] J. Lurie, Higher Algebra. Here is where you'll find the basics of stable infinity categories (with a complete proof of the Dold-Kan correspondence), as well as all the gory details behind symmetric monoidal infinity categories and commutative algebra objects in such.
[Cis19] D.-C. Cisinski, Higher Categories and Homotopical Algebra. A more accessible and somewhat different perspective than Lurie's on the construction and properties of infinity categories.
[MG19] A. Mazel-Gee, A user's guide to co/cartesian fibrations. A user's guide to co/cartesian fibrations : A nice primer on how the Grothendieck construction works, and how it generalizes to the infinity category theoretic context.
[MG16] A. Mazel-Gee, UC Berkeley Thesis. The introductory Chapter 0 of this thesis is very well-written and lays out the general paradigm of infinity categorical thinking, albeit in sometimes amusingly breathless terms.
[Mao20] Y. Mao, Revisiting derived crystalline cohomology. A very clear exposition of how the process of animation works as well as how to make calculations in it.
[Kha16] A. Khan, Descent in algebraic K-theory. The first few lectures here give a nice exposition of the basics of derived algebraic geometry

Fall 2018

MATH 8821: Number Theory I, $p$-adic Hodge theory

An introduction to $p$-adic Hodge theory and the Fargues-Fontaine curve.

Notes by Dalton Fung