Covariance

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Covariance

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00-09 Atlas

🗺️ Map of Content - Algebra Qual

🗺️ Map of Content - Algebra Theory

🗺️ Map of Content - Category Theory

🗺️ Map of Content - Differential Equations

🗺️ Map of Content - Modules

🗺️ Map of Content - Ring Theory

10-19 Teaching

11 Classes

MATH 561 - Graduate Algebra

2025 - Fall

Homework

Study Guides

Study Guide for Final Exam

Study Guide for Midterm Exam

Daily class summaries

Midterm Exam Solutions

Exercises

A criterion for a module to be finitely generated

Alternate characterization of rank

An intramural isomorphism

Annihilators of torsion modules

Cokernels in the category of abelian groups

Direct sum of free modules

Direct sums and injective, projective, flat

Equalizers and coequalizers of matrices

Equivalence relations on sets

Finite abelian groups are neither injective nor projective

Group morphisms that cannot define module morphisms

Groups as categories

Initial and terminal objects

Irreducible modules

Monics and epics

Power set functors

Primes and annihilators

Quotients of cyclic modules

Rank and quotients

Rational and Jordan canonical forms

Ring property from module property

Submodules via ideals

Tensor product of projective modules is projective

Tensor product of rings is a coproduct

The evaluation map

The pullback functor is an adjoint

The Short Five Lemma

There is no 'center' functor on Grp

Torsion Z-modules

Torsion submodules

14 Algebra Qual

Previous exams

Algebra Qual 2014-03

Algebra Qual 2014-09

Algebra Qual 2015-03

Algebra Qual 2015-09

Algebra Qual 2016-03

Algebra Qual 2016-09

Algebra Qual 2017-03

Algebra Qual 2017-09

Algebra Qual 2018-06

Algebra Qual 2018-09

Algebra Qual 2019-01

Algebra Qual 2019-06

Algebra Qual 2019-09

Algebra Qual 2020-01

Algebra Qual 2020-06

Algebra Qual 2020-09

Algebra Qual 2021-01

Algebra Qual 2021-06

Algebra Qual 2021-09

Algebra Qual 2022-01

Algebra Qual 2022-06

Algebra Qual 2022-09

Algebra Qual 2023-01

Algebra Qual 2023-06

Algebra Qual 2023-09

Algebra Qual 2024-01

Algebra Qual 2024-06

Algebra Qual 2024-09

Algebra Qual 2025-01

Problem bank

Master lists

Pool problems in group theory

Pool problems in linear algebra

Pool problems in ring theory

Template problems in group theory

Template problems in linear algebra

Template problems in ring theory

The Problem Bank (List)

Pool problems

Group theory

A condition to be non-cyclic

A condition under which a group must be abelian (2)

A dihedral group that is not an internal direct product

A group isomorphic to a subgroup of a direct product of quotient groups

A group isomorphic to an internal direct product

A group of upper-triangular matrices

A group with a trivial automorphism group (2)

A group with a trivial automorphism group

A property of the order of an element

An automorphism of a group of odd order

Another condition for a group to be abelian

Automorphisms of a finite cyclic group

Centralizers in symmetric groups

Closely related subgroups of a finite group

Comparing cosets

Computations with inner automorphisms

Elements of finite order

Elements of order 2

Existence of a normal subgroup of finite index

Existence of an identity element in a group

Existence of automorphisms

Image of a normal subgroup and induced morphisms

Image of the identity is the identity

Indices and intersections

Inner automorphisms and the center of a group

Nonexistence of a simple group of a given order

Nonexistence of small nonabelian groups

Normal subgroups with trivial intersection

Normality and the operation on cosets (defunct)

Normality and the operation on cosets

Normalizer of a subgroup

Normalizers and centralizers

Order of a power of an element

Order of an element in a finite group

Orders of elements in a quotient group

Preimage of a subgroup

Product of two subgroups

Products of quotient groups

Projection onto a quotient

Properties of the center of a group

Property of the order of an element

Proving Lagrange's Theorem

Stabilizer of a coset

Subgroups of a group of even order

The integers as a subgroup of the rationals

The Third Isomorphism Theorem

Linear algebra

A condition ensuring diagonalizability

A property of surjective linear transformations

An upper bound on the number of nonzero eigenvalues

Analyzing an unusual matrix

Direct sums and idempotent transformations

Dot product and cross product as linear transformations

Eigenvalues and eigenspaces of a matrix with a given property

Eigenvectors and linear independence

Eigenvectors of commuting linear transformations

Eigenvectors with distinct eigenvalues are linearly independent

Linear dependence and relations

Linear endomorphism of a vector space of matrices

Matrix representing a linear transformation

Numerical range of a linear transformation

Projections and adjoints

Properties of transpose

Skew-symmetric matrices

Sum and union of subspaces

Ring theory

A ring in which all prime ideals are maximal

Automorphisms of a ring

Boolean algebras

Boolean rings are commutative

Characteristic of a ring (2)

Characteristic of a ring

Dimension of a PID

Euclidean domains are PIDs

Existence of an identity element in a finite ring

Generator for a field extension

Ideals in a polynomial ring (2)

Ideals in a polynomial ring

Idempotent elements in a ring (2)

Idempotent elements in a ring

Image of the identity element under a ring morphism

Maximal ideals in a PID

Nilpotent elements in a ring

Nilpotent elements

Nonzero prime ideals are maximal in a PID

Polynomials with even constant term

Prime and irreducible elements in a commutative ring

Prime ideals and quotient rings

Properties of Boolean rings

Properties of the annihilator

Proving an ideal is prime

Quotienting out nilpotent elements

Quotients and direct products

The nilradical of a ring

The structure of the integers as both a group and a ring

Template problems

Group theory

A condition under which a group must be abelian

Computations in symmetric groups

Computing an automorphism group

Counting morphisms between specified groups

Directly proving the existence of an element of a desired order

Finding all morphisms between two groups

Inner and outer automorphisms

Inner automorphisms of an alternating group

Nonexistence of morphisms between two groups

Order of elements in a symmetric group

The cyclic group of order 2020

Verifying axioms of a group

Working modulo 11

Linear algebra

A linear transformation from a vector space of polynomials

A vector space of square matrices

Bases for a subspace and its orthogonal complement

Computations with a given linear transformation

Diagonalization and matrix powers

Diagonalization of a given matrix (2)

Diagonalization of a given matrix

Dimension of a subspace and its orthogonal complement

Jordan canonical form of a matrix (2)

Jordan canonical form of a matrix

Linear transformation from a vector space of polynomials

Matrix and characteristic polynomial for a given linear transformation

Matrix and eigenvalues of a given linear transformation

Orthogonal complements

Orthogonal projection and scaling

Orthogonal projection onto a line (2)

Orthogonal projection onto a line (3)

Orthogonal projection onto a line

Orthogonal projection onto a plane (2)

Orthogonal projection onto a plane

Projection onto a plane

Radial expansion from a fixed line

Reflection across a plane

Rotation around an axis

Ring theory

A maximal ideal in a function ring

An evaluation morphism

An isomorphism of rings

Constructing a field extension

Constructing the field with eight elements

Evaluation at i

Existence of certain ring morphisms

Group of units of a product

Image of an evaluation morphism

Morphism from the Gaussian integers (2)

Morphism from the Gaussian integers

The field with eight elements

The field with nine elements

The kernel of an evaluation morphism

Using the Chinese Remainder Theorem

The Algebra Qual

20-29 Research

24 Summer REUs

2024

Summer REU 2024 - Tropical conic duality

2025

Categorical representation theory

Summer REU 2025 - Categorical representation theory

Project Single Arrow

Summer REU 2025 - Project arrow

40-49 Knowledge

41 Mathematics

Algebra theory

Exterior algebras

Symmetric algebras

Symmetric and alternating tensors

Tensor algebras

Algebraic geometry

Examples of classical conic duality

Calculus

Determining and classifying local extrema

Determining constrained extrema

Category theory

Abelian categories

Abelian categories

Additive categories

Chain complexes

Diagram chases without elements

Double complexes and mural maps

Exact sequences and chain homology

Spectral sequences

The Salamander Lemma

Adjoints

Examples of adjoints

Properties of adjoints

Basic structures

Functor categories

Natural transformations

Special morphisms

Kan extensions

Universal properties

Universal arrows and elements

Universal Properties I - Inspiring Examples

Universal Properties II - Commutative diagrams, cones and limits

Universal Properties III - Yoneda's Lemma

Differential equations

Fourier series

Fourier series solutions I - A problem with power series

Fourier series solutions II - Periodic functions

Fourier series solutions III - Series representations of periodic functions

Fourier series solutions IV - Computing Fourier coefficients

Fourier series solutions V - Computing some Fourier series

Fourier series solutions VI - Inner product spaces

Fourier series solutions VII - Solving differential equations with Fourier series

Fourier transform

Fourier transform I - Pushing Fourier series to the limit

Fourier transform II - The Fourier transform and inverse transform

Fourier transform III - Properties of the Fourier transform

Fourier transform IV - Derivatives

Fourier transform V - Convolution

Fourier transform VI - Solving differential equations

Frobenius series solutions

Frobenius Series Solutions I - Slightly generalizing power series

Frobenius Series Solutions II - Some illustrative examples

Frobenius Series Solutions III - Frobenius theory

Laplace transform

Laplace transform I - Desperate times

Laplace transform II - The inverse Laplace transform

Laplace transform III - The Shifting Theorems

Laplace transform IV - Convolution redux

Laplace transform V - Solving differential equations

Power series solutions

Power Series Solutions I - From polynomials to power series

Power Series Solutions II - First steps with power series

Power Series Solutions III - A more representative example

Power Series Solutions IV - Analytic functions and ordinary points

Field theory

A question about finite fields

Group theory

Basic definitions and examples

Normal subgroups

Module theory

Basic definitions and examples

Module morphisms and submodules

Module morphisms

Bimodules

Bimodule morphisms

The 2-category of bimodules

Constructions on modules

Direct products of modules

Direct products vs. direct sums vs. sums

Direct sums of modules

Examples of free modules

Generators for modules and submodules

Quotient modules

Sums of submodules

The Isomorphism Theorems for Modules

Exact sequences

Exact Sequences I - Illustrative Examples

Exact Sequences II - Exact Sequences

Exact Sequences III - Morphisms of Exact Sequences

Exact Sequences IV - Exact Sequences and Functors

Modules over a PID

Jordan Canonical Form I - Definition

Jordan Canonical Form II - Computation

Linear independence, rank and the structure of free modules

Modules over a PID - The Fundamental Theorem

Noetherian modules

Rational Canonical Form I - Definition

Rational Canonical Form II - Additional Properties

Rational Canonical Form III - Computation

Tensor products of modules

Tensor Products I - Extending scalars

Tensor Products II - Tensor products of bimodules

Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product

Tensor Products IV - The Adjoint Property

Precalculus

MATH 118 - Section 1.6 Hints

Representation theory

Basic definitions and examples

Representations of groups

Ring theory

Main theorems

Chinese Remainder Theorem

Special types of rings

Principal ideal domains (PIDs)

Least common multiples

Tropical algebraic geometry

Investigation into tropical tangency loci

Linear tropical varieties

Quadratic tropical varieties

Tropical varieties

42 Software

LaTeX

Systems of equations

46 People

Assistant to the Dean

50-59 Logs

51 Class summaries

2024 - Fall

MATH 561

2024-09

2024-10

2024-11

2024-12

2025 - Fall

MATH 561

2025-09

2025-10

2025-11

2025-12

52 Research meetings

2024 - Summer

REU Meeting - 2024-07-02

REU Meeting - 2024-07-05

REU Meeting - 2024-07-09

REU Meeting - 2024-07-12

REU Meeting - 2024-07-16

REU Meeting - 2024-07-19

REU Meeting - 2024-07-23

REU Meeting - 2024-07-26

REU Meeting - 2024-07-30

REU Meeting - 2024-08-02

REU Meeting - 2024-08-06

REU Meeting - 2024-08-09

REU Meeting - 2024-08-13

REU Meeting - 2024-08-20

REU Meeting - 2024-08-23

REU Meeting - 2024-08-27

2025 - Summer

REU 1 - Aaron and Mark

REU Meeting - 2025-06-26

REU Meeting - 2025-06-30

REU Meeting - 2025-07-03

REU Meeting - 2025-07-07

REU Meeting - 2025-07-10

REU Meeting - 2025-07-31

REU Meeting - 2025-08-04

REU Meeting - 2025-08-06

REU Meeting - 2025-08-15

REU Meeting - 2025-08-20

REU Meeting - 2025-08-29

REU 2 - Lena and Liya

REU Meeting - 2025-07-01

REU Meeting - 2025-07-08

REU Meeting - 2025-08-06

REU Meeting - 2025-08-11

REU Meeting - 2025-08-20

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Hom(R,R) is R

Suppose $R$ is a commutative ring. Prove that ${Hom}_{R} (R, R)$ and $R$ are isomorphic as rings.

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