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Fall term 2019/2020
Material from earlier semesters

Math 542: Modern Algebra, Spring 2016

Math 542 Syllabus Syllabus

Math 542 Homework Homework
(To be updated as we go along)

Lecture 1, W 1/20
Rings of fractions

Lecture 2, F 1/22
The field of fractions

Lecture 3, M 1/25
The Chinese remainder theorem; example

Lecture 4, W 1/27
The Chinese remainder theorem

Lecture 5, F 1/29
Euclidean domains

Lecture 6, M 2/1
Euclidean domains and universal side divisors

Lecture 7, W 2/3
Principal ideal domains

Lecture 8, F 2/5
Unique factorization domains

Lecture 9, M 2/8
A PID is a UFD

Lecture 10, W 2/10
Primes in the Gaussian integers

Lecture 11, F 2/12
Expressing an integer as the sum of two squares

Lecture 12, M 2/15
Polynomial rings

Lecture 13, W 2/17
The Gauss lemma

Lecture 14, F 2/19
Irreducibility criteria

Lecture 15, M 2/22
The Eisenstein irreducibility condition

Lecture 16, W 2/24
For a finite field the group of units is cyclic

Lecture 17, F 2/26
The group of units for Z mod N

Lecture 18, M 2/29
Modules for rings

Lecture 19, W 3/2
Homomorphisms of Rmodules

Lecture 20, F 3/4
Quotients of Rmodules

Lecture 21, M 3/7
Direct products of Rmodules

Lecture 22, W 3/9
Free Rmodules

Lecture 23, F 3/11
Vector spaces

Lecture 24, M 3/14
Vector spaces and linear transformations

Midterm W 3/16
Midterm exam questions

Midterm Solutions W 3/16
Midterm exam solutions

Lecture 25, F 3/18
Transition matrices

Lecture 26, M 3/28
The tensor product of vector spaces

Lecture 27, W 3/30
The dual of a vector space

Lecture 28, F 4/1
The transpose of a linear transformation

Lecture 29, M 4/4
Determinants

Lecture 30, W 4/6
Properties of the determinant

Lecture 31, F 4/8
The inverse of a matrix

Lecture 32, M 4/11
Modules over a PID

Lecture 33, W 4/13
Torsion modules

Lecture 34, F 4/15
Rank and torsion

Lecture 35, M 4/18
Submodules of free modules over a PID

Lecture 36, W 4/20
Submodules of free modules over a PID, cont.

Lecture 37, F 4/22
Finitely generated modules over a PID

Lecture 38, M 4/25
Elementary divisors and invariant factors

Lecture 39, W 4/27
The rational canonical form

Lecture 40, F 4/29
The Smith normal form

Lecture 41, M 5/2
The Jordan canonical form

Lecture 42, W 5/4
Examples involving the Jordan canonical form

Lecture 43, F 5/6
Finding the eigenvalues of some tridiagonal matrices using hidden symmetry

Final Exam, S 5/8
Final exam questions

Final Exam Solutions, S 5/8
Final exam solutions


Math 475: Combinatorics, Spring 2014.

Math 475 Syllabus Syllabus

Math 320: Linear Algebra and Differential Equations, Spring 2014.

Math 320 Syllabus Syllabus

Lecture 1, W 1/22 Section 1.1

Lecture 2, F 1/24 Section 1.2

Lecture 3, M 1/27 Section 1.3

Lecture 4, W 1/29 Section 1.4

Lecture 5, F 1/31 Section 1.5

Lecture 6, M 2/3 Section 2.1

Lecture 7, W 2/5 Section 2.2

Lecture 8, F 2/7 Section 2.4

Lecture 9, M 2/10 Section 3.1

Lecture 10, W 2/12 Section 3.2

Lecture 11, F 2/14 Section 3.3

Lecture 12, M 2/17 Section 3.4

Lecture 13, W 2/19 Section 3.5

Lecture 14, F 2/21 Sections 3.5,3.6

Lecture 15, M 2/24 Section 3.6

Lecture 16, W 2/26 Section 3.6 (loose ends)

Lecture 17, M 3/3 Section 4.1

Lecture 18, W 3/5 Section 4.2

Lecture 19, F 3/7 Section 4.3

Lecture 20, M 3/10 Section 4.4

Lecture 21, W 3/12 Section 4.5

Lecture 22, F 3/14 Section 4.6

Lecture 23, M 3/24 Section 4.7

Lecture 24, W 3/26 Section 5.1

Lecture 25, F 3/28 Section 5.2

Lecture 26, M 3/31 Section 5.3

Lecture 27, W 4/2 Section 5.5

Lecture 28, F 4/4 Section 6.1

Lecture 29, M 4/7 Section 6.2

Lecture 30, W 4/9 Section 6.3

Lecture 31, F 4/11 Section 6.3 (loose ends)

Lecture 32, W 4/16 Section 7.1

Lecture 33, F 4/18 Section 7.2

Lecture 34, M 4/21 Section 7.3

Lecture 35, W 4/23 Section 7.5

Lecture 36, F 4/25 Section 7.5 (continued)

Lecture 37, M 4/28 Section 8.1

Lecture 38, W 4/30 Section 8.1 (continued)

Lecture 39, F 5/2 Section 8.2

Lecture 40, M 5/5 Section 8.2 (continued)

Lecture 41, W 5/7 Section 8.3

Lecture 42, F 5/9 Section 8.3 (continued)

Math 210: What follows is a list of Math 210 Homework solutions.

Math 210 syllabus for Fall 2012

Typos contains a list of typos in
the text "Finite Mathematics" 5th Edition, by Maki and Thompson.

Section 1.1 HW solutions

Section 1.2 HW solutions

Section 1.3 HW solutions

Section 1.4 HW solutions

Section 1.R HW solutions

Section 2.1 HW solutions

Section 2.2 HW solutions

Section 2.3 HW solutions

Section 2.4 HW solutions

Section 2.R HW solutions

Section 3.1 HW solutions

Section 3.2 HW solutions

Section 3.3 HW solutions

Section 3.4 HW solutions

Section 3.5 HW solutions

Section 3.R HW solutions

Section 4.1 HW solutions

Section 4.2 HW solutions

Section 4.R HW solutions

Section 5.1 HW solutions

Section 5.2 HW solutions

Section 5.3 HW solutions

Section 5.R HW solutions

Section 6.1 HW solutions

Section 6.2 HW solutions

Section 6.R HW solutions

Section 7.1 HW solutions

Section 7.2 HW solutions

Section 7.3 HW solutions

Section 7.R HW solutions

Section 8.1 HW solutions

Section 8.2 HW solutions

Section 8.3 HW solutions

Section 8.R HW solutions

Section 9.1 HW solutions

Section 9.2 HW solutions

Section 9.3 HW solutions

Some of my recent talks are given below.

spinBled2019.pdf
Leonard pairs, spin models, and distanceregular graphs

terwilli2019.pdf
The alternating PBW basis for the positive part of the quantum affine sl2
algebra

catalan2018.pdf
Catalan words and qshuffle algebras

infdim.pdf
An infinitedimensional Square_q module obtained from the qshuffle algebra
for affine sl2

lusztigAutTalk2018.pdf
The Lusztig automorphisms of the qOnsager algebra

tdpairs2018.pdf
Tridiagonal pairs and applications

tbtdpair.pdf
Totally bipartite tridiagonal pairs

eltalk.pdf
Tridiagonal pairs of qRacah type, the Bockting operator, and Loperators
for Uq(L(sl2))

www2016.pdf
Leonard triples of qRacah type; Combinatorics seminar talk Fall 2016

lrtriple60min.pdf
LoweringRaising triples and Uq(sl2); Colloquium talk at DePaul U. 2016

mathclub2015.pdf
Finding the eigenvalues of some tridiagonal matrices using hidden symmetry

skeinUAW2015.pdf
Topological aspects of the Z3symmetric AskeyWilson relations

lrtTalk2014.pdf
LoweringRaising triples and Uq(sl2)

billiard.pdf
Billiard Arrays and finitedimensional irreducible Uqsl2modules

tdpInAlgGrTh.pdf
Tridiagonal pairs in Algebraic Graph Theory

lpAndQtet.pdf
Leonard pairs and the qtetrahedron algebra

univAWalgebra.pdf
The universal AskeyWilson algebra

classTDpair.pdf
A classification of sharp tridiagonal pairs

rahmanSl3.pdf
The Rahman polynomials and the Lie algebra sl3

The pdf file for the following paper is given below.
P. Terwilliger. The incidence algebra of a uniform poset.
Coding theory and design theory Part I. 193212 IMA Vol. Math.
Appl. 20 Springer New York, 1990.

uniformposet.pdf

The pdf file for the following paper is given below.
P. Terwilliger. Leonard pairs and dual polynomial sequences.
The paper was submitted to LAA on May 15, 1987, but never published.

leonardpair.pdf

The pdf file for the following paper is given below.
Counting 4vertex configurations in P and Q polynomial association schemes.
Algebras, Groups, and Geometries 2 (1985) 541554.

counting4vertex.pdf

The following contains the course notes from Math 846 Algebraic Graph Theory,
Spring term 2009.

Part 1 846 notes

Part 2 846 notes

Part 3 846 notes

Part 4 846 notes

Part 5 846 notes

Part 6 846 notes

The following contains the notes from six lectures on distanceregular
graphs given at
De La Salle University in Manila, from May 24 to June 5, 2010.

Lecture 1 Manila notes

Lecture 2 Manila notes

Lecture 3 Manila notes

Lecture 3 appendix Manila notes

Lecture 4 Manila notes

Lecture 5 Manila notes

Lecture 6 Manila notes

The following contains the course notes from Math 805 Special Functions,
Fall term 2010. The course was about the
Askey scheme of orthogonal polynomials. We focussed on the terminating
branch and its relationship to Leonard pairs.

Part 1 Intro/Hermite polynomials

Part 2 General orthogonal polynomials

Part 3 The Kratchouk polynomials

Part 4 The Krawtchouk polynomials cont.

Part 5 Leonard systems and their
classification

Part 6 The terminating branch of the Askey
scheme via Leonard systems

Part 7 Leonard systems of qRacah type

The following contains the course notes from Math 846 The Double
Affine Hecke Algebra,
Fall term 2011. The course was about the
DAHA of rank 1. We focussed on the ring theoretic aspects
as well as the connection to the AskeyWilson polynomials.

Part 1 Intro/linear basis for
DAHA

Part 2 DAHA and the universal
AskeyWilson algebra

Part 3 The centralizer of the
generator t0

Part 4 The center of DAHA

Part 5 Modules for DAHA

Part 6 Connection to the AskeyWilson
polynomials.

Part 7 The action of DAHA on
the space of Laurent polynomials in one variable

The following contains the course notes from Math 475: Combinatorics,
for Fall term 2012.
The textbook was:
Richard Brualdi. Introductory Combinatorics, 5th Ed.

Math 475 Syllabus Syllabus

Chapters 1,2
Permutations and Combinations

Chapter 3 The Pigeonhole Principle

Chapter 4 Generating Permutations
and Combinations

Chapter 5 The Binomial Coefficients

Chapter 6 The InclusionExclusion
Principle and Applications

Chapter 7 Recurrence Relations and
Generating Functions

Chapter 8 Special Counting Sequences

Chapter 14 Polya Counting

The following contains the course notes from Math 846: Tridiagonal
Pairs and Related Topics,
for Fall term 2013.

Lecture 1, W 9/4 Introduction

Lecture 2, F 9/6 TD pairs and graphs

Lecture 3, M 9/9 The dual adjacency matrix

Lecture 4, W 9/11 The path graph

Lecture 5, F 9/13 Distanceregularity
with respect to a vertex

Lecture 6, M 9/16 The hypercubes

Lecture 7, W 9/18 The hypercubes and sl2

Lecture 8, F 9/20 The hypercubes and
the tetrahedron algebra

Lecture 9, M 9/23 The equitable basis
for sl2

Lecture 10, W 9/25 Evaluation modules
for the tetrahedron algebra

Lecture 11, F 9/27 The Dirac basis for sl2

Lecture 12, M 9/30 The Drinfeld polynomial

Lecture 13, W 10/2 The Krein
parameters of a DRG

Lecture 14, F 10/4 The Krein
parameters of a DRG, cont.

Lecture 15, M 10/7 TD pairs
and Leonard pairs; preliminaries

Lecture 16, W 10/9 Totally
bipartite TD pairs

Lecture 17, F 10/11 Totally
bipartite TD pairs, cont.

Lecture 18, M 10/14 Totally
bipartite TD pairs, cont.

Lecture 19, W 10/16 Totally
bipartite TD pairs and Uq(sl2)

Lecture 20, F 10/18
The equitable presentation of Uq(sl2)

Lecture 21, M 10/21
The split decomposition of a TD system

Lecture 22, W 10/23
The shape vector and the tetrahedron diagram

Lecture 23, F 10/25
The tridiagonal relations

Lecture 24, M 10/28
Double lowering maps

Lecture 25, W 10/30
Double lowering maps, cont.

Lecture 26, F 11/1
Double lowering maps and the qexponential function

Lecture 27, M 11/4
Double lowering maps; the operators K,B,M.

Lecture 28, W 11/6
Double lowering maps; the classification

Lecture 29, F 11/8
Double lowering maps and recurrent sequences

Lecture 30, M 11/11
Leonard pairs of qRacah type

Lecture 31, W 11/13
The algebra Uq(sl2)

Lecture 32, F 11/15
Rotators for Uq(sl2) modules

Lecture 33, M 11/25
Rotators and Lusztig operators

Lecture 34, W 11/27
The dual space of a Uq(sl2)module

Lecture 35, W 12/4
Uq(sl2)modules from the equitable point of view

Lecture 36, F 12/6
Uq(sl2)modules from the equitable point of view, cont.

Lecture 37, M 12/9
Twelve bases for an irreducible Uq(sl2)module

Lecture 38, W 12/11
Irreducible Uq(sl2)modules; the bilinear form

Lecture 39, F 12/13
Irreducible Uq(sl2)modules; transition matrices

The following contains the course notes from Math 846: Quantum groups
and Hopf algebras,
for Fall term 2015. We used the textbook by Kassel.

Math 846 Syllabus Syllabus

Lecture 1, W 9/2 Introduction

Lecture 2, F 9/4 First examples

Lecture 3, W 9/9 M(2) and GL(2)

Lecture 4, F 9/11 SL(2) and gradings

Lecture 5, M 9/14 Filtrations

Lecture 6, W 9/16 Ore extensions

Lecture 7, F 9/18 Noetherian rings

Lecture 8, M 9/21 Tensor products

Lecture 9, W 9/23 Tensor products and Hom

Lecture 10, F 9/25 Free Amodules

Lecture 11, M 9/28 Dual vector spaces

Lecture 12, W 9/30 The transpose map

Lecture 13, F 10/2 tensor products of algebras

Lecture 14, M 10/5 tensor algebras and free algebras

Lecture 15, W 10/7 Coalgebras

Lecture 16, F 10/9 Algebras and coalgebras

Lecture 17, M 10/12 Coalgebra morphisms

Lecture 18, W 10/14 Bialgebras

Lecture 19, F 10/16 Primitive elements

Lecture 20, M 10/19 The antipode S

Lecture 21, W 10/21 Hopf algebras

Lecture 22, F 10/23 Some Hopf algebra morphisms

Lecture 23, M 10/26 The Hopf algebras
GL(2), SL(2)

Lecture 24, W 10/28 Hopf algebra modules

Lecture 25, F 10/30 Morphisms for Hopf algebra modules

Lecture 26, M 11/2 Comodules

Lecture 27, W 11/4 Examples of comodules

Lecture 28, F 11/6 Hcomodule algebras

Lecture 29, M 11/9 The quantum plane

Lecture 30, W 11/11 The qdeterminant

Lecture 31, F 11/13 The Hopf algebras GLq(2), SLq(2)

Lecture 32, M 11/16 Lie algebras

Lecture 33, W 11/18 Hmodule algebras

Lecture 34, F 11/20 Bialgebra dualities

Lecture 35, M 11/30 A duality between U(sl2) and M(2)

Lecture 36, M 11/23 The algebra Uq(sl2) (talk by Mao Li)

Lecture 37, M 11/23 The YangBaxter equation (talk by Zhaochen Wang)

Lecture 38, W 11/25 Modules for Uq(sl2)
(talk by Zheng Lu)

Lecture 39, W 12/2 The Hopf algebra Uq(sl2)
(talk by Jason Steinberg)

Lecture 40, F 12/4 Uq(sl2) and the quantum plane
(talk by Dongxi Ye)

Lecture 41, M 12/7
A duality between Uq(sl2) and Mq(2)

Lecture 42, W 12/9
A duality between Uq(sl2) and SLq(2)

Lecture 43, F 12/11
A bilinear form for Uq(sl2)modules

Lecture 44, M 12/14
Quantum groups in statistical mechanics (talk by Hans Chaumont)

Lecture 45, M 12/14
The equitable presentation of Uq(sl2)

The following contains the course notes from Math 846:
Hopf algebras in Combinatorics, Fall 2016.

Math 846 Syllabus Syllabus

Lecture 1, W 9/7/16 Introduction

Lecture 2, F 9/9/16 coalgebras

Lecture 3, M 9/12/16 bialgebras

Lecture 4, W 9/14/16 bialgebra examples

Lecture 5, F 9/16/16 Gradings

Lecture 6, M 9/19/16 The antipode

Lecture 7, W 9/21/16 Hopf algebra examples

Lecture 8, F 9/23/16 The Takeuchi formula

Lecture 9, M 9/26/16 Subcoalgebras

Lecture 10, W 9/28/16 Primitive elements

Lecture 11, F 9/30/16 Lie algebras and Hopf algebras

Lecture 12, M 10/3/16 Commutative Hopf algebras

Lecture 13, W 10/5/16 Duality

Lecture 14, F 10/7/16 The shuffle algebra

Lecture 15, M 10/10/16 The shuffle algebra, cont.

Lecture 16, W 10/12/16 Symmetric functions

Lecture 17, F 10/14/16 The Hopf algebra
structure

Lecture 18, M 10/17/16 The dominance order and applications

Lecture 19, W 10/19/16 Schur functions

Lecture 20, F 10/21/16 Schur functions
and column strict tableaux

Lecture 21, M 10/24/16 Skew Schur functions

Lecture 22, W 10/26/16 A generating function

Lecture 23, F 10/28/16 The power symmetric functions revisited

Lecture 24, M 10/31/16 The Hall inner product

Lecture 25, W 11/2/16 An alternate view of Schur funcions

Lecture 26, F 11/4/16 An alternate view of Schur funcions, cont.

Lecture 27, M 11/7/16 LittlewoodRichardson coefficients

Lecture 28, W 11/9/16 The Pieri rule

Lecture 29, F 11/11/16 The skew Pieri rule

Lecture 30, M 11/14/16 Positive self dual
Hopf algebras

Lecture 31, W 11/16/16 Primitive elements and symmetric algebras

Lecture 32, F 11/18/16 Primitive elements
in a PSH

Lecture 33, M 11/21/16 Decomposing a PSH
into Hopf subalgebras

Lecture 34, W 11/23/16 PSH algebras
and symmetric functions

Lecture 35, M 11/28/16 PSH algebras
and symmetric functions, cont.

Lecture 36, W 11/30/16 Zhaochen Wang lecture

Lecture 37, F 12/2/16 Shuai Shao lecture

Lecture 38, M 12/5/16 David Wanger lecture

Lecture 39, W 12/7/16 Michael Powers lecture

Lecture 40, F 12/9/16 Hang Huang lecture

Lecture 41, M 12/12/16 Rebecca Eastham lecture

Lecture 42, W 12/14/16 Benjamin Branman lecture

The following contains the course notes from Math 846:
Introduction to quantum groups, Spring 2019. We use the book
by Janzten.

Lecture 1, W 1/23/19 A basis for Uq(sl2)

Lecture 2, F 1/25/19 The Casimir element C for Uq(sl2)

Lecture 3, M 1/28/19 The center of Uq(sl2)

Lecture 4, W 1/30/19 Highest weight modules for Uq(sl2)

Lecture 5, F 2/1/19 The fd irred Uq(sl2)modules

Lecture 6, M 2/4/19 When is a fd Uq(sl2)module semi simple?

Lecture 7, W 2/6/19 Some infinite diml Uq(sl2)modules

Lecture 8, F 2/8/19 The quantum adjoint action for Uq(sl2)

Lecture 9, M 2/11/19 The HarishChandra map for Uq(sl2)

Lecture 10, W 2/13/19 The coproduct and counit for Uq(sl2)

Lecture 11, F 2/15/19 The antipode for Uq(sl2)

Lecture 12, M 2/18/19 Hopf algebras

Lecture 13, W 2/20/19 Some canonical maps

Lecture 14, F 2/22/19 The quantum trace

Lecture 15, M 2/25/19 More canonical maps

Lecture 16, W 2/27/19 The R matrix

Lecture 17, F 3/1/19 The R matrix, continued

Lecture 18, M 3/4/19 Link diagrams and the R matrix

Lecture 19, W 3/6/19 Loperators for Uq(sl2)

Lecture 20, F 3/8/19 Finite diml semisimple Lie algebras g

Lecture 21, M 3/11/19 The quantum group Uq(g)

Lecture 22, W 3/13/19 The square of the
antipode S for Uq(g)

Lecture 23, F 3/15/19 The square of the
antipode S for Uq(g), cont.

Lecture 24, M 3/25/19 The quantum adjoint
action

Lecture 25, W 3/27/19 A basis for tilde Uq

Lecture 26, F 3/29/19 The map tilde Uq >Uq

Lecture 27, M 4/1/19 The triangular decomposition of Uq

Lecture 28, W 4/3/19 Representations of Uq

Lecture 29, F 4/5/19 The quantum trace for Uq

Lecture 30, M 4/8/19 The Uqmodules M(lambda)

Lecture 31, W 4/10/19 the Uqmodules L(lambda)

Lecture 32, F 4/12/19 fd irred Uqmodules

Lecture 33, M 4/15/19 classification of fd irred Uqmodules

Lecture 34, W 4/17/19 The braid group action

Lecture 35, F 4/19/19 The braid group action, cont.

Lecture 36, M 4/22/19 Kevin Kristensen talk

Lecture 37, W 4/24/19 Peter Nielsen talk

Lecture 38, F 4/26/19 Dadong Peng talk

Lecture 39, M 4/29/19 Peter Ruan talk

Lecture 40 W 5/1/19 Ben Branman talk

Lecture 41 F 5/3/19 Seth Davis talk

The following contains the course notes from Math 846:
Crystal Bases in Algbraic Combinatorics, Fall 2019. We use the book
by Bump and Schilling.

Lecture 1, W 9/4/19 Root systems

Lecture 2, F 9/6/19 The simple roots

Lecture 3, M 9/9/19 The Weyl group

Lecture 4, W 9/11/19 The simple root systems

Lecture 5, F 9/13/19 Kashiwara crystals

Lecture 6, M 9/16/19 Examples of crystals

Lecture 7, W 9/18/19 More examples of crystals

Lecture 8, F 9/20/19 The tensor product of crystals