Teaching
Vienna
(VDS) Summer School (Sep 16-22, 2018)
Ultraproducts at the cross-roads of model-theory, algebra
and geometry
- Here is a short
introduction to the course, which also contains further
readings, as well as a very brief introduction to the model-theory
(of rings).
- Some additional material
- Lecture Notes from a master
course at the KULeuven on Topics in Commutative Algebra;
it contains all concepts (and more) that we will use.
- Tentative schedule
- Class I: Monday 11:00-12:30
- Class II: Tuesday, 9:00-10:30
- Class II: Thursday, 9:00-10:30
- Problem session I: Tuesday, 16:30-17:30: some problems
we might discuss.
- Problem session II: Friday, 9:00-10:00
Summer
2018
NYC College of Technology:
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- The free textbook we will use is APEX
Calculus (Version 3): go to the webpage
and scroll down to Version 3 (there are many different formats;
choose the one that is appropriate for you).
- For the midterm (Wed, June 20th), from the following review
sheet, do questions 3-5, 7-17; for extrema problems, look at
the home works for 3.1.
- On the final (Th, July 5), you are
allowed one formula sheet and your calculator.
- For review, look at all your old homeworks, quizes, and
midterm, as well as the following review
sheet.
- For problems of De l'Hopital, see your home works.
- On Tuesday, I will provide the answers to the following "mock
final", you should try it out before then, to see where
you're standing.
Spring
2018
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:45PM, Namm 826
- Course outline
- Weekly
assignments
- We are now using Version 3 of APEX
Calculus: go to the webpage
and scroll down to Version 3 (there are many different formats; choose
the one that is appropriate for you)
- Test 1: Tue, Feb 27
- Review exercises:
- 1.3 (p. 28): #27-32
- 1.6 (p. 56): #19-24
- 2.3 (p. 84): #11-19, 33-38
- 2.4 (p. 93): #15-21, 23, 24, 26, 27, 30-33 (not normal
line).
- from the following review
sheet, do questions 1, 2, 4c, 4e, 6.
- Solutions
of test 1
- Test 2: Tue, mar 27
- Review material (apart from the regular homeworks)
- from the following review
sheet,
do questions 1c, 1d, 1e, 3, 4, 5, 7
- Review exercises
- 2.5 (p. 102): 14-16, 24-26
- 2.6 (p. 112): 26a, 32
- 2.7 (p. 119): 18-23
- 6.7 (p. 320): 16-21, 26, 27, 29-31.
- 3.1: assigned homeworks
- Solutions
of test 2
- Test 3: Th, May 3
- Review material (apart from the regular homeworks)
- On the final (Th, May 24), you are
allowed one formula sheet and your calculator. For review, look at all
your old homeworks and midterms, as well as the following review
sheet.
- MAT1575 (Calculus II)
- Office hours: Tues/Thur 5:45-6:45PM, Namm 826
- Course outline
- Weekly
assignments
- We are now using Version 3 of APEX
Calculus: go to the webpage
and scroll down to Version 3 (there are many different formats; choose
the one that is appropriate for you)
- Test 1: Tue, Feb 27
- Apart from the homeworks:
- Test 2: Tue, Mar 27
- Review material for the midterm apart from all homeworks:
- Test 3: Th, May 10
- Review material besides the homeworks:
- On the final (Th, May 24), you are
allowed one formula sheet and your calculator. For review, look at all
your old homeworks and midterms, as well as the following review
sheet.
Spring
2017
KULeuven:
GOB63a: Topics in Commutative Algebra (room
E200 01.209)
CUNY: I
am on sabbatical leave for the entire 2017 calendar year
Fall
2016
NYC College of Technology:
- Office hours: Tues/Thur 5:45-645PM, Namm 826
- Each section has its own course outline and grading policy, so be
sure to download the correct one:
- Weekly assignments
- We are now using Version 3 of APEX
Calculus: go to the webpage
and scroll down to Version 3 (there are many different formats; choose
the one that is appropriate for you)
- Test 1: Tue, Sep 22
- Apart from the homeworks:
- Test 2: Th, Oct 20
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do #6, 7, 11, 12.
- for improper integrals, also review the homework
- Solutions
of test 2
- Test 3: Th, Dec 1
- Review material besides the homeworks:
- Final: Tues Dec. 20:
- You can use one formula sheet (back-and-front) on the final.
- Review material beside homeworks and previous midterms:
Summer
2016
NYC College of Technology:
- MATH
1275 (College Algebra and Trigonometry)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- To prepare for the midterm (Mon, June 20),
apart from the regular homework assignments, also do from the
following review
sheet questions: 1, 2, 3, 4, 5, 7,
From the book on p. 399, do 72, 73, 74, 89d. On p. 800, do 52, 53.
For problems on rational and radical equations, look at the
homeworks and quizzes.
- Solutions for the
midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final (Th, June 30), you are
allowed one formula sheet and your calculator. For review, look at
all your old homeworks, quizes, and midterm, as well as the
following review
sheet (except #6).
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- The free textbook we will use is APEX
Calculus (Version 3): go to the webpage
and scroll down to Version 3 (there are many different formats;
choose the one that is appropriate for you).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- Review material for the midterm (Mon, June 20):
- from the following review
sheet, do questions 1ab, 2, 3, 4, 5, 7, 8, 11, 12, 13, 14,
15
- Solutions
for the midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final (Th, June 30), you are
allowed one formula sheet and your calculator. For review, look at
all your old homeworks, quizes, and midterm, as well as the
following review
sheet.
Spring
2016
NYC College of Technology:
- Office hours: Tues/Thur 5:45-645PM, Namm 826
- Course requirements and
policy
- Weekly
assignments
- We are now using Version 3 of APEX
Calculus (you used Version 2 in Calculus I last semester, so
make sure to get the newer version): go to the webpage
and scroll down to Version 3 (there are many different formats; choose
the one that is appropriate for you)
- Test 1: Mar 3
- Test 2: Mar 24
- Review material for the midterm apart from all homeworks:
- Test 3: Tue, May 10
- Review material besides the homeworks:
- Final: Tues May 24:
- You can use one formula sheet (back-and-front) on the final.
- Review material beside homeworks and previous midterms:
- MAT3080 (Modern Algebra)
- Office hours: Tues/Thur 5:45-645PM, Namm 826
- Course requirements and
policy
- Weekly assignments
- The textbook is free and can be downloaded from the Abstract
Algebra website (you can also purchase a hardcopy for about
$25, see their site).
- Slides from the first
lesson (preliminaries)
- Homework assigment on Chapter 3: 2, 6, 7, 10, 31, 33, 40, 41, 42,
46, 48.
- Homework assigment on Chapter 4: 2, 3a, 3k, 15, 20, 21, 23, 31,
36, 39, 44
- Homework assigment on Chapter 5: 1, 2, 3, 14, 27, 35
- Slides for chapter 12: on Matrix
Groups and Symmetry
and on Wallpaper
groups
- Written assignment: due
after the break. If you need intructions on how to write
down or present the answers (espccially the geomtry part), you can
also go to Prof. Douglass' office hours Wed. 12-2pm (room N707).
- The Mock Midterm, which we
will discuss on Tue (Apr 19), in perparation for the midterm on Th,
(Apr 21).
- Review materila for the final (in addition to the review material
for the midterm): review on rings.
About LaTeX:
At some point in your math carreer you will, most probably, have
to write math assiginments or perhaps even a thesis. Now is
as good a time as ever to start learning it, so that you can
produce assignments in a professional format. You are of
course free to hand write assignments or use some other word
processor, but TeX produces so much more legible and neat
documents that you will regret in hindsight not having used it
earlier. Rather than learning the "bare bones" language TeX,
most people nowadays opt for the more user friendly LaTeX (the
most popular dialect presently is pdflatex which outputs a PDF
file). There are plenty free programs out there to write
(La)TeX, for almost all imaginable platforms. Here are some
links to start you off:
Fall
2015
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:25PM, Namm 628; Frid
3:30-4:00PM,GC 4432
- Course requirements and
policy
- Weekly
assignments
- Textbook
(color version); for other versions, go to their main webpage
and look for Version 2.0.
- Test 1: Tue, Sep 29
- Review exercises:
- 1.3 (p. 26): #27-32
- 1.6 (p. 52): #19-28
- 2.3 (p. 82): #11-16, 26-30
- 2.4 (p. 91): #16-20, 23, 24, 26, 27.
- from the following review
sheet, do questions 1a, 1b, 2, 4c, 4e, 6.
- Solutions of
test 1
- Test 2: Tue, Oct 27
- Review material (apart from the regular homeworks)
- from the following review
sheet, do questions 1c, 1d, 1e, 3, 4, 5, 7
- Review exercises
- 2.5 (p. 102): 14-16, 24-26
- 2.6 (p. 112): 26a, 32
- 2.7 (p. 119): 18-23
- 6.7 (p. 320): 16-21, 26, 27, 29-31.
- Solutions
of test 2
- Test 3: Tuesday, Dec 1
- Review material besides the homeworks:
- for 3.1, do homework problems 15-19 on p. 128
- from the following review
sheet, do questions 8-19 (for the last two questions only
make the table of signs and indicate the several pieces).
- to test graphing questions, do this QUIZ;
check your answers with Quiz
Solutions
- Solutions
of test 3
- Final: Thursday Dec 17 (must be taken to
pass course!)
- Office hours: Tues/Thur 5:45-6:25PM, Namm 628; Frid
3:30-4:00PM, GC 4432
- Course requirements and
policy
- Weekly
assignments
- Test 1: Th, Oct 1
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 5.
- Solutions of
test 1
- Test 2: Th, Oct 22
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do #4a,c and #7.
- on p. 463: #12, 17, 35, 45, 78, 79, 80, 83, 84, 85.
- on p. 499: #21, 23, 24, 25, 30, 32.
- Solutions
of test 2
- Test 3: Th, Dec 3
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96, 103.
- from the following review
sheet, do 8-10.
- additional review
sheet.
- Solutions
of test 3
- Final: Thursday Dec 17 (must be taken to
pass course!)
CUNY Graducate
Center:
- MATH 71100 (Logic I): Tues/Thur: 6:30PM-8:00PM, Room 5417 (GC)
- Office hours: Frid 3:30-4:00PM, GC 4432
- Syllabus
- Assignment #1 on definable
subsets.
- Assignment #2: study the following
proof of Tarski's theorem by Muchnik.
- Final: as agreed, you have until Dec
21 (your local time--so if you're running out of time, you can move
to Hawaii for a couple of extra hours) to submit, by email, your
solution (preferably in TeX but you can instead scan your notes as
well). I will confirm receipt, so if you don't get this from me,
then something went wrong with the submission (too large a file is
often the reason). You are supposed to work on these problems by
yourself and not discuss it with the others! Here is the final.
About LaTeX:
At some point in your graduate carreer you will, most probably, have
to learn TeX to write math papers or a dissertation. Now is as
good a time as ever to start learning it, so that you can produce
assignments in a professional format. You are of course free to
hand write assignments or use some other word processor, but TeX
produces so much more legible and neat documents that you will regret
in hindsight not having used it earlier. Rather than learning
the "bare bones" language TeX, most people nowadays opt for the more
user friendly LaTeX (the most popular dialect presently is pdflatex
which outputs a PDF file). There are plenty free programs out
there to write (La)TeX, for almost all imaginable platforms.
Here are some links to start you off:
Summer
2015
NYC College of Technology:
- MATH
1275 (College Algebra and Trigonometry)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- To prepare for the midterm (Mon, June 15),
apart from the regular homework assignments, also do from the
following review
sheet questions: 1, 2, 3, 4, 5, 7, 8.
From the book on p. 399, do 72, 73, 74, 89d. On p. 800, do 52, 53.
For problems on rational and radical equations, look at the
homeworks and quizzes.
- Solutions for the
midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- On the final, you are allowed one formula sheet and your
calculator. For review, look at all your old homeworks, quizes, and
midterm, as well as the following review
sheet (except #6).
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- Review material for the midterm (Mon, June 15):
- from the following review
sheet, do questions 1ab, 2, 3, 4, 5, 7, 8, 11, 12, 13, 14,
15
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Related rates: 118, 119, 123,
- Linear Approximations: p. 282, 4-6
- Absolute Max/Min: p. 282, 31, 32, 35
- Solutions
for the midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final, you are allowed one formula sheet and your
calculator. For review, look at all your old homeworks, quizes, and
midterm, as well as the following review
sheet. Some extra problems on limits: textbook, P. 99, # 33,
39 (without using L'Hopital's rule), and some of the problems we
will do in class.
Spring
2015
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tue, Mar 3
- Review exercises:
- (2.10) p. 117: # 14, 15, 16, 17, 19, 21, 23, 29, 41. Also
59, 61 but only the limits.
- (3.12) p. 203: #5, 7, 9, 1, 29, 30, 32, 33, 34, 41
- from the following review
sheet, do questions 2 (limits only), 3, 4c, 4e, 6.
- Solutions
of test 1
- Test 2: Tuesday, Mar 24
- Review material (apart from the regular homeworks)
- from the following review
sheet, do questions 4, 5, 7
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Solutions
of test 2
- Test 3: Tuesday, May 5
- Review material besides the homeworks:
- from the book, do the following problems on p. 203ff
- Related rates: 118, 119, 123,
- from the following review
sheet, do questions 1, 2, 8, 11, 12, 13, 14, 15, 18, and
19 (ignore symmetry and intercepts).
- Solutions
of test 3.
- Final: Thursday May 21 (must be taken to
pass course!)
- Review material
- review
sheet
- additional problems: textbook, P. 99, # 33, 39 (without
using L'Hopital's rule).
- Offfce hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tue, Mar 3
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 5.
- Solutions of
test 1
- Test 2: Tuesday, Mar 24
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do 6, 7, 11, 12 (without the graphing but instead
with the error estimate!).
- on p. 463: #12, 17, 35, 45, 78, 79, 80, 83, 84, 85.
- on p. 499: #21, 23, 24, 25, 30, 32.
- Solutions of
test 2
- Test 3: Tuesday, May 5
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96, 103.
- from the following review
sheet, do 8-10.
- additional review
sheet.
- Solutions
of test 3
- Final: Thursday May 21(must be taken to
pass course!)
Fall
2014
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Th, Oct 2
- Review exercises:
- (2.10) p. 117: # 14, 15, 16, 17, 19, 21, 23, 29, 41.
- (3.12) p. 203: #5, 7, 9, 1, 29, 30, 32, 33, 34, 41
- from the following review
sheet, do questions 2 (limits only), 3, 4c, 6.
- Solutions
of test 1
- Test 2: Tuesday, Oct 28
- Review material (apart from the regular homeworks)
- from the following review
sheet, do questions 4, 5, 7
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67,
,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Solutions
of test 2
- Test 3: Thursday, Dec 4
- Review material besides the homeworks:
- from the book, do the following problems on p. 203ff
- Related rates: 118, 119, 123,
- from the following review
sheet, do questions 1, 2, 8, 11, 12, 13, 14, 15, 18, and
19 (ignore symmetry and intercepts).
- Solutions
of test 3.
- Final: Thursday Dec 18 (must be taken
to pass course!)
- Review material
- review
sheet
- additional problems: textbook, P. 99, # 33, 39 (without
using L'Hopital's rule).
- Offfce hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1:
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 5.
- Solutions of
test 1
- Test 2: Tuesday, Oct 28
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do 6, 7, 11, 12 (without the graphing but instead
with the error estimate!).
- on p. 463: #12, 17, 35, 45, 78, 79, 80, 83, 84, 85.
- on p. 499: #21, 23, 24, 25, 30, 32.
- Solutions
of test 2
- Test 3: Tuesday, Dec 2
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96,
103.
- from the following review
sheet, do 8-10.
- additional review
sheet.
- Solutions
of test 3
- Final: Thursday Dec 18 (must be taken
to pass course!)
Summer
2014
NYC College of Technology:
- MATH
1275 (College Algebra and Trigonometry)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- To prepare for the midterm (Mon, June 16), apart from the regular
homework assignments, also do from the following review
sheet questions: 1, 2, 3, 4, 5, 7, 8.
From the book on p. 399, do 72, 73, 74, 89d.
- Solutions for the
midterm.
- Solutions for Quiz 3.
- On the final Wednesday, you are allowed
one formula sheet and your calculator. For review, look at all your
old homeworks, quizes, and midterm, as well as the following review
sheet (except #6).
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only).
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- Review material for the midterm (Mon, June 16):
- from the following review
sheet, do questions 2, 3, 5, 7, 8, 11, 12, 13, 14, 15
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Related rates: 118, 119, 123,
- Linear Approximations: p. 282, 4-6
- Absolute Max/Min: p. 282, 31, 32, 35
- Solutions
for the midterm.
- Solutions for Quiz 3.
- On the final Wednesday, you are allowed
one formula sheet and your calculator. For review, look at all your
old homeworks, quizes, and midterm, as well as the following review
sheet. Some extra problems on limits: textbook, P. 99, # 33,
39 (without using L'Hopital's rule), and some of the problems we
will do in class.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
Spring
2014
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tuesday, Feb 25
- Review exercises:
- (2.10) p. 117: # 14, 15, 16, 17, 19, 21, 23, 29, 41.
- (3.12) p. 203: #5, 7, 9, 1, 29, 30, 32, 33, 34, 41
- Solutions
of test 1
- Test 2: Tuesday, Mar 25
- Review material:
- from the following review
sheet, do questions 4, 5, 7, 11, 12, 13, 14, 15
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Related rates: 118, 119, 123,
- Solutions
of test 2
- Test 3:
- Review material for Exam 3, apart from all homeworks:
- from the following review
sheet, do questions 1, 2, 8, 18, and 19 (ignore symmetry and
intercepts).
- Solutions of
test 3
- Final: Thursday May 22 (must be taken to
pass course!)
- Review material
- review
sheet
- additional problems: textbook, P. 99, # 33, 39 (without
using L'Hopital's rule).
- Offfce hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tuesday, Feb 25
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 3.
- Solutions
of test 1
- Test 2: Tuesday, Mar 18
- Review material for the midterm apart from all homeworks:
- on p. 463: #12, 17, 35, 45, 78, 79, 80, 83, 84, 85.
- on p. 499: #21, 23, 24, 25, 30, 32.
- Solutions
of test 2
- Test 3:
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96, 103.
- from the following review
sheet, do 8-10.
- additional review
sheet.
- Solutions
of test 3.
- Final: Thursday May 22 (must be taken to
pass course!)
Fall
2013
NYC
College
of Technology:
- Office hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Thursday, Sep 26
- Review exercises:
- (2.10) p. 117: # 14, 15, 16, 17, 19, 21, 23, 29, 41.
- (3.12) p. 203: #5, 7, 9, 1, 29, 30, 32, 33, 34, 41
- Solutions of
test 1
- Test 2: Thursday, Oct 31
- Review material:
- from the following review
sheet, do questions 4, 5, 7, 11, 12, 13, 14, 15
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Related rates: 118, 119, 123,
- Solutions of
Exam 2
- Test 3: Tuesday, Dec 3: Review material for Exam 3, apart from all
homeworks:
- from the following review
sheet, do questions 1, 2, 8, 18, and 19 (ignore symmetry and
intercepts).
- Solutions of
Exam 3
- Final: Thursday Dec 19 (must be taken to
pass course!)
- Review material
- review
sheet
- additional problems: textbook, P. 99, # 33, 39 (without
using L'Hopital's rule).
- Offfce hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Thursday, Sep 26
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 3.
- Test 2: Tuesday, Oct 22
- Review material for the midterm apart from all homeworks:
- on p. 463: #12, 17, 35, 37, 45, 79, 85, 88.
- on p. 499: #21, 23, 24, 25, 30.
- Test 3: Tues, Dec 3
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96, 103.
- from the following review
sheet, do 8-10.
- additional review
sheet.
- Solutions of
test 3
- Final: Thursday Dec 19 (must be taken to
pass course!)
- Review material
- review
sheet
- Additional problems: textbook P. 381, # 17, P. 389, # 9, P.
578, # 10, 13, 39.
Summer
2013
NYC College of Technology:
- MATH
1275 (College Algebra and Trigonometry)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only)
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- To prepare for the midterm (Wed, June 19), apart from the regular
homework assignments, also do from the following review
sheet questions: 1, 2, 3, 4, 5, 7, 8.
From the book on p. 399, do 72, 73, 74, 89d.
- Solutions for the
midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final Wednesday, you are allowed
one formula sheet and your calculator. For review, look at all your
old homeworks, quizes, and midterm, as well as the following review
sheet.
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only)
- Solutions for Quiz 1.
- Solutions for Quiz 2.
- Review material for the midterm (Wed, June 19):
- from the following review
sheet, do questions 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14,
15
- from the book, do the following problems on p. 203ff
- Derivatives: 5-8, 29-41, 45-48, 52, 55-60, 64-67, ,95-97
- Implicit differentation: 101-106
- Logarithmic differentation: 74, 75, 109, 110
- Related rates: 118, 119, 123,
- Linear Approximations: p. 282, 4-6
- Absolute Max/Min: p. 282, 31, 32, 35
- Solutions
for the midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final Wednesday, you are allowed
one formula sheet and your calculator. For review, look at all your
old homeworks, quizes, and midterm, as well as the following review
sheet.
Spring
2013
NYC College of Technology:
- Office hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tuesday, Feb 26
- Review exercises:
- (2.10) p. 117: # 14, 15, 16, 17, 19, 21, 23, 29, 41.
- (3.12) p. 203: #5, 7, 9, 1, 29, 30, 32, 33, 34, 41.
- Solutions of Exam 1
- Test 2: Thursday, Apr 4
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do questions 3, 4, 5, 6, 7.
- from review problems on p. 203, do 64, 65, 66, 83 (only
compute the derivative of g at the given value), 97, 101, 103,
105, 109, 110, 113
- to practice the chain rule, take the following quiz
; here are the solutions.
- Solutions of
Exam 2
- Test 3: Tuesday, May 7: Review material for Exam 3, apart from all
homeworks:
- from the following review
sheet, do questions 1, 2, 8, 11, 12, 13, 14, 18, and 19
(ignore symmetry and intercepts).
- Solutions of
Exam 3.
- Review material for the final: review
sheet.
- MAT1575 (Calculus II)
- Offfce hours: Tues/Thur 5:45-6:35PM, Namm 628 and by
appointment
- Course requirements and
policy
- Weekly
assignments
- Test 1: Tuesday, Feb 26
- Review exercises:
- (5.9) p. 353: #24, 25, 26, 34, 39, 40, 44, 47, 56, 57,
- (7.9) p.463: #4, 13, 14, 16, 23, 29, 37, 48, 49,
- from the following review
sheet, do 1,2 and 3.
- Solutions of Exam 1
- Test 2: Thursday, Apr 4
- Review material for the midterm apart from all homeworks:
- on p. 463: #12, 17, 35, 37, 45, 79, 85, 88.
- on p. 499: #21, 23, 24, 25, 30.
- Solutions of
Exam 2
- Test 3: Thursday, May 9
- Review material besides the homeworks:
- on p. 603: #29, 39, 43, 45, 55, 63, 73, 81, 93, 95, 96, 103.
- Write 1.2343434... as a fraction.
- from the following review
sheet, do 8-10.
- Solutions of
Exam 3.
- Review material for the final: review
sheet.
Fall
2012
NYC
College
of Technology:
- Office hours: Tues/Thur 1:50-2:15PM and 5:45-6:15PM,
Namm 628 and by appointment
- Course requirements and
policy
- Weekly assignments
- Solutions of Exam
1.
- Review material for the midterm apart from all homeworks:
- from the following review
sheet, do questions 3, 4, 6, 7, 8, 9, 10, 11.
- to practice the chain rule, take the following quiz
; here are the solutions.
- Solutions of Exam
2
- Review material for Exam 3, apart from all homeworks:
- from the following review
sheet, do questions 1, 14, and 15 (ignore symmetry).
- Solutions of Exam
3
- Updated review sheet
for the final exam (Thursday, Dec 20)
- MAT1575 (Calculus II)
- Offfce hours: Tues/Thur 1:50-2:15PM and 5:45-6:15PM,
Namm 628 and by appointment
- Course requirements and
policy
- Weekly assignments
- Solutions of Exam
1.
- Review for Exam 2 (Th, Oct 25), apart from all the homeworks:
- from the following final exam
review sheet, do questions 1), 2), 5), 6) and 7)
- from the review in the book on p. 518-519, do questions #4, 5,
6, 7,10 ,11, 15, 20 ,21, 23, 38, 41, 43, and 45.
- Solutions of Exam2
- Review for Exam 3, apart from the homeworks:
- Solutions of Exam
3
- Updated review sheet
for the final exam (Thursday, Dec 20).
CUNY
Graducate
Center:
- MATH 71100 (Logic I): Tues/Thur: 6:30PM-8:00PM, Room 6417 (GC)
- Syllabus
- Assignment #1 on definable
subsets.
- Assignment #2: study the following
proof of Tarski's theorem by Muchnik.
- Assignment #3: notes on
o-minimality (latest version, hopefully now without any gaps).
- The final exam will be due
Saturday 11:59pm (one day extension). You must send me the answers
by email (refrain
from sending oversized files).
- Model-theory seminar
(GC 6417, Fridays 12:30-1:45PM; organized by R. Kossak, P. Rothmaler and
H. Schoutens):
About LaTeX:
At some point in your graduate carreer you will, most probably, have
to learn TeX to write math papers or a dissertation. Now is as
good a time as ever to start learning it, so that you can produce home
work solutions in a professional format. You are of course free
to hand write assignments or use some other word processor, but TeX
produces so much more legible and neat documents that you will regret
in hindsight not having used it earlier. Rather than learning
the "bare bones" language TeX, most people nowadays opt for the more
user friendly LaTeX (the most popular dialect presently is pdflatex
which outputs a PDF file). There are plenty free programs out
there to write (La)TeX, for almost all imaginable platforms.
Here are some links to start you off:
Summer
2012
NYC College of Technology:
- MATH
1275 (College Algebra and Trigonometry)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only)
- Solutions to Quiz 2
- To prepare for the midterm, apart from the regular homework
assignments, also do from the following review
sheet questions: 1, 2, 3, 4, 5, 7, 8 (for the correct
equations in 8, see this file).
From the book on p. 399, do 72, 73, 74, 89d.
- Solutions for the
midterm.
- Solutions of Quiz 3.
- Solutions of Quiz 4.
- Solutions for Quiz 5.
- On the final Tuesday, you are allowed one formula sheet and your
calculator. For review, look at all your old homeworks, quizes, and
midterm, as well as the following review
sheet.
- MATH 1475 (Calculus I)
- Course
requirements and policy
- Weekly
assignments (please ignore the numbering by session, as these
pertain to regular terms only)
- Solutions to Quiz 2
- Review material for the midterm (Mon, June 18):
- from the following review
sheet, do questions 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
- from the book, do the following problems on p. 262ff
- Derivatives: 1-9, 11-14, 17, 20-24, 26, 29, 31, 32, 36-39,
50, 57-59
- Implicit differentation: 15, 18, 25, 40, 60-61 (tangent
line only)
- Logarithmic diffrentation: 28, 30, 41
- Related rates: 97, 98, 100
- Linear Approximations: 102a, 103a
- Solutions
for the midterm.
- Solutions for Quiz 3.
- Solutions for Quiz 4.
- Solutions for Quiz 5.
- On the final Tuesday, you are allowed one formula sheet and your
calculator. For review, look at all your old homeworks, quizes, and
midterm, as well as the following
review sheet.
Spring
2012
NYC
College
of Technology:
- MATH1272
(Statistics)
- MATH1475 (Calculus I)
- Office hours: Tues 1:50-2:30PM; Thur 2:00-2:30PM and
4:45-6:30PM, Namm 628
- Course requirements and
policy
- Weekly
assignments
- Solutions to
the first exam.
- Review problems for Exam 2 (BESIDES THE HOMEWORK PROBLEMS): for
review, do the following problems on p. 262ff
- Derivatives: 1-9, 11-14, 17, 20-24, 26, 29, 31, 32, 36-39, 50,
57-59
- Implicit differentation: 15, 18, 25, 40, 60-61 (tangent line
only)
- Logarithmic diffrentation: 28, 30, 41
- Related rates: 97, 98, 100
- Linear Approximations: 102a, 103a
- Solutions for
the second test
- Review problems for the third test
- p. 167: #13, 14, and 16
- p. 348-349: #1-12, 19-23, 27, 28, 50, 52, 58, and 59.
- Solutions for
the third test
- Review
material for the final
CUNY Graducate Center:
- MATH 88700: The
use of ultraproducts in commutative algebra
- Independent study (MATH 89904)
- Model-theory seminar
(GC 6417, Fridays 12:30-1:45PM; organized by R. Kossak, P. Rothmaler and
H. Schoutens):
Fall
2011
NYC
College
of Technology:
CUNY Graducate Center:
- Independent study (MATH 89904)
The model theory seminar in the Fall of 2011 will not have a one
unified theme. Instead we will continue several topics introduced in
previous semesters. We will begin with a four week introduction/tutorial
of types in Peano Arithmetic with applications to the classification of
elementary pairs. There will also be a connection to the theory of
Abstract Elementary Classes.
Spring
2011
NYC College of Technology:
- Office hours: Tues/Thur 3:10PM-4:00PM, Namm 628
- Course requirements and policy
- Weekly
assignments
- Solutions for
the first test
- Review problems for Exam 2 (BESIDES THE HOMEWORK PROBLEMS): for
review, do the following problems.on p. 262ff
- Derivatives: 1-9, 11-14, 17, 20-24, 26, 29, 31, 32, 36-39, 50,
57-59
- Implicit differentation: 15, 18, 25, 40, 60-61 (tangent line
only)
- Logarithmic diffrentation: 28, 30, 41
- Related rates: 97, 98, 100
- Linear Approximations: 102a, 103a
- Solutions for
the second test
- Solutions for
the third test
- Review
for the final exam
- Fimal grades
- MAT1575 (Calculus II)
CUNY Graducate Center:
- Model Theory II (MATH 80200, T/Th, 2pm-3:30pm);
Prof. Rothmaler is offering an advanced model-thory course
(see his syllabus), which is a
continuation of the current class and the model-theory seminars of the
last few semesters. He told me that he will start with a quick
introduction covering the remaining chapters of his book (Chapters
11-14.5), for anybody who has not been exposed to this material through
the seminars. I will assist those that are interested in this course
through an IS, see next item:
- Independent study (MATH 89904, meeting hours tba):
if you are interested in the above advanced model-theory course by Prof.
Rothmaler (see previous item), but have not passed the Logic Qualifier,
you can instead register for an IS with me, take the course, and we then
will get together to "fill in the gaps" and discuss the material. You
wouldn't be responsible for any course work, but will receive three
credits for working with me.
- Model-theory
seminar (GC 6417, Fridays 12:30-1:45PM; organized by P.
Rothmaler and H. Schoutens):
Various topics in model theory.
Some of the topics to be studied in this semester's seminar are
described below. They vary in degree of model-theoretic
sophistication and prerequisites. Topic 1 and largely also Topic 3
require no more than some general understanding of first-order
semantics (as provided in Logic I). Topic 2 will be rather
complementary to Model Theory II, also offered this semester (see
above). Other topics may be added when appropriate.
- 1. Interpretations. The method of interpreting
one kind of structure syntactically in another is a very old one. As
an example, think of the field of complex numbers as interpreted in
the field of real numbers or of the field of rational numbers in the
ring of integers (coincidentally in both cases by pairs of elements
of the ambient structure). Quite obviously, these interpretations
work on a syntactic level using first-order formulas. The goal is to
establish some well-known classical features of such interpretations
that play a fundamental role in all areas of mathematical logic. We
will start with the elementary [R, Sect. 6.4] and may go on to apply
this to Malcev's local theorems, the very first application of model
theory to classical mathematics, if enough interest in (purely)
group theoretic applications can be mustered [R, Sect. 6.5] (other
applications could be studied instead). Ref: [Mal], [M, Sect. 1.3],
[R].
- 2. Pairs of stable structures. By a pair of
L-structures, (N, M), we mean an L-structure N considered in an
expansion L(P) of L by a unary predicate P whose interpretation in N
constitutes a proper elementary substructure M of N (as
L-structure). There are many natural examples of such pairs, e.g.
any extension of algebraically closed fields K/k can be thought of
as such a pair (K, k). Interestingly, we will see that the latter,
though of infinite Morley rank (as opposed to rank 1 for single
algebraically closed fields), are still omega-stable. We will study
a much more general result due to Casanovas and Ziegler [CZ] showing
that under a reasonably general extra condition (satisfied by
strongly minimal theories such as algebraically closed fields) pairs
of stable structures are stable. Ref: [P1], [CZ].
- 3. Model theory of modules. One of the simplest
examples of strongly minimal structure are vector spaces over fields
(even division rings). So vector spaces are at the lowest level of
the stability hierarchy. Surprisingly, even if we replace the field
or division ring by the ring of integers (to obtain all abelian
groups) or just any, even non-commutative ring (to obtain arbitrary
modules), we do not leave the realm of stable theories. By one of
Shelah's fundamental theorems of stability theory, this means that
despite the vast richness of the class of abelian groups or even all
modules, no infinite linear order, not even in the most rudimentary
sense, can be first-order definable in a module. The standard
reference is [Pr], but we may start with [P2, Sect. 6.5] and [H,
Appendices 1 and 2] and continue with the unpublished [Rot] as
preparation before we go into other sources. Ref: [H], [P2], [Pr],
[Rot], [R, Ch.15].
- Literature
- [CZ] E. Casanovas and M. Ziegler, Stable theories with a new
predicate, J. Symbolic Logic 66(3), (2001), 1127-1140.
- [H] W. Hodges, Model Theory, Encycl. Math. and Appl. 42,
Cambridge Univ. Press, 1993.
- [Mal] A. I. Malcev, A general method for obtaining local
theorems in group theory (1941), in his collected papers
1936-1967: The metamathematics of algebraic systems, pp.15-21,
North-Holland 1971.
- [M] D. Marker, Model Theory: an introduction, GTM 217,
Springer 2002.
- [P1] B. Poizat, Paires de structures stables, J. Symbolic
Logic 48(3), (1983), 239-249.
- [P2] B. Poizat, A Course in Model Theory, An Introduction to
Contemporary Mathematical Logic, Universitext, Springer 2000.
- [Pr] M. Prest, Model Theory of Modules, London Math Soc Lect
Note Ser 130, Cambridge Univ. Press 1988.
- [Rot], Ph. Rothmaler, introduction to elementary duality of
modules, unpublished.
- [R] Ph. Rothmaler, Introduction to Model Theory, Gordon &
Breach 2000.
About LaTeX:
At some point in your graduate carreer you will, most probably, have
to learn TeX to write math papers or a dissertation. Now is as
good a time as ever to start learning it, so that you can produce home
work solutions in a professional format. You are of course free
to hand write assignments or use some other word processor, but TeX
produces so much more legible and neat documents that you will regret
in hindsight not having used it earlier. Rather than learning
the "bare bones" language TeX, most people nowadays opt for the more
user friendly LaTeX (the most popular dialect presently is pdflatex
which outputs a PDF file). There are plenty free programs out
there to write (La)TeX, for almost all imaginable platforms.
Here are some links to start you off:
Summer
2010
at KULeuven
Chromatic
products in algebra and geometry
Ultraproducts of rings, and variant
constructions, called chromatic
products, can be used to prove some deep theorems in (pure) algebra and
algebraic geometry. One such application, which I have already lectured
about in Leuven a few years back, is tight closure theory in
characteristic zero. In this lecture series, I will describe further
applications, obtained by similar principles.
Lecture I (Mon. June 21, 14:00) "Ultrarings and uniform bounds"
- (Ia) Definition of ultrarings: algebraic, sheaf-theoretic,
and geometric constructions;
(Ib) Lefschetz hulls as adjoints of the forgetful functor: flat
embeddings of rings inside ultrarings,
(Ic) Application 1: uniform Artin Approximation
(Id) Application 2: uniform bounds in algebra for linear properties
(Ie) (optional) Application 3: tight closure in characteristic zero
(If) (optional) Application 4: big Cohen-Macaulay algebras in
characteristic zero
Lecture II (Wed. June 23, 14:00) "Protoproducts and homological
conjectures"
(Ic) (continuation) uniform Artin
Approximation
(IIa) Protoproducts as subrings of ultrarings
(IIb) Application 1: uniform bounds for the etale protograde
(IIc) (optional) Ax-Kochen-Ershov principle and transfer in mixed
characteristic
(IId) (optional) Application 2: Asymptotic homological conjectures
(example: the Monomial Conjecture)
Lecture III "Cataproducts and classification of singularities"
(IIIa) Cataproducts and catapowers:
definition and properties
(IIIb) Similarity relation on germs of singularities.
(IIIc) Application 1: Borel classifications and the jet metric.
(IIId) Application 2: ring properties via uniform arithmetic.
(IIIe) (optional) Application 3: Asymptotic homological conjectures
(example: the New Improved Intersection Conjecture)
Lecture IV "Ultra-Frobenius and
rational singularities"
(IVa) The action of the ultra-Frobenius on
ultra-cohomology
(IVb) F-singularities versus "birational" singularities
(IVc) Application 1: quotient singularities are rational
(IVd) Application 2: vanishing theorems a la Kodaira and Kawata-Viehweg
Additional reading material from "
The
use of ultraproducts in commutative algebra", Hans Schoutens,
Lecture Notes in Mathematics
1999, Springer
(2010)
NYC College of Technology:
Fall
2010
Fall
2009-Spring 2010 (on leave)
Spring 2009
Fall
2008
Spring 2008
Fall 2006
Spring 2006 (on leave)
Fall 2005
CUNY Graducate Center:
Fall
2010
Fall 2008
- MATH 71100 (Logic I): Mon-Wed. 6:30PM-8:00PM, Room 4422 (GC)
Spring 2008
Fall
2007
Fall 2006
- MATH 88700: The
use
of ultraproducts in commutative algebra
Spring
2006
(on leave)
Fall
2005