Director of the Center
(This page is like a cathedral: always under reconstruction ;-)
And speaking of photos: here's a link to Rob's old class photos.
This Semester
In the Spring of 2021, Rob continues remote-teaching from Coronavirus
University (which – like Garrison Keillor's whimsical Lake Woebegone – is an
institution where the professors are strong, the administrators are
good-looking, and all the students are above-average ;-)!
For those taking
Linear Algebra (Math 235.06) with Rob, we expect that the
incoming course chair will follow what Rob developed with the other
instructors in Fall 2020 (see the coursewide webpage below for what we
did then and expect to do again in Spring 2021, but with obvious
changes in the dates).
For students
taking Honors
Differential Geometry of Curves & Surfaces (Math 563H) with
me, please have a look (below) at what we did last spring — the
texts (required and recommended) remain the same.
Welcome back from what we all hope was a wonderful winter break – and
please stay healthy!
Last Semester
M235
Rob was chairing (again, yet again) and
teaching Math
235: Introduction to Linear Algebra during the 2020 Fall semester.
All students should have received email
with instructions
how to get the e-text and sign up for on-line homework for
Rob's section (M235.03) only. Course-wide logistical details and
important links
appear here.
For those of you who have yet to officially enroll: please be patient, keep an eye on SPIRE for openings, and in a
short time we should be able to officially enroll you (we don't want
to deal with over-rides until we know they're necessary).
You may enjoy reviewing
Basic Linear Algebra in 7 Easy Pages prepared by my former
student (2011-14)
Andrew Maurer (he's now in grad school at the University of Georgia; and his advisor
Dan Nakano took my linear algebra course at Berkeley in the early 1980's - years later Dan told me that my course made him want to become a mathematician).
For you hardcore linear algebra students: the usual multiplication algorithm has complexity O(n^3) for n × n matrices; here's a recent paper on the latest complexity bound: O(n^2.3726...). (Conjecture: O(n^2) is best.)
Note: The last part of the course uses the relation AE=ED (for any
matrix E whose columns form a basis of real eigenvectors of a real 2
× 2 matrix A with distinct real eigenvalues) to factor
A=EDE^{-1}, where D is the real diagonal 2 × 2 matrix whose nonzero
entries are the eigenvalues, i.e. A is similar to D. This is useful
to compute powers of A and the exponential of A. But what if A has
repeated real eigenvalue (a, a), or if the eigenvalues are complex
conjugate pairs (a+bi, a-bi) with nonzero b? How does one decompose
A=aI+bJ where J^2=-I in the latter case, or decompose A=aI+N where
N^2=0 in the former? Some texts (e.g. Bretscher) find a 2 × 2 matrix
aI+bJ_o to which A is similar; here J_o is the standard 90-degree
rotation matrix.
Here's
another approach, suggested by my emeritus
colleague Arunas
Rudvalis, which seems simpler - and more general since it also
deals with the N (nilpotent) case.
Last Year
Rob taught Honors
Differential Geometry of Curves & Surfaces (Math 563H). Here's a
link to the S20 Math
563H ASCII Notes and HW Problems.
We moved to a remote format during spring break, sponsored by Rob's new
institution Coronavirus University, courtesy of YouTube and Zoom. [So
far, all the links were shared privately, but the best YouTube
lectures and final projects will be posted here eventually!]
Rob chaired Rob (yet again) and
taught Math
235: Introduction to Linear Algebra during the Fall 2019 semester.
Course-wide logistical details and links to individual learning sections
appear here.
He also (again!)
taught Advanced
Multivariate Calculus (Math 425) in Fall 2019 [I'm still updating things from last fall, but please see my comments below].
Advice (for those of you in Math 425):
The texts are as before: search for Calculus on Manifolds to
find a .pdf of Spivak's beautiful text since it seems to be out of
print (outside of China); the Marsden & Tromba text is a good resource
and various recent editions should be available in used form; you
should try to get - and begin reading! - both of these before the
semester begins. And my former student, Kate Donoghue (now at grad school in Princeton) worked
with me on a what we hope will become a helpful set of course notes and
exercises - when ready, you'll find
them here!
Before taking Math 425, students are expected to have a good
understanding of basic linear algebra (Math 235) and vector calculus
(Math 233); so if you're feeling a bit "rusty" in these areas, please
"brush up" before the semester begins. Here's what I wrote to one
prospective student:
Hi Dxxxx,
You should take a look at Marsden & Tromba's book and be sure you know
about partial derivatives and multiple integrals; how to find critical
points of functions of 2 or more variables, and test whether they're
max, min, saddles or otherwise; and maybe a bit about finding critical
points when the variables are constrained to the level set of another
function (Lagrange multipliers). Although we'll (quickly!) review
some of this in the first few weeks, and "remind" (I hope) you of the
differential operators div, grad & curl, it's best if you've made an
effort over the break!
Best,
Rob
Recent
Rob was chairing and
teaching Math 235: Introduction to Linear Algebra during the 2019
Spring semester. This was how students got
the e-text and sign up for
on-line homework; course-wide logistical details
appear here.
Rob was also
teaching Advanced
Multivariate Calculus (Math 425) in Fall 2018.
Rob
taught Advanced
Multivariate Calculus (Math 425.1) in Spring 2018. He also was
teaching a large section of Ordinary
Differential Equations (Math 331.7), a
course chaired by Jinguo Lian.
Rob also
taught Advanced
Multivariate Calculus (Math 425) in the Fall 2017 semester.
Rob chaired and taught
Introduction to Linear Algebra (Math 235) during the 2017 Spring semester.
Logistical details appear here.
Less Recent
Rob taught Advanced Multivariate Calculus (Math 425) in the Fall 2016 semester.
During AY 2015-16, Rob was on sabbatical: the winter, spring and summer at
MSRI in Berkeley; and the fall at Penn, where he taught
Geometric Variational Problems (Math 299/550), a topics course for advanced
undergraduates and interested graduate students.
In the spring of 2015 Rob was
teaching
Honors Multivariable Calculus (Math 233H). Here's the ASCII page
for Math 233H
homework schedule and notes.
During the fall of 2014 Rob chaired - and taught two (double) sections (3
and 4) of -
Introduction to Linear Algebra (Math 235). The next few paragraphs
reflect past and foreshadow future versions of that course.
Here's a link to both our
sections'
S14 Math 235 WeBWorK. There will be roughly one problem set each
week, usually due Tuesday evenings. Please start with the "Orientation" set to help you learn WeBWorK syntax. Over the years WeBWorK has
improved, but occasionally you'll find a bug! If you're sure that
you've figured out a problem, but WeBWorK won't swallow your answer,
please don't waste time guessing and plugging -- each problem is
"worth" so little anyway -- better if you've developed the skill and
confidence to KNOW you've found a real bug, and not just made a silly
mistake: such is life!
Here is the (draft ASCII) Math 235 (Linear
Algebra) homework schedule for Rob's section.
Here's an evolving link to common course guidelines, suggested homework and practice exam problems.
During Spring 2014, Rob taught honors Differential
Geometry of Curves & Surfaces (Math 563H).
Here is the (quick-loading, even on a slow connection) ASCII file for Math 563H (Curves
and Surfaces) topics & problems. We are also working up a
.tex/.pdf version....
Rob also taught (a double) section 2
of
Introduction to Linear Algebra (Math 235) in Spring 2014.
In Fall 2013, Rob taught (a double)
section 3 of Introduction
to Linear Algebra (Math 235).
Here's a link to Rob's F13 Math 235 WeBWorK.
Here's a link to Math 235 Common Midterm Review Problems which Prof. Paul Hacking will go over with you from 7-8:30 on Tuesday evening 12 November 2013 in Thompson 102.
Here are links to Math 235 Common Final Review Problems and Solutions which Prof. Paul Hacking went over from 7-8:30 on Tuesday evening 10 December 2013.
And here is a link to Fall 2013
Linear Algebra (Math 235) common guidelines.
In Spring 2013, Rob taught honors Differential
Geometry of Curves & Surfaces (Math 563H).
Here is the (quick-loading, even on a slow connection) ASCII file for Math 563H (Curves
and Surfaces) topics & problems .
Rob also served as course chair for, and taught (a double)
section 3 of Introduction
to Linear Algebra (Math 235).
For the past several semesters, instructors have followed these Math 235
(Linear Algebra) common guidelines.
You can find some practice final
exams below, but the following four may be the most helpful to review,
along with your WeBWorK:
Math 235 Practice Final Problems from
Fall 2008
, ...
Spring 2009 , ...
Fall 2009
and from
Fall
2011 .
MAKE-UPS (only pre-approved according to UMass guidelines) will be
handled by individual instructors.
Here's a link to the S13 235 midterm solutions (written up by Geri Jennings).
Here's a link to the S13 235
Mini-midterm2 solutions.
[Please continue to watch here for important developments and announcements!]
Rob was on leave at the Isaac Newton Institute for Mathematical Sciences
for the Fall 2012 semester.
During Spring 2012, Rob is taught honors Differential
Geometry of Curves & Surfaces (Math 563H).
Here is the Spring 2012 ASCII file for Math 563H (Curves
and Surfaces) topics & problems .
Rob also served as course chair for, and taught both (a double) section 4 and (a single) section 7 of Introduction
to Linear Algebra (Math 235) in Spring 2012.
Here are notes on
Inner Products,
Gram-Schmidt and Ortho'matrices by Jenn Koonz (who gave a few lectures
when Rob was in Germany at the end of March 2012) as well as notes on
Determinants and on
Eigenstuff by Arunas Rudvalis.
Here are links to Math 235 MiniMidtermII solutions (front) and (back) from Tuesday 17 April 2012.
And here are links to Math 235 Midterm solutions (page 1,
2,
3,
4,
5,
and 6)
from Wednesday 7 March 2012.
And links to Math 235 MiniMidtermI solutions (front) and (back) from Monday 13 February 2012.
In Fall 2011 Rob taught (a double) section 2 of Introduction
to Linear Algebra (Math 235), with links to:
Fall 2011 Math 235 Practice Final Problems
Scanned pages of Fall
Math 235 Common Midterm Solutions (find the sign error)!
Fall 2011 Math 235 Practice Midterm Problems.
Not as Ancient as You May Think...(Events of the Past Year Make Me "The Last Oak Standing Between Here and Eternity")
In Spring 2011, Rob taught the computational linear algebra (Math313/513-CS513) course while on
sabbatical at Penn, and last fall he taught -- you guessed it! -- Introduction
to Linear Algebra (Math 235) at UMass....
During Spring 2010, Rob taught honors Differential
Geometry of Curves & Surfaces (Math 563H) and a double section of
Linear Algebra (Math 235). Here is the ASCII file for Math 563H (Curves
and Surfaces) topics & problems . (Andrew Reiter and Nicky Reyes
are helping to TeX these now, so please stay tuned!)
Because he was the course chair during a semester when all the other
instructors were graduate students, he and the other instructors
established the following Math 235
(Linear Algebra) common guidelines, which may be useful to future
course chairs.
Here's a link to the Fall 2009 Math 235 Practice Midterm and to sketches of answers!
Here's a link to Fall 2009 Math 235 Midterm Answers.
And here is a link to last year's
Math 235 Practice Midterm.
Finally, here's the Fall 2009 Math 235 practice final , along with Peter Norman's handwritten answers , a link to an old Math 235 practice final and links to older Math 235 practice finals or even older
Math 235 practice finals, including one similar to our practice final with answers ....
Here are some
Quick
and Dirty Differential Equations notes prepared by Peter Norman
last fall (with a little help from me) for use in Math 235.
During Spring 2009, Rob taught a special version of
Math 462
(Geometry and the Imagination) - please visit here for
the
Math 462 topics and comments! He also taught a (double) section of Math 235
(Linear Algebra).
In Fall 2008, Rob taught a double-section of Math 235 (Linear Algebra).
Rob was on sabbatical in Spring 2008, but he taught back-to-back sections
of Math 235 in Fall 2007 -- Linear Algebra is one of his favorite and most popular classes!
In Spring 2007 Rob taught Math 425
(Advanced Calculus) and a section of Math 235
(Linear Algebra). Here are links to the Math 425
(Advanced Calculus) homework schedule. Also,
please have a look at my colleague Roman Federov's
problems for enrichment and review.
Rob organized the undergraduate TAP seminar (Math 191) during the Fall
2006 semester. It met in GANG (LGRT 1535) at lunchtime most Mondays and some Wednesdays. The following people kindly
contributed to the
2006 TAP
Topics. Many thanks to all of you!
And since this is Greek to me, here's a link to symbols like ε.
More
Watch here
for further developments.
Address
Phone 413 545 6022
Secretary 413 545 2762
Fax 413 545 1801
Electronic Mail kusner "at" math "dot" umass "dot" edu
Department Webpage www.math.umass.edu/directory/faculty/rob-kusner
Kusner's Math Classes
About the photo: Rob's hair is now a lot greyer and a bit shorter, in
part because he was elected to the Amherst Select Board several years
ago (having also led a concerted and successful effort to preserve
Amherst Town Meeting in 2005). However, unlike in mathematics, where
truth is (effectively) eternal, in politics the voters decide: 2018
saw a slim majority adopt a new charter replacing Amherst's
259-year-old Town Meeting with a city form of government led by
13=(2+3i)(2-3i) Councilors. Rob was a candidate for one of the
at-large Council seats: he made it through the 4 September 2018
primary, broke his leg in a bike accident on campus the next day, and
2 months later garnered 2114 votes (not enough to be elected, yet not
bad for someone who could barely walk - let alone run). None of this
should impact the mathematics you can learn with him if you come to
class, pay attention, and ask good questions, but (occasionally) a wry
comment about how well (or not) we are governing ourselves may slip
from his lips....
While a lot of useful coursewide things are posted at our own learning
section page (above), there is now also an
official coursewide
webpage for Spring 2021 Math 235.
Some of this information is mirrored at Moodle, where you'll
also find Rob's Math 235.06 syllabus, but the pages above are the
definitive ones.
For your covenience, here's where you'll
find instructions
for how to get the e-text and sign up for on-line homework (Rob's
section only).
I've begun posting
the
Math 563H ASCII Notes and HW Problems.
I'm also making more videos to share my own approach to curves and
surfaces – I'll email students directly when I start to post these to
YouTube, and later I'll curate
a small
library of my 563H videos (and others about differential
geometry)....
A bit of this information is also mirrored at Moodle....
In academic year 2006-07 Rob again planned to teach a
two-semester course on
Manifolds
(Math 703-4) for graduate students. Due to unforeseen and unfortunate
circumstances, Rob was not able to offer Math 704 in the spring.
But here is where to find homework problems and informal notes for
that course:
Manifolds I (703) and
Manifolds II
(704), along with older notes from Fall 2002
(703),
Spring 2003 (704) and Spring 2000 (Math 704).
In the Spring 2006 semester Rob taught Advanced
Linear Algebra (Math 545) ; here are the updated problems/notes
for this course. (Please compare with Rob's Spring 2002 advanced linear
algebra problems/notes.)
Rob also pinch-hit all semester for Ordinary
Differential Equations (Math 331) in Spring 2006; here is the Math 331 general
topics list and the Math 331
homework schedule for Rob's section. (This was an ambitious
syllabus, covering more than the other sections did, and not without
some ill-tempered jeers from the bleachers in the final innings -- it
didn't help that Select Board meetings went on till after midnight on
many, many occasions before this early morning course ;)
In the Fall 2005 semester Rob taught
Introduction to Abstract Algebra (Math 411); here are the notes and
homework schedule .
In the Spring 2005 semester Rob was teaching an experimental
version of Geometry and the
Imagination. This was a very popular course, and Rob hopes to
have the opportunity to offer it again soon, perhaps in conjunction
with Math 563 (Differential Geometry of Curves and Surfaces)!
Rob was on sabbatical for 2003-04. During academic year 2002-03 Rob taught a two-semester course on
Manifolds (Math
703-4) for graduate students.
Here is where to find homework problems and notes from that course: Manifolds I (703) and
Manifolds II (704),
along with older notes from my Spring 2000 Math 704.
Rob has taught the basic graduate topology course several times, most recently in Fall 2000,
Click here for the postscript
or the
rawtext
version of the final Topology exam for Fall 2000.
1435G Lederle Graduate Research Tower
Department of Mathematics
University of Massachusetts at Amherst
Amherst MA 01003
USA
All material on this website is Copyleft* by Rob Kusner.
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