5/6/16: Hey guys, thank you for a great semester of learning together!
It has been a pleasure. I will post the grades over the weekend. Have
a refreshing summer, and I will see you in the Fall.
5/6/16: Hey guys, thank you for a great semester of learning together!
It has been a pleasure. I will post the grades over the weekend. Have
a refreshing summer, and I will see you in the Fall.
4/26/16: Zeshen Gu noticed that, in the formula
for K(θ,θ) in Exercise 8.7(4), the right-hand side
should be 1/(dual Coxeter number). There are other minor errors in the
relevant sections of the text, which I have sent to the author, such
as in some places, -λ-1 must be -λ-2, n-1 must be n-2,
h* must be hR*, but these are
pretty obvious.
4/25/16: I also have to reschedule my office hours for Wednesday,
April 27, to 1:30-2:20 instead of the standard Wednesday time
10:10-11:00.
4/22/16: Because of a thesis defense, I have to reschedule my office
hours for Monday, April 25, to 1:30-2:20 instead of the standard
Monday time 11:15-12:05.
4/21/16: Homework 6, due April 29, is now posted. I apologize for the
delay. The good news is that it has six problems and that it is the
last homework due.
4/1/16: I have slightly modified Homework 5: added a clarification to
Problem 3 and removed Section 8.3 from the reading assignment.
4/8/16: I have slightly reworded Problem 3 on Homework 5 to avoid
possible confusion.
4/7/16: Finished Homework 5. Sorry it took me so long. Do not hesitate
to ask me, should you have any questions.
4/7/16: Part of Homework 5, due, April 15, has been posted. I will add
more sections and problems on Thursday.
3/31/16: There is some support available for graduate students through
an NSF conference grant to attend a one-day meeting ALGECOM 13, which
will be held Saturday April 30, 2016 at the University of Notre Dame.
ALGECOM is an NSF supported one-day meeting on Algebra, Geometry and Combinatorics. The speakers are:
Information (including titles, abstracts, and special rates for hotels) can be found here: https://sites.google.com/site/algecomday/algecom-13-1. They will have a subsidized dinner, hence they ask that you register by sending an email to Alexander Diaz-Lopez (adiaz4@nd.edu) before April 15.
3/30/16: Folks, I apologize about the delay in getting the homework
graded for you. Our grader Amit has promised to do it sometime
today. I will put your homework in your mailboxes as soon as I get it.
3/24/16: Ben has noticed an error in the statement of Problem 7.7,
assigned in the current homework. In the displayed formula, w should
be replaced with w-1. I have sent a note to the author and
am correcting the homework accordingly.
3/19/16: I hope you have been getting productive rest during the
Spring Break. Reminder: Since I am at a conference in Germany, I have
canceled our first Monday and Wednesday classes after the break. We
will reconvene on Friday, March 25.
3/11/16: Corrections to today's example of the longest element in the
Weyl group W = Sn+1 for the root system An:
The positive and negative Weyl chambers are C+ = {(λ1, ..., λn+1) | λ1 + ... + λn+1 = 0, λ1 > ... > λn+1} and C- = {(λ1, ..., λn+1) | λ1 + ... + λn+1 = 0, λ1 < ... < λn+1}. I had the sum of coordinates equal to 1 in class, while it is supposed to be 0 -- after all, the ambient space E should be a vector space;
The longest element w0 in Sn+1 is
| 1 | 2 | ... | n+1 |
| n+1 | n | ... | 1 |
The length l(w0), or the number |R+| of positive roots, is n(n+1)/2, rather than n(n-1)/2.
3/9/16: Homework 4, due April 1, is now posted. Remember that the
Monday and Wednesday classes after the Spring Break are canceled and
we will reconvene after the break on Friday.
3/3/16: I have just realized that to prove uniqueness in Problem 4, we
need to know that the Weyl group acts freely on the set of
polarizations, which we will prove later. I apologize. You may assume
that if an element w of the Weyl group maps positive roots to
positive roots then w = id.
3/3/16: When doing Problem 7.3, do not forget to use the corrected
version of Lemma 7.17, as per the online Errata.
3/3/16: Reminder: unless stated to the contrary, root systems are
assumed to be reduced in the text and on the homework. In particular,
as Zeshen noticed, Problem 7.2(2) would not be true for the
non-reduced system of Problem 7.1. However, the statement of 7.2(2) is
correct, if the root system is assumed to be reduced.
3/2/16: Given how much we know, Problem 4 on HW3 is a hard problem.
Here is a hint: First, prove that for a reflection
sα in a simple root α, sα
(R+) = R+ \ {α} U {-α}. Even though
this statement appears further on in the text as a consequence of more
advanced stuff, it is easy to prove this by methods we have learned so
far. Then use a suitable reflection sα to increase
the number of positive roots in S, if need be, by replacing S with
sα (S).
3/2/16: As Eric pointed out to me, in the hint to Problem 7.2, (7.17)
must be replaced with (7.7). That was the only update in HW3 since the
last update on February 27.
2/27/16: I have changed the wording of Problem 4: it was totally
messed up. I apologize. Thanks to Christian for noticing that. Make
sure you have an update dated 2/27/2016 when doing the homework.
2/26/16: I plan to attend
a conference in
Germany March 18-22. This means that we will have an extended Spring
Break in this class, from March 14 through March 23, that is to say,
we will start on Friday, March 25 after the Spring Break.
2/22/16: Homework 3, due March 4, is now posted.
2/18/16: The author of our textbook, after a bit of discussion,
suggested that Problem 6.4 should be simply removed from the
textbook. Apparently, there is no simple condition on the root
decomposition to add, which would make the problem sensible. Please
disregard it for the homework. I do apologize about the confusion: the
problem appeared correct and innocent to me on the first glance.
2/17/16: Zeshen Gu pointed to me that Problem 6.4 was not literally
true. Indeed, if we do not require much from the root decomposition,
just as stated in the problem, we can easily create a solvable Lie
algebra, which would be as far from a semisimple one, as you can
go. If you require certain properties of a root system, then it is the
construction of a semisimple Lie algebra from a root system, which is
hard, see Theorem 7.54. I have asked Kirillov what he meant and am
waiting for his response. Please hold off from solving that HW problem
for the time being.
2/11/16, 1:30 a.m.: I have posted the second homework on our class web
page.
2/4/16: Too late, but I have reworded Problem 4 on HW 1 to look more
like a research project. Christian pointed out to me that the original
wording was misleading, for what I apologize.
2/1/16: I am rescheduling my Wednesday office hours to 10:10-11:00 for
the rest of the term, to be able to attend the Topology Reading
Seminar. I apologize for the inconvenience.
2/1/16: I forgot about the case when V was the trivial
representation C in the proof of Whitehead's
Lemma: H1(g, V) = 0. I will start with
that on Wednesday.
1/29/16: I have added a problem to the homework. Now the count of
problems is 8, all fitting on one page. I am not going to add any more
problems.
1/27/16: Here is something that may be useful for those interested in
an industrial job after graduation. Next week there will be a huge
Spring CSE Career Fair. Attending it is a way to explore opportunities
for jobs, internships, and co-ops. It will happen on Tuesday, Feb. 2,
2016, 10 a.m.-4 p.m., in Mariucci Arena.
Employers attending the Career Fair include many top national and regional companies such as Amazon, Boston Scientific, Cargill, Dell, Eaton Corporation, Ford Motor Company, General Mills, Honeywell, Hormel, Intel, Medtronic, Schlumberger, St. Jude Medical, Target, Wells Fargo, Xcel Energy, and many others. To see a full list of employers attending the Spring 2016 CSE Career Fair, visit z.umn.edu/employerssp16.
In addition, several employers will be hosting information sessions in our building and conducting on-campus interviews over the next few weeks.
For more information about the Spring 2016 CSE Career Fair, including prep workshops, the career fair app or resources for students, visit cse.umn.edu/careerfair.
1/25/16: Corrected a typo and added a clarification to the
homework. There are still seven problems on the homework, but now it
spans over two pages.
1/25/16: I have slightly reworded Problem 5 on Homework 1 and also
added a problem. Now there are seven problems on the homework.
1/25/16: I have corrected some errors that I originally made in the
homework. Thanks to Aran for pointing them to me. One of the errors
came from a misprint in the textbook, an incorrect sign for the
Casimir in Problem 6.1.
1/24/16: There are six problems on Homework 1 so far. I will be adding
one or two more problems along the way, supporting the material
discussed in class.
1/24/16: Yesterday I posted the first homework on our class web page,
but plan to add a couple more problems.
1/20/16: I have to reschedule my office hours just for Monday, January
25: I will be available right after class till 10:30 a.m.
1/20/16: I plan to post the first homework sometime later this
week. It will cover the material of the first two weeks of the
semester and be due Friday, February 5.
1/19/16: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. After each
class review your notes and study the corresponding part of the
text. You can find out which part of the text at
the
Class Outlines page. Then do the assigned homework problems
pertinent to that material. Some students find it helpful to read the
material before it is covered in class, some prefer to do reading
after class.
1/19/16: If you need to register for this class, please, send a
message to Stephanie Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2016-05-06 12:51:45 CDT)