Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Prerequisites:
Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology.
Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017.
While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order):
Algebraic Geometry (Dover Books on Mathematics) by Solomon Lefschetz (Less likely based on level and age, but Dr. Miller does love inexpensive Dover editions)
Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.)
The topic for Mike Miller’s UCLA Winter math course isn’t as much a surprise as is often the case. During the summer he had announced he would be doing a two quarter sequence on complex analysis, so this Winter, we’ll be continuing on with our complex analysis studies.
I do know, however, that there were a few who couldn’t make part of the Fall course, but who had some foundation in the subject and wanted to join us for the more advanced portion in the second half. Toward that end, below are the details for the course:
Introduction to Complex Analysis: Part II | MATH X 451.41 – 350370
Course Description
Complex analysis is one of the most beautiful and practical disciplines of mathematics, with applications in engineering, physics, and astronomy, to say nothing of other branches of mathematics. This course, the second in a two-part sequence, builds on last quarter’s development of the differentiation and integration of complex functions to extend the principles to more sophisticated and elegant applications of the theory. Topics to be discussed include conformal mappings, Laurent series and meromorphic functions, Riemann surfaces, Riemann Mapping Theorem, analytical continuation, and Picard’s Theorem. The course should appeal to those whose work involves the application of mathematics to engineering problems, and to those interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.
Winter 2017 Days: Tuesdays Time: 7:00PM to 10:00PM Dates: Jan 10, 2017 to Mar 28, 2017 Contact Hours: 33.00 Location: UCLA, Math Sciences Building Course Fee(s): $453.00 Available for Credit: 3 units Instructors: Michael Miller
No refund after January 24, 2017. Class will not meet on one Tuesday to be announced.
For many who will register, this certainly won’t be their first course with Dr. Miller–yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension
If you missed the first quarter and are interested in the second quarter but want a bit of review or some of the notes, let me know in the comments below.
I look forward to seeing everyone in the Winter quarter!
Michael Miller making a “handwaving argument” during a lecture on Algebraic Number Theory at UCLA on November 15, 2015. I’ve taken over a dozen courses from Mike in areas including Group Theory, Field Theory, Galois Theory, Group Representations, Algebraic Number Theory, Complex Analysis, Measure Theory, Functional Analysis, Calculus on Manifolds, Differential Geometry, Lie Groups and Lie Algebras, Set Theory, Differential Geometry, Algebraic Topology, Number Theory, Integer Partitions, and p-Adic Analysis.
For those who missed the first class of Introduction to Complex Analysis on 09/20/16, I’m attaching a link to the downloadable version of the notes in Livescribe’s Pencast .pdf format. This is a special .pdf file but it’s a bit larger in size because it has an embedded audio file in it that is playable with the more recent version of Adobe Reader X (or above) installed. (This means to get the most out of the file you have to download the file and open it in Reader X to get the audio portion. You can view the written portion in most clients, you’ll just be missing out on all the real fun and value of the full file.) [Editor’s note: Don’t we all wish Dr. Tao’s class was recording his lectures this way.]
With these notes, you should be able to toggle the settings in the file to read and listen to the notes almost as if you were attending the class live. I’ve done my best to write everything exactly as it was written on the board and only occasionally added small bits of additional text.
If you haven’t registered yet, you can watch the notes as if you were actually in the class and still join us next Tuesday night without missing a beat. There are over 25 people in the class not counting several I know who had to miss the first session.
Hope to see you then!
Viewing and Playing a Pencast PDF
Pencast PDF is a new format of notes and audio that can play in Adobe Reader X or above.
You can open a Pencast PDF as you would other PDF files in Adobe Reader X. The main difference is that a Pencast PDF can contain ink that has associated audio—called “active ink”. Click active ink to play its audio. This is just like playing a Pencast from Livescribe Online or in Livescribe Desktop. When you first view a notebook page, active ink appears in green type. When you click active ink, it turns gray and the audio starts playing. As audio playback continues, the gray ink turns green in synchronization with the audio. Non-active ink (ink without audio) is black and does not change appearance.
Audio Control Bar
Pencast PDFs have an audio control bar for playing, pausing, and stopping audio playback. The control bar also has jump controls, bookmarks (stars), and an audio timeline control.
Active Ink View Button
There is also an active ink view button. Click this button to toggle the “unwritten” color of active ink from gray to invisible. In the default (gray) setting, the gray words turn green as the audio plays. In the invisible setting, green words seem to write themselves on blank paper as the audio plays.
Dr. Michael Miller has announced his Autumn mathematics course, and it is…
Introduction to Complex Analysis
Course Description
Complex analysis is one of the most beautiful and useful disciplines of mathematics, with applications in engineering, physics, and astronomy, as well as other branches of mathematics. This introductory course reviews the basic algebra and geometry of complex numbers; develops the theory of complex differential and integral calculus; and concludes by discussing a number of elegant theorems, including many–the fundamental theorem of algebra is one example–that are consequences of Cauchy’s integral formula. Other topics include De Moivre’s theorem, Euler’s formula, Riemann surfaces, Cauchy-Riemann equations, harmonic functions, residues, and meromorphic functions. The course should appeal to those whose work involves the application of mathematics to engineering problems as well as individuals who are interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.
Prerequisites
Basic calculus or familiarity with differentiation and integration of real-valued functions.
For many who will register, this certainly won’t be their first course with Dr. Miller — yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension
I often recommend people to join in Mike’s classes and more often hear the refrain: “I’ve been away from math too long”, or “I don’t have the prerequisites to even begin to think about taking that course.” For people in those categories, you’re in luck! If you’ve even had a soupcon of calculus, you’ll be able to keep up here. In fact, it was a similar class exactly a decade ago by Mike Miller that got me back into mathematics. (Happy 10th math anniversary to me!)
(Note that there’s another introductory complex analysis textbook from Silverman that’s offered through Dover, so be sure to choose the correct one.)
As always in Dr. Miller’s classes, the text is just recommended (read: not required) and in-class notes are more than adequate. To quote him directly, “We will be using as a basic guide, but, as always, supplemented by additional material and alternate ways of looking at things.”
The bonus surprise of his email: He’s doing two quarters of Complex Analysis! So we’ll be doing both the Fall and Winter Quarters to really get some depth in the subject!
Alternate textbooks
If you’re like me, you’ll probably take a look at some of the other common (and some more advanced) textbooks in the area. Since I’ve already compiled a list, I’ll share it:
R. A. Silverman, Complex Analysis with Applications, 1st ed. Dover Publications, Inc., 2010, pp. 304–304 [Online]. Available: http://amzn.to/2c7KaQy
[2]
J. Bak and D. J. Newman, Complex Analysis, 3rd ed. Springer, 2010, pp. 328–328 [Online]. Available: http://amzn.to/2bLPW89
[3]
T. Gamelin, Complex Analysis. Springer, 2003, pp. 478–478 [Online]. Available: http://amzn.to/2bGNQct
[4]
J. Brown and R. V. Churchill, Complex Variables and Applications, 8th ed. McGraw-Hill, 2008, pp. 468–468 [Online]. Available: http://amzn.to/2bLQWcu
[5]
E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics, 3rd ed. Pearson, 2003, pp. 563–563 [Online]. Available: http://amzn.to/2f3Nyj6
[6]
L. V. Ahlfors, Complex Analysis, 3rd ed. McGraw-Hill, 1979, pp. 336–336 [Online]. Available: http://amzn.to/2bMXrxm
[7]
S. Lang, Complex Analysis, 4th ed. Springer, 2003, pp. 489–489 [Online]. Available: http://amzn.to/2c7OaR0
[8]
J. B. Conway, Functions of One Complex Variable, 2nd ed. Springer, 1978, pp. 330–330 [Online]. Available: http://amzn.to/2cggbF1
[9]
El. M. Stein and R. Shakarchi, Complex Analysis. Princeton University Press, 2003, pp. 400–400 [Online]. Available: http://amzn.to/2bGOG9c
I know you’ve all been waiting for the announcement with bated breath! We’ve known for a while that Mike Miller’s Winter course would be a follow-on course to his Algebraic Number Theory course this Fall, but it’s been officially posted, so now you can register for it: Algebraic Number Theory: The Sequel.
I’m sure, as always, that there are a few who are interested, but who couldn’t make the Fall lectures. Never fear, there’s a group of us that can help you get up to speed to keep pace with us during the second quarter. Just drop us a note and we’ll see what we can do.
Algebraic Number Theory: The Sequel
In no field of mathematics is there such an irresistible fascination as in the theory of numbers. This course, the second in a two-quarter sequence, is an introductory, yet rigorous, survey of algebraic number theory, which evolved historically through attempts to prove Fermat’s Last Theorem. Beginning with a quick review of the previous quarter’s work, the course continues discussions on the structure of algebraic number fields, focusing particular attention on primes, units, and roots of unity in quadratic, cubic, and cyclotomic fields. Topics to be discussed include: norms and traces; the ideal class group; Minkowski’s Translate, Convex Body, and Linear Forms theorems; and Dirichlet’s Unit Theorem.
UCLA: 5137 Math Sciences
Tuesday, 7-10pm,
January 5 – March 15
11 meetings total
MATH X450.9
3.00 units
Recommended Textbook
We’ll be using Introductory Algebraic Number Theory by Saban Alaca and Kenneth S. Williams (Cambridge University Press, 2003, ISBN: 978-0521183048).
Like a kid anxiously awaiting Christmas morning, I spent some time refreshing UCLA Extension’s web page over the weekend in hopes of seeing the announcement of Mike Miller’s Fall math course with no results.
I checked again a half hour ago and their site was down!
My salivating hit a fever pitch!
Refreshing, refreshing, refreshing… and now it’s live again with:
Register quickly before it fills up. And let the pool for the guesses about which textbook he’ll recommend begin!
Algebraic Number Theory
MATH X 450.8 | 3.00 units
In no field of mathematics is there such an irresistible fascination as in the theory of numbers. This course, the first in a two-quarter sequence, is an introductory, yet rigorous, survey of algebraic number theory, which evolved historically through attempts to prove Fermat’s Last Theorem. Beginning with a quick review of primality and unique factorization for ordinary integers, the course extends these notions to more exotic domains: quadratic, cubic, cyclotomic, and general number fields. This development is then applied to the representation of integers as sums of squares and, more generally, to classic Diophantine equations. Topics to be discussed include: Euclidean, principal ideal, and Noetherian domains; integral bases; binary quadratic forms; algebraic field extensions; and several remarkable theorems/conjectures of Ramanujan.
UCLA: 5137 Math Sciences
Tuesday, 7-10pm,
September 22 – December 8
11 meetings total (no mtg 11/17)
See you all in just a few weeks!
A Note For the Reticent
Exercise Your Brain
As many may know or have already heard, Dr. Mike Miller, a retired mathematician from RAND and long-time math professor at UCLA, has been offering incredibly inexpensive upper level undergraduate and graduate level math courses for 30+ years through UCLA Extension.
Whether you’re a professional mathematician, engineer, physicist, physician, or simply a hobbyist interested in mathematics you’ll be sure to get something interesting out of this course, not to mention the camaraderie of 20-30 other “regulars” with widely varying backgrounds (actors to surgeons and evolutionary theorists to engineers) who’ve been taking almost everything Mike has offered over the years. Once most new students have taken one class, they’re incredibly prone to want to take them all (and yes, he’s THAT good — we’re sure you’ll be addicted too.)
“Beginners” Welcome!
Even if it’s been years since you last took calculus or linear algebra, Mike (and usually the rest of the class) will help you get quickly back up to speed to delve into what is often a very deep subject. Though there are a handful who will want to learn the subject for specific applications, naturally, it’s simply a beautiful and elegant subject for those who just want to meander their way through higher mathematics for the fun of it (this will probably comprise the largest majority of the class by the way.)
Whether you’ve been away from serious math for decades or use it every day or even if you’ve never gone past calculus, this is bound to be the most entertaining thing you can do with your Tuesday nights in the fall. If you’re not sure what you’re getting into (or are scared a bit by the course description), I highly encourage to come and join us for at least the first class before you pass up on the opportunity – there’s no need to preregister or prepay if you’re unsure. I’ll mention that the greater majority of new students to Mike’s classes join the ever-growing group of regulars who take almost everything he teaches subsequently.
For the reticent, I’ll mention that one of the first courses I took from Mike was Algebraic Topology which generally requires a few semesters of Abstract Algebra and a semester of Topology as prerequisites. I’d taken neither of these prerequisites, but due to Mike’s excellent lecture style and desire to make everything comprehensible to the broadest number of students, I was able to do exceedingly well in the course. Also keep in mind that you can register to take the class for a grade, pass/fail, or even no grade at all to suit your needs/lifestyle.
My classes have the full spectrum of students from the most serious to the hobbyist to those who are in it for the entertainment and ‘just enjoy watching it all go by.’
Mike Miller, Ph.D.
Textbook: Introductory Algebraic Number Theory
Update (8/19/15) Per my email conversation with Dr. Miller, despite that neither the Extension website nor the bookstore have a book listed for the class yet, he’s going to be recommending Introductory Algebraic Number Theory by Saban Alaca and Kenneth S. Williams (Cambridge University Press, 2003, ISBN: 978-0521183048).
