# Math 20e Homework Solutions

Instructor | |

Name: | Daniel J Hoff |

Email: | d1hof 9Y78VTID8KAf@math.ucsd.eKUWRFP2Idu |

Office: | APM 6331 (map) |

Office Hours: | M, W: 8:15AM-9:15AMM, W: 11:00AM - 12:00PM |

Teaching Assistants | |

Name: | Thada (Kim) Udomprapasup |

Email: | tudompr 8R2SSSFZF6512Y9Sa@ucsd.e9MHH2PDDdu |

Office: | APM 6444 (map) |

Office Hours: | W, F: 12:00PM-2:00PM |

Name: | Zihao Li |

Email: | zil10 HTX7KZTJK8CKZNMY8@ucsd.eCFFC7N68du |

Office: | APM 5412 (map) |

Office Hours: | T, Th: 11:00AM-12:00PMF: 10:00AM-12:00PM |

### Announcements

- 08/22/15: In order to allow everyone enough space, the final exam will be held in PETER 108 (map). This is NOT the usual lecture room, and it is also NOT the room where the midterm exams were held. See the exams page for all exam information.
- 08/28/15: I've typed up the rest of the example from the end of Wednesday's lecture, including an explanation of why the boundary is oriented that way, since we didn't have much time to explain that. Take a look: Stokes' Theorem Example
- 08/22/15: In order to allow everyone enough space, the second midterm will also be held in WLH 2001 (map). This is not the usual lecture room, but it is adjacent. See the exams page for all exam information.
- 08/11/15: In order to allow everyone enough space, the first midterm will be held in WLH 2001 (map). This is not the usual lecture room, but it is adjacent. See the exams page for all exam information.
- 08/11/15: Solutions to the review problems for Midterm Exam 1 have now been posted on the exams page.
- 08/10/15: Details and review problems for Midterm Exam 1 have now been posted on the exams page. Check them out!
- 08/10/15: Kim has very kindly typed up solutions to two of the harder problems on Homework 1: 5.3.11 and 5.3.15.
- 08/06/15: The homework dropboxes have now been set up in basement of APM (map). Homework should be turned in to your discussion section TA's box. The two 20E TA's boxes are both in the double-column of boxes marked "5" and are both labeled with yellow tags.
- 08/06/15: I have created a Piazza page for the course. Piazza provides a nice forum for course questions and discussion. It is, however, a private company not run by UCSD, so sign-up and use is completely optional. The Piazza page can be accessed from the menu bar above after signing up here.
- 08/05/15: This course will be using the TED system for grades; you can find a link to TED in the menu bar above.
- July: Welcome to Math 20E! Content will be added to the course website as the term approaches; stay tuned.

## Math 20E: Vector Calculus

## Summer Session II, 2015

## Lars Kadison's Homepage

**Stilling/Position:**investigador/researcher/lecturer in mathematics**Email addresses:**fc[doT]up[peRiod]pt broadpark[pOinT]no**Brugernavn:**lkadison**Homepage since:**1997

WELCOME. Aloha, welcome to my academic homepage. Below is my central network system of links and documents. Mailing address in Europe: Departamento de Matematica, Faculdade de Ciencias da Universidade do Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal, six year investigador and invited scientist position from 2010. Teaching Fall 2006 and Spring 2007: Dept. of Math, Univ. of Pennsylvania, Philadelphia, PA; currently associated as visiting scholar. Old friends, family and acquantainces should contact me on facebook.

PUBLICATION LIST. Pdf version

CO-ORGANIZER SPECIAL SESSIONS. Recent past: (with A.A. Stolin) Representation Theory and Quantum Groups (#51), afternoon sessions June 10, 11, 12, Porto, the first session with a focus on representation theory, the second on representation theory and quantum groups, the third on quantum groups. Confirmed Speakers, 1st day: Burkhard Kuelshammer, Jena (45 minutes); Alberto Hernandez (-Kadison,Young,Szamotulski), Porto; Sebastian Burciu, Bucharest (45 minutes). 2nd day: Arvid Siqveland, B.U.C., Norway (45 minutes); Marcin Szamotulski, U. Lisbon; Yorck Sommerhauser, S.U.N.Y. Buffalo; Mio Iovanov, U. Iowa. 3rd day: Volodomyr Mazorchuk, Uppsala (45 minutes); Anthony Giaquinto, Loyola, Chicago; Alexander Stolin (- Efim Zelmanov), Gothenburg(-San Diego); Arturo Pianzola, Edmonton.

Past: (with A.A. Stolin) Actions of Quantum Algebras, AQuA at the Louisiana State University at Baton Rouge, Southeastern Spring Sectional Meeting, March 28-30, 2008.

Ph.D. STUDENTS as faculty member 2010-1016 of Universities of Porto and Coimbra Ph.D. Program. Christopher J. Young, Thesis: Hopf Algebra Depths in Smash Products. Granted: September 19, 2014 in Porto. Alberto Hernandez A. Thesis: The Quotient Module, Coring Depth and Factorization Algebras, granted September 30, 2016 in Porto.

WHAT'S NU. A preprint An in-Depth look at Quotient Modules (of Hopf subalgebras) and another, Symmetric, separable equivalence of rings, an addendum to Hokkaido Mathematical Journal 24 (1995), 527-549. A paper with Sam Lopes et al on A quantum subgroup depth has appeared in Acta Mathematica Hungarica. The 2016 articles A tower condition characterizing normality in Hokkaido Mathematical Journal Vol. 45 (2016) p. 243 - 262, Algebra depth in tensor categories in Vol. 23, no. 5 (2016) of the Bulletin Belgian Math. Soc.-Simon Stevin, pp. 721-753 and Subgroup depth and twisted coefficients with Hernandez and Szamotulski in volume 44, issue 8, of Communications in Algebra.

RESEARCH. Mainly in the areas of Representation Theory, Quantum Algebra, and Associative Ring Theory. More specifically, noncommutative Galois theory; Hopf algebroids, weak and ordinary Hopf algebras, groups and their actions; modules, subrings and induced representations; subgroup depth and Frobenius extensions.

VISITS, TALKS AND CONFERENCES. L.S.U. talk, MSRI, Brussels, Budapest, V.U.B. Workshoptalk, Odense colloquium, Oslo, Swanseatalk, Leeds talk, Cambridge Mass., Bergen, Karlstadtalk, Mainztalk, Spa Belgium, Oslo colloq., Algebra, Geometry and Math. Phys., G�teborgtalk, Norwegian algebra meeting, Noncommutative Structures in Math and Physicstalk. U.S.A. algebra seminar.Jena, Summer meeting, Canadian Math Society, June 6-8, 2009Talk. Texas A & M University, Algebra and Combinatorics Seminar. University of Porto Algebra Seminar. AGMP 2010 talk, AGMP 2011 Mulhouse talk. AGMP2013 Kongsberg, Norway. Coimbra, Ghent talk Workshop, Special Session, 2014. Brussels Tensor Category meeting, June 2-6, 2015 talk.

RECENT PREPRINTS. It's an honor to have my paper Subgroups of depth three with Sebastian Burciu in Surv. Diff. Geom. XV. Runner-up honors to Odd H-depth and H-separable extensions in Cent. Eur. J. Math. 10 (2012). " Subring depth, Frobenius extensions and towers in Int. J. Math. & Math. Sci. (2012),

The article Algebraic quotient modules and subgroup depth with Hernandez and Young in Abhandlungen aus dem Mathematischen Seminar der Universitaet Hamburg (October 2014). A well-cited article On subgroup depth with Sebastian M. Burciu and Burkhard Kuelshammer in I.E.J.A.. My best article in this period is perhaps Hopf subalgebras and tensor powers of generalized permutation modules in Journal of Pure and Applied Algebra (January 2014).

OPEN PROBLEMS for the curious. In the Oslo, Karlstad, Mainz, Leeds and Canada talks above, some background for several problems in depth two (and more): 1) Are left D2 extensions right D2? 2) Are D2 Hopf subalgebras normal? Yes, J.Alg. 2010 Boltje-Kuelshammer 3) Do D2 extensions of simple algebras have a Galois correspondence between subextensions and Hopf subalgebroids? 4) Is a fin. gen. bialgebroid the endomorphism ring of a D2 algebra extension? (5) Are there subgroups of minimimum depth 2n where n > 3? (6) Find a characterization of depth n subgroup (e.g. n = 2 <=> normal subgroup). See the paper in (9) below for combinatorial depth of a subgroup and its depth n characterization. (7) How to efficiently compute the matrix of the zeroeth K-theory mapping K(R) ---> K(H) of a subalgebra pair (R,H) of semisimple Hopf algebras? Find an alternative way to compute depth not involving full character and induction-restriction tables. (8) Determine if depth is finite for any Hopf subalgebra of any finite dimensional Hopf algebra. See the papers of Boltje-Danz-Kuelshammer (J.Alg. 335 (2011)), the talk talk in Lausanne for the case of a pair of group algebras, the JPAA (2014) and Abh. Hamburg papers mentioned above for some additional nonsemisimple Hopf algebras.

POSTS. Aarhus, Denmark, Luminy, Roskilde: IMFUFA (now defunct institute), Copenhagen: Professorable, Heidelberg University (see snapshot below, a related story given in the second paragraph of Cuntz's piece in Notices AMS), Trondheim, Norway, (see snapshot to your right), Munich, Duesseldorf, NorFA researcher at Chalmers in Gothenburg, University of New Hampshire in Durham. University of Pennsylvania in Philadelphia. Lousiana State University in Baton Rouge. U.C. San Diego. Research position-faculty member, Universities of Porto and Coimbra Ph.D. Program. Brief CV, Research Statement. Teaching Statement.

UNDERGRADUATE and GRADUATE TEACHING. Graduate algebra course in the fall of 2013 on finite-dimensional algebras and their representation theory using the book by Drozd and Kirichenko. A doctoral course Math 558 on "Quantum Groups," given in the spring of 2012 and of 2014. A master's course in group representations using Alperin-Bell's Springer GTM textbook (Spring 2011). Math 20E, vector calculus, and Math 20B, second quarter Calculus, at U.C. San Diego. Math 4201, Galois theory, in the spring of 2008 at LSU as well as Math 1441. Calculus I and Calculus III at Penn in the fall, and Math 180 in the spring of 2007. At UNH, Introduction to mathematical proof, Calculus I, Mathematical modelling for life sciences graduates, Intro. to PDE's, an evening class in Calculus I, Abstract Algebra, Diff. Eqs. with Linear Algebra, Abstract Algebra with Projective Geometry. Nine undergraduate courses in Norway/Denmark in linear algebra, projective geometry, differential equations, algebra and mathematics education.

STARRED PROBLEM for abstract algebra course. Following the proof of Fermat's two-square theorem (a prime congruent to 1 module 4 is the sum of two squares) using factorization of sum of two squares over the complex numbers and unique factorization in euclidean domains, assign the following problem to the honors students. Find a three-square theorem and its somewhat similar proof using factorization of the sum of three squares over the quaternions and a fact from an introductory book on number theory such as Hardy and Wright. SolutionProof

MY UCSD GRADING POLICY. I have been on a teaching visitorship at the University of California at San Diego from September 2009 to January 2010. Grading in both my courses has been on a curve suggested on p. 10 in the visitor guidebook, instructor resources of the math department's website. The weighting in Math 20B (from the first day) is 20% Midterm 1, 25% Midterm 2, 15% Homework (and/or quizzes, here the detailed policy set by your TA), 40% Final. In Math 20E it is 20% each midterm, 10% Homework, and 50% Final.

MATH 20E. (198 students, Warren Lecture Hall 2005, 11 a.m. TA's: r1gomez, htn005, jmiddleton. Grader: Swartz.) The median of the weighted scores including the final was a 65 with a standard deviation of 24. The solutions are given here, but ignore the front page which is from midterm 2 by a glitch.

MATH 20B. (202 students. Center Hall 101, 3 p.m. TA's: pcompeau, mtiefenbruck, ashakeel, vkungurt. Grader: Schilz.) The average for the weighted scores including the final was a 68 with a median of 72 and a standard deviation of 19. The solutions to the final are here, except for problem 1a) whose answer is a_n = n sin (1/n) goes to 1 since sin x/x goes to 1 as x goes to zero; and problem 1b) whose answer is sum e to the minus n from n = 1 is equal to 1/(e-1) by a slight modification of the sum of a geometric series with ratio less than one.

TEXTBOOK.Projective Geometry and Modern Algebra with Matthias Buch-Kromann. A textbook for undergraduates that mixes group theory, ring theory and projective geometry: has served to motivate students to learn abstract algebra.

MONOGRAPH. New Examples of Frobenius Extensions, volume 14 in the American Mathematical Society's University Lecture Series.

Since its publication, there have been many more developments that would have fit well in this book, including depth two and reconstruction of weak Hopf algebras on separable centralizers, Hopf algebras on one-dimensional centralizers, Hopf algebroids and Galois theory; antipodes, depth, and double algebra structures for Frobenius extensions, subgroup depth, Frobenius algebras in categories and separability. See the link to updated and annotated list of references to "New Examples..."

RECENT, FUTURE AND MAIN PUBLICATIONS. Simplicial Hochschild cochains as an Amitsur complex is published in J. Gen. Lie Theory in 2008 (arXiv has the better version of this paper). Finite depth and Jacobson-Bourbaki correspondence, appeared in the Journal of Pure and Applied Algebra in July 2008. Pseudo-Galois extensions and Hopf algebroids appeared July 4, 2008, in Birkhauser Trends in Math, Proceedings of Conf. on Modules and Comodules, Portugal, Sept. 2006, on the occasion of Robert Wisbauer's birthday. Bialgebroid Actions on Depth Two Extensions and Duality with coauthor Kornel Szlachanyi appeared in October 2003 in volume 179 of Advances in Mathematics. With coauthor Dmitri Nikshych, Frobenius extensions and weak Hopf algebras appeared in volume 244 (2001), 312-342, in Journal of Algebra, and our paper, Hopf algebra actions on strongly separable extensions of depth two appeared in no. 2, vol. 163 (2001) in Adv. in Math. The rest of my publications, or anyone's for that matter, are summarized by Zentralblatt or MR lookup.

EDUCATION. Bachelor's degree from Princeton University with a PBK, and a Ph.D. from UC Berkeley in 1989 with a dissertation on relative cyclic homology.

MATHEMATICAL ANCESTORS. According to the math geneology project, my Ph.D. supervisor and his supervisor are in reverse iteration (anno 1999): John Wagoner, Princeton, 1966; William Browder, Princeton, 1958; John C. Moore, Brown, 1952; George Whitehead, Jr., Chicago, 1941; Norman Steenrod, Princeton, 1936; Solomon Lefschetz, Clark, 1911; William Story, Leipzig, 1875; Wilhelm Scheibner, Halle, 1848; Karl Jacobi, Berlin, 1825.

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