Teaching Fall 2006 and Spring 2007: Dept. of Math, Univ. of Pennsylvania, Philadelphia, PA; currently associate v.scholar. I have been on a teaching visitorship at the University of California at San Diego from September 2009 to January 2010.
WHAT'S NEW. Odd H-depth and H-separable extensions. Subgroups of depth three with Sebastian Burciu has appeared in the S.D.G. XV in honor of Isadore Singer. A recent follow-up paper: "On subgroup depth" with Sebastian M. Burciu and Burkhard Kuelshammer is available in I.E.J.A. vol. 9 (2011).
Depth of a subgroup H of a finite group G is based on its induction-restriction table, an r by s matrix M if H has r irreducible characters and G has s irreducible characters. For example, if H is the product of permutation groups S_n and S_m, G is the permutation group on n+m letters, the entries of M are the Littlewood-Richardson coefficients. Define powers of M by applying the transpose matrix M^t as follows: M-squared is MM^t, M-cubed is MM^tM, etc. The matrix M may have several zero entries, then the matrix M-cubed has fewer and just in the same entries: this sequence of odd powers of M stabilizes w.r.t. number of zero entries, as does the sequence of even powers. The depth of subgroup H < G is the least point of stabilization, or the minimum n such that the n + 1'st power of M is less than or equal to a multiple of the n - 1'st power. This depth is amusing to compute from character tables of G and H. If the depth is one or two, the subgroup is normal, and conversely (here the zeroth power of M should stand for identity matrix of order r). Frobenius complements in Frobenius groups are depth three. If the depth of a Sylow p-subgroup is five or more, it is nonabelian. For more information turn the pages of the article above, look at the lectures in St. John's, Mobile, Porto or College Station below, or link above to the papers of Boltje-Kuelshammer, the latest of which with Danz (J.Alg. to appear) proposes a depth of an arbitrary subring and shows that subgroups over any commutative ring have finite depth (bounded by a G-set theoretic combinatorial depth, in turn twice the index of the normalizer).
RESEARCH. Mainly in the areas of Ring Theory, Quantum Algebra, and Quantum Groupoids. More specifically, noncommutative Galois theory; Hopf algebras, Hopf algebroids, weak Hopf algebras, groups and their actions; modules, subrings and induced representations; Frobenius extensions and depth.
Secondary interests are in other aspects of homological algebra, cyclic homology, knot theory, commutative algebra, subfactors, K-theory, algebraic topology, representation theory, noncommutative geometry, Lie algebroids and Poisson geometry.
VISITS, TALKS AND CONFERENCES. L.S.U. talk, MSRI, Brussels, Budapest, V.U.B. Workshop talk, Odense colloquium, Oslo, Swansea talk, Leeds talk, Cambridge Mass., Bergen, Karlstad talk, Mainz talk, Spa Belgium, Oslo colloq., Algebra, Geometry and Math. Phys., Göteborg talk, Norwegian algebra meeting, AQuA, Noncommutative Structures in Math and Physics talk. U.S.A. algebra seminar. Jena, Summer meeting, Canadian Math Society, June 6-8, 2009 Talk. Texas A & M University, Algebra and Combinatorics Seminar. University of Porto Algebra Seminar. AGMP 2010 talk, AGMP 2011 Mulhouse Talk.
RECENT PREPRINTS. The arXiv preprint Ideal depth of QF extensions has been divided into two papers: "Subring depth below an ideal" to appear in the Journal of Physics : Conference Series, and "Subring depth, Frobenius extensions and their towers" submitted with several new pages.
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 compute the induction-restriction table of a subalgebra pair of finite groupoid algebras or semisimple Hopf algebras? (8) Find an alternative way to compute depth not involving full character and induction-restriction tables. (9) Obtain a bound (if there is one) on the depth of a Hopf subalgebra of a finite dimensional Hopf algebra. See the paper of Boltje-Danz-Kuelshammer (J.Alg. 335 (2011), 258-281) for the case of a pair of group algebras.
POSTS. Aarhus, Denmark, Luminy, Roskilde: IMFUFA (now defunct institute), Copenhagen: professorable, Heidelberg University (see snapshot below), 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.
UNDERGRADUATE TEACHING. Scheduled to teach a course in Hopf algebras in the spring of 2012. 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. Solution Proof
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.
MY UCSD GRADING POLICY. 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,.
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. When weak Hopf algebras are Frobenius with Mio C. Iovanov has appeared in the Proceedings of the American Math Society Simplicial Hochschild cochains as an Amitsur complex has appeared 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, 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.